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

Percentage Accurate: 61.4% → 90.4%

Time: 9.3s

Alternatives: 12

Speedup: 0.9×

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.4% 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

real(8) function code(x, y, z, t, a, b)
    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: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y}\\ t_2 := t + \left(y + x\right)\\ t_3 := \frac{y}{t\_2}\\ t_4 := \mathsf{fma}\left(t\_3 + \frac{1}{\frac{t\_2}{x}}, z, a - b\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{t\_2} + t\_3, a, \frac{\mathsf{fma}\left(x, z, \left(z - b\right) \cdot y\right)}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) (+ (+ t x) y)))
        (t_2 (+ t (+ y x)))
        (t_3 (/ y t_2))
        (t_4 (fma (+ t_3 (/ 1.0 (/ t_2 x))) z (- a b))))
   (if (<= t_1 (- INFINITY))
     t_4
     (if (<= t_1 5e+232)
       (fma (+ (/ t t_2) t_3) a (/ (fma x z (* (- z b) y)) t_2))
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / ((t + x) + y);
	double t_2 = t + (y + x);
	double t_3 = y / t_2;
	double t_4 = fma((t_3 + (1.0 / (t_2 / x))), z, (a - b));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_1 <= 5e+232) {
		tmp = fma(((t / t_2) + t_3), a, (fma(x, z, ((z - b) * y)) / t_2));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / Float64(Float64(t + x) + y))
	t_2 = Float64(t + Float64(y + x))
	t_3 = Float64(y / t_2)
	t_4 = fma(Float64(t_3 + Float64(1.0 / Float64(t_2 / x))), z, Float64(a - b))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_1 <= 5e+232)
		tmp = fma(Float64(Float64(t / t_2) + t_3), a, Float64(fma(x, z, Float64(Float64(z - b) * y)) / t_2));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t + N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 + N[(1.0 / N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + N[(a - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$4, If[LessEqual[t$95$1, 5e+232], N[(N[(N[(t / t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision] * a + N[(N[(x * z + N[(N[(z - b), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 4.99999999999999987e232 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 9.6%

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

   3. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. div-sub N/A

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

   5. Applied rewrites 41.3%

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

   7. Applied rewrites 41.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 81.5%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999987e232

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. div-sub N/A

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

   5. Applied rewrites 99.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t + \left(y + x\right)} + \frac{1}{\frac{t + \left(y + x\right)}{x}}, z, a - b\right)\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{t + \left(y + x\right)} + \frac{y}{t + \left(y + x\right)}, a, \frac{\mathsf{fma}\left(x, z, \left(z - b\right) \cdot y\right)}{t + \left(y + x\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t + \left(y + x\right)} + \frac{1}{\frac{t + \left(y + x\right)}{x}}, z, a - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+150}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{-128}:\\ \;\;\;\;\frac{z \cdot x - b \cdot y}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ t x) y))
        (t_2 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) t_1))
        (t_3 (- (+ a z) b)))
   (if (<= t_2 -2e+150)
     t_3
     (if (<= t_2 1e-128)
       (/ (- (* z x) (* b y)) t_1)
       (if (<= t_2 5e+232) (/ (fma a t (* z x)) (+ t x)) t_3)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + x) + y;
	double t_2 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / t_1;
	double t_3 = (a + z) - b;
	double tmp;
	if (t_2 <= -2e+150) {
		tmp = t_3;
	} else if (t_2 <= 1e-128) {
		tmp = ((z * x) - (b * y)) / t_1;
	} else if (t_2 <= 5e+232) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + x) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / t_1)
	t_3 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_2 <= -2e+150)
		tmp = t_3;
	elseif (t_2 <= 1e-128)
		tmp = Float64(Float64(Float64(z * x) - Float64(b * y)) / t_1);
	elseif (t_2 <= 5e+232)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+150], t$95$3, If[LessEqual[t$95$2, 1e-128], N[(N[(N[(z * x), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 5e+232], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $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)) < -1.99999999999999996e150 or 4.99999999999999987e232 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 16.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -1.99999999999999996e150 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.00000000000000005e-128

   1. Initial program 99.7%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 1.00000000000000005e-128 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999987e232

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. div-sub N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 68.7

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

   8. Applied rewrites 68.7%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -2 \cdot 10^{+150}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 10^{-128}:\\ \;\;\;\;\frac{z \cdot x - b \cdot y}{\left(t + x\right) + y}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := t + \left(y + x\right)\\ t_4 := \mathsf{fma}\left(\frac{y}{t\_3} + \frac{1}{\frac{t\_3}{x}}, z, a - b\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t + y, a, \mathsf{fma}\left(x, z, \left(z - b\right) \cdot y\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ t x) y))
        (t_2 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) t_1))
        (t_3 (+ t (+ y x)))
        (t_4 (fma (+ (/ y t_3) (/ 1.0 (/ t_3 x))) z (- a b))))
   (if (<= t_2 (- INFINITY))
     t_4
     (if (<= t_2 5e+232)
       (/ (fma (+ t y) a (fma x z (* (- z b) y))) t_1)
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + x) + y;
	double t_2 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / t_1;
	double t_3 = t + (y + x);
	double t_4 = fma(((y / t_3) + (1.0 / (t_3 / x))), z, (a - b));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_2 <= 5e+232) {
		tmp = fma((t + y), a, fma(x, z, ((z - b) * y))) / t_1;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + x) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / t_1)
	t_3 = Float64(t + Float64(y + x))
	t_4 = fma(Float64(Float64(y / t_3) + Float64(1.0 / Float64(t_3 / x))), z, Float64(a - b))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_2 <= 5e+232)
		tmp = Float64(fma(Float64(t + y), a, fma(x, z, Float64(Float64(z - b) * y))) / t_1);
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t + N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(y / t$95$3), $MachinePrecision] + N[(1.0 / N[(t$95$3 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z + N[(a - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$4, If[LessEqual[t$95$2, 5e+232], N[(N[(N[(t + y), $MachinePrecision] * a + N[(x * z + N[(N[(z - b), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 4.99999999999999987e232 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 9.6%

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

   3. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. div-sub N/A

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

   5. Applied rewrites 41.3%

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

   7. Applied rewrites 41.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 81.5%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999987e232

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t + \left(y + x\right)} + \frac{1}{\frac{t + \left(y + x\right)}{x}}, z, a - b\right)\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t + y, a, \mathsf{fma}\left(x, z, \left(z - b\right) \cdot y\right)\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t + \left(y + x\right)} + \frac{1}{\frac{t + \left(y + x\right)}{x}}, z, a - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t + y, a, \mathsf{fma}\left(x, z, \left(z - b\right) \cdot y\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ t x) y))
        (t_2 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) t_1))
        (t_3 (- (+ a z) b)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 1e+282)
       (/ (fma (+ t y) a (fma x z (* (- z b) y))) t_1)
       t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + x) + y;
	double t_2 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / t_1;
	double t_3 = (a + z) - b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 1e+282) {
		tmp = fma((t + y), a, fma(x, z, ((z - b) * y))) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + x) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / t_1)
	t_3 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 1e+282)
		tmp = Float64(fma(Float64(t + y), a, fma(x, z, Float64(Float64(z - b) * y))) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $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+282], N[(N[(N[(t + y), $MachinePrecision] * a + N[(x * z + N[(N[(z - b), $MachinePrecision] * y), $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 1.00000000000000003e282 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 5.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 73.8

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

   5. Applied rewrites 73.8%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.00000000000000003e282

   1. Initial program 99.7%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. sub-neg N/A

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

      4. associate--l+ N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -\infty:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 10^{+282}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t + y, a, \mathsf{fma}\left(x, z, \left(z - b\right) \cdot y\right)\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+194}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) (+ (+ t x) y)))
        (t_2 (- (+ a z) b)))
   (if (<= t_1 -1e+194)
     t_2
     (if (<= t_1 5e+232) (/ (fma a t (* z x)) (+ t x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / ((t + x) + y);
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -1e+194) {
		tmp = t_2;
	} else if (t_1 <= 5e+232) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / Float64(Float64(t + x) + y))
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_1 <= -1e+194)
		tmp = t_2;
	elseif (t_1 <= 5e+232)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+194], t$95$2, If[LessEqual[t$95$1, 5e+232], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.99999999999999945e193 or 4.99999999999999987e232 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 14.8%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 73.3

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

   5. Applied rewrites 73.3%

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

   if -9.99999999999999945e193 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999987e232

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. div-sub N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 61.8

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

   8. Applied rewrites 61.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -1 \cdot 10^{+194}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 5 \cdot 10^{+232}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{t + x}, \frac{z}{t + x} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -8.2e+79)
     t_1
     (if (<= y 9e+99) (fma a (/ t (+ t x)) (* (/ z (+ t x)) 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 <= -8.2e+79) {
		tmp = t_1;
	} else if (y <= 9e+99) {
		tmp = fma(a, (t / (t + x)), ((z / (t + x)) * 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 <= -8.2e+79)
		tmp = t_1;
	elseif (y <= 9e+99)
		tmp = fma(a, Float64(t / Float64(t + x)), Float64(Float64(z / Float64(t + x)) * x));
	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, -8.2e+79], t$95$1, If[LessEqual[y, 9e+99], N[(a * N[(t / N[(t + x), $MachinePrecision]), $MachinePrecision] + N[(N[(z / N[(t + x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.2e79 or 8.9999999999999999e99 < y

   1. Initial program 32.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 88.1

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

   5. Applied rewrites 88.1%

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

   if -8.2e79 < y < 8.9999999999999999e99

   1. Initial program 74.5%

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

   3. Taylor expanded in z around 0

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. div-sub N/A

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

   5. Applied rewrites 84.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.5%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+79}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+99}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{t + x}, \frac{z}{t + x} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y + x}{t + \left(y + x\right)} \cdot z\\ \mathbf{if}\;z \leq -4.4 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+68}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ (+ y x) (+ t (+ y x))) z)))
   (if (<= z -4.4e+89) t_1 (if (<= z 5.4e+68) (- (+ a z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((y + x) / (t + (y + x))) * z;
	double tmp;
	if (z <= -4.4e+89) {
		tmp = t_1;
	} else if (z <= 5.4e+68) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 = ((y + x) / (t + (y + x))) * z
    if (z <= (-4.4d+89)) then
        tmp = t_1
    else if (z <= 5.4d+68) then
        tmp = (a + z) - b
    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 = ((y + x) / (t + (y + x))) * z;
	double tmp;
	if (z <= -4.4e+89) {
		tmp = t_1;
	} else if (z <= 5.4e+68) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((y + x) / (t + (y + x))) * z
	tmp = 0
	if z <= -4.4e+89:
		tmp = t_1
	elif z <= 5.4e+68:
		tmp = (a + z) - b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(y + x) / Float64(t + Float64(y + x))) * z)
	tmp = 0.0
	if (z <= -4.4e+89)
		tmp = t_1;
	elseif (z <= 5.4e+68)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((y + x) / (t + (y + x))) * z;
	tmp = 0.0;
	if (z <= -4.4e+89)
		tmp = t_1;
	elseif (z <= 5.4e+68)
		tmp = (a + z) - b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(y + x), $MachinePrecision] / N[(t + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -4.4e+89], t$95$1, If[LessEqual[z, 5.4e+68], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4e89 or 5.39999999999999982e68 < z

   1. Initial program 43.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 68.7

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

   5. Applied rewrites 68.7%

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

   7. Applied rewrites 72.6%

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

   if -4.4e89 < z < 5.39999999999999982e68

   1. Initial program 69.0%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 55.8

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

   5. Applied rewrites 55.8%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{y + x}{t + \left(y + x\right)} \cdot z\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+68}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{t + \left(y + x\right)} \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{+100}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -3.8e+80) t_1 (if (<= y 1.08e+100) (+ 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 <= -3.8e+80) {
		tmp = t_1;
	} else if (y <= 1.08e+100) {
		tmp = a + z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    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 <= (-3.8d+80)) then
        tmp = t_1
    else if (y <= 1.08d+100) 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 <= -3.8e+80) {
		tmp = t_1;
	} else if (y <= 1.08e+100) {
		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 <= -3.8e+80:
		tmp = t_1
	elif y <= 1.08e+100:
		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 <= -3.8e+80)
		tmp = t_1;
	elseif (y <= 1.08e+100)
		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 <= -3.8e+80)
		tmp = t_1;
	elseif (y <= 1.08e+100)
		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, -3.8e+80], t$95$1, If[LessEqual[y, 1.08e+100], N[(a + z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.79999999999999997e80 or 1.07999999999999996e100 < y

   1. Initial program 32.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 88.1

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

   5. Applied rewrites 88.1%

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

   if -3.79999999999999997e80 < y < 1.07999999999999996e100

   1. Initial program 74.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 35.3

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

   5. Applied rewrites 35.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 45.1%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+80}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{+100}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+159}:\\ \;\;\;\;z - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -4.5e+159) (- z b) (+ a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.5e+159) {
		tmp = z - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-4.5d+159)) then
        tmp = z - b
    else
        tmp = a + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -4.5e+159) {
		tmp = z - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -4.5e+159:
		tmp = z - b
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -4.5e+159)
		tmp = Float64(z - b);
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -4.5e+159)
		tmp = z - b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -4.5e+159], N[(z - b), $MachinePrecision], N[(a + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.50000000000000026e159

   1. Initial program 33.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 83.8%

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

   if -4.50000000000000026e159 < y

   1. Initial program 63.8%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 47.1

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 50.6%

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

Alternative 10: 51.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+195}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -8.5e+195) (- a b) (+ a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.5e+195) {
		tmp = a - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-8.5d+195)) then
        tmp = a - b
    else
        tmp = a + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.5e+195) {
		tmp = a - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -8.5e+195:
		tmp = a - b
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -8.5e+195)
		tmp = Float64(a - b);
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -8.5e+195)
		tmp = a - b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -8.5e+195], N[(a - b), $MachinePrecision], N[(a + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5e195

   1. Initial program 29.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 76.7%

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

   if -8.5e195 < y

   1. Initial program 63.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 47.8

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 50.4%

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

Alternative 11: 48.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+196}:\\ \;\;\;\;-b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= y -8.5e+196) (- b) (+ a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.5e+196) {
		tmp = -b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (y <= (-8.5d+196)) then
        tmp = -b
    else
        tmp = a + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -8.5e+196) {
		tmp = -b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if y <= -8.5e+196:
		tmp = -b
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -8.5e+196)
		tmp = Float64(-b);
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (y <= -8.5e+196)
		tmp = -b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -8.5e+196], (-b), N[(a + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.50000000000000041e196

   1. Initial program 29.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 61.5%

      \[\leadsto -b \]

   if -8.50000000000000041e196 < y

   1. Initial program 63.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 47.8

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 50.4%

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

Alternative 12: 13.4% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 54.1

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

5. Applied rewrites 54.1%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 15.5%

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

Developer Target 1: 82.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024248 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

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

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