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

Percentage Accurate: 59.9% → 90.8%

Time: 12.9s

Alternatives: 10

Speedup: 0.7×

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: 59.9% 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: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ t_2 := y + \left(x + t\right)\\ t_3 := a \cdot \mathsf{fma}\left(y, \frac{b}{a \cdot \left(\left(-y\right) - \left(x + t\right)\right)}, \mathsf{fma}\left(\frac{x + y}{t\_2}, \frac{z}{a}, \frac{y + t}{t\_2}\right)\right)\\ \mathbf{if}\;a \leq -1.9 \cdot 10^{+22}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t\_1} + \frac{y}{t\_1}, \frac{\mathsf{fma}\left(a, y + t, b \cdot \left(-y\right)\right)}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y)))
        (t_2 (+ y (+ x t)))
        (t_3
         (*
          a
          (fma
           y
           (/ b (* a (- (- y) (+ x t))))
           (fma (/ (+ x y) t_2) (/ z a) (/ (+ y t) t_2))))))
   (if (<= a -1.9e+22)
     t_3
     (if (<= a 3.3e-63)
       (fma z (+ (/ x t_1) (/ y t_1)) (/ (fma a (+ y t) (* 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 = y + (x + t);
	double t_3 = a * fma(y, (b / (a * (-y - (x + t)))), fma(((x + y) / t_2), (z / a), ((y + t) / t_2)));
	double tmp;
	if (a <= -1.9e+22) {
		tmp = t_3;
	} else if (a <= 3.3e-63) {
		tmp = fma(z, ((x / t_1) + (y / t_1)), (fma(a, (y + t), (b * -y)) / t_1));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	t_2 = Float64(y + Float64(x + t))
	t_3 = Float64(a * fma(y, Float64(b / Float64(a * Float64(Float64(-y) - Float64(x + t)))), fma(Float64(Float64(x + y) / t_2), Float64(z / a), Float64(Float64(y + t) / t_2))))
	tmp = 0.0
	if (a <= -1.9e+22)
		tmp = t_3;
	elseif (a <= 3.3e-63)
		tmp = fma(z, Float64(Float64(x / t_1) + Float64(y / t_1)), Float64(fma(a, Float64(y + t), Float64(b * Float64(-y))) / t_1));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(a * N[(y * N[(b / N[(a * N[((-y) - N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x + y), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(z / a), $MachinePrecision] + N[(N[(y + t), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.9e+22], t$95$3, If[LessEqual[a, 3.3e-63], N[(z * N[(N[(x / t$95$1), $MachinePrecision] + N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(y + t), $MachinePrecision] + N[(b * (-y)), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.9000000000000002e22 or 3.29999999999999994e-63 < a

   1. Initial program 46.5%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 53.9%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 86.1%

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

   10. Applied rewrites 97.0%

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

   12. Applied rewrites 97.3%

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

   if -1.9000000000000002e22 < a < 3.29999999999999994e-63

   1. Initial program 77.1%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 88.7%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{+22}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(y, \frac{b}{a \cdot \left(\left(-y\right) - \left(x + t\right)\right)}, \mathsf{fma}\left(\frac{x + y}{y + \left(x + t\right)}, \frac{z}{a}, \frac{y + t}{y + \left(x + t\right)}\right)\right)\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-63}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}, \frac{\mathsf{fma}\left(a, y + t, b \cdot \left(-y\right)\right)}{t + \left(x + y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \mathsf{fma}\left(y, \frac{b}{a \cdot \left(\left(-y\right) - \left(x + t\right)\right)}, \mathsf{fma}\left(\frac{x + y}{y + \left(x + t\right)}, \frac{z}{a}, \frac{y + t}{y + \left(x + t\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+288}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+257}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* z (+ x y)) (* a (+ y t))) (* y b)) (+ y (+ x t))))
        (t_2 (- (+ a z) b)))
   (if (<= t_1 -4e+288) t_2 (if (<= t_1 2e+257) t_1 t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / (y + (x + t));
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -4e+288) {
		tmp = t_2;
	} else if (t_1 <= 2e+257) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: tmp
    t_1 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / (y + (x + t))
    t_2 = (a + z) - b
    if (t_1 <= (-4d+288)) then
        tmp = t_2
    else if (t_1 <= 2d+257) then
        tmp = 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 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / (y + (x + t));
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -4e+288) {
		tmp = t_2;
	} else if (t_1 <= 2e+257) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / (y + (x + t))
	t_2 = (a + z) - b
	tmp = 0
	if t_1 <= -4e+288:
		tmp = t_2
	elif t_1 <= 2e+257:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(a * Float64(y + t))) - Float64(y * b)) / Float64(y + Float64(x + t)))
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_1 <= -4e+288)
		tmp = t_2;
	elseif (t_1 <= 2e+257)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / (y + (x + t));
	t_2 = (a + z) - b;
	tmp = 0.0;
	if (t_1 <= -4e+288)
		tmp = t_2;
	elseif (t_1 <= 2e+257)
		tmp = 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[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+288], t$95$2, If[LessEqual[t$95$1, 2e+257], t$95$1, 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)) < -4e288 or 2.00000000000000006e257 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 9.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 71.9

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

   5. Applied rewrites 71.9%

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

   if -4e288 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.00000000000000006e257

   1. Initial program 99.1%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -4 \cdot 10^{+288}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+150}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+85}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{x + t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* z (+ x y)) (* a (+ y t))) (* y b)) (+ y (+ x t))))
        (t_2 (- (+ a z) b)))
   (if (<= t_1 -4e+150)
     t_2
     (if (<= t_1 1e+85) (/ (fma a t (* x z)) (+ x t)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / (y + (x + t));
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -4e+150) {
		tmp = t_2;
	} else if (t_1 <= 1e+85) {
		tmp = fma(a, t, (x * z)) / (x + t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(a * Float64(y + t))) - Float64(y * b)) / Float64(y + Float64(x + t)))
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_1 <= -4e+150)
		tmp = t_2;
	elseif (t_1 <= 1e+85)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(x + t));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+150], t$95$2, If[LessEqual[t$95$1, 1e+85], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(x + t), $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)) < -3.99999999999999992e150 or 1e85 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 36.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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. lower-+.f64 72.3

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

   5. Applied rewrites 72.3%

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

   if -3.99999999999999992e150 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1e85

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 69.8

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

   5. Applied rewrites 69.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -4 \cdot 10^{+150}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 10^{+85}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{x + t}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{a}{x + t}, t, x \cdot \frac{z}{x + t}\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-127}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;x \leq 0.205:\\ \;\;\;\;a + \mathsf{fma}\left(y, \frac{z}{y + t}, \frac{b \cdot \left(-y\right)}{y + t}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ a (+ x t)) t (* x (/ z (+ x t))))))
   (if (<= x -2.7e+154)
     t_1
     (if (<= x -1.1e-127)
       (- (+ a z) b)
       (if (<= x 0.205)
         (+ a (fma y (/ z (+ y t)) (/ (* b (- y)) (+ y t))))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((a / (x + t)), t, (x * (z / (x + t))));
	double tmp;
	if (x <= -2.7e+154) {
		tmp = t_1;
	} else if (x <= -1.1e-127) {
		tmp = (a + z) - b;
	} else if (x <= 0.205) {
		tmp = a + fma(y, (z / (y + t)), ((b * -y) / (y + t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(a / Float64(x + t)), t, Float64(x * Float64(z / Float64(x + t))))
	tmp = 0.0
	if (x <= -2.7e+154)
		tmp = t_1;
	elseif (x <= -1.1e-127)
		tmp = Float64(Float64(a + z) - b);
	elseif (x <= 0.205)
		tmp = Float64(a + fma(y, Float64(z / Float64(y + t)), Float64(Float64(b * Float64(-y)) / Float64(y + t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a / N[(x + t), $MachinePrecision]), $MachinePrecision] * t + N[(x * N[(z / N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e+154], t$95$1, If[LessEqual[x, -1.1e-127], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], If[LessEqual[x, 0.205], N[(a + N[(y * N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision] + N[(N[(b * (-y)), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.70000000000000006e154 or 0.204999999999999988 < x

   1. Initial program 50.3%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 68.2%

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

   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 56.9%

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

   10. Applied rewrites 69.7%

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

   if -2.70000000000000006e154 < x < -1.1000000000000001e-127

   1. Initial program 62.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -1.1000000000000001e-127 < x < 0.204999999999999988

   1. Initial program 68.8%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 81.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{x + t}, t, x \cdot \frac{z}{x + t}\right)\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-127}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;x \leq 0.205:\\ \;\;\;\;a + \mathsf{fma}\left(y, \frac{z}{y + t}, \frac{b \cdot \left(-y\right)}{y + t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{x + t}, t, x \cdot \frac{z}{x + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \frac{t}{x + t}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+195}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (/ t (+ x t)))))
   (if (<= t -2.4e+92) t_1 (if (<= t 7.2e+195) (- (+ a z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (t / (x + t));
	double tmp;
	if (t <= -2.4e+92) {
		tmp = t_1;
	} else if (t <= 7.2e+195) {
		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 = a * (t / (x + t))
    if (t <= (-2.4d+92)) then
        tmp = t_1
    else if (t <= 7.2d+195) 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 = a * (t / (x + t));
	double tmp;
	if (t <= -2.4e+92) {
		tmp = t_1;
	} else if (t <= 7.2e+195) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (t / (x + t))
	tmp = 0
	if t <= -2.4e+92:
		tmp = t_1
	elif t <= 7.2e+195:
		tmp = (a + z) - b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(t / Float64(x + t)))
	tmp = 0.0
	if (t <= -2.4e+92)
		tmp = t_1;
	elseif (t <= 7.2e+195)
		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 = a * (t / (x + t));
	tmp = 0.0;
	if (t <= -2.4e+92)
		tmp = t_1;
	elseif (t <= 7.2e+195)
		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[(a * N[(t / N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.4e+92], t$95$1, If[LessEqual[t, 7.2e+195], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.40000000000000005e92 or 7.1999999999999997e195 < t

   1. Initial program 60.3%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 59.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 59.2%

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

   if -2.40000000000000005e92 < t < 7.1999999999999997e195

   1. Initial program 60.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 66.1

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

   5. Applied rewrites 66.1%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+92}:\\ \;\;\;\;a \cdot \frac{t}{x + t}\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+195}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \frac{t}{x + t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -8 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -8e+129) t_1 (if (<= y 1.45e-11) (+ 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 <= -8e+129) {
		tmp = t_1;
	} else if (y <= 1.45e-11) {
		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 <= (-8d+129)) then
        tmp = t_1
    else if (y <= 1.45d-11) 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 <= -8e+129) {
		tmp = t_1;
	} else if (y <= 1.45e-11) {
		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 <= -8e+129:
		tmp = t_1
	elif y <= 1.45e-11:
		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 <= -8e+129)
		tmp = t_1;
	elseif (y <= 1.45e-11)
		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 <= -8e+129)
		tmp = t_1;
	elseif (y <= 1.45e-11)
		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, -8e+129], t$95$1, If[LessEqual[y, 1.45e-11], N[(a + z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8e129 or 1.45e-11 < y

   1. Initial program 39.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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. lower-+.f64 75.3

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

   5. Applied rewrites 75.3%

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

   if -8e129 < y < 1.45e-11

   1. Initial program 76.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 43.2

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

   5. Applied rewrites 43.2%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 52.9%

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

Alternative 7: 52.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.65 \cdot 10^{+103}:\\ \;\;\;\;a \cdot 1\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -2.65e+103) (* a 1.0) (+ a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -2.65e+103) {
		tmp = a * 1.0;
	} 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 (t <= (-2.65d+103)) then
        tmp = a * 1.0d0
    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 (t <= -2.65e+103) {
		tmp = a * 1.0;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -2.65e+103:
		tmp = a * 1.0
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -2.65e+103)
		tmp = Float64(a * 1.0);
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -2.65e+103)
		tmp = a * 1.0;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -2.65e+103], N[(a * 1.0), $MachinePrecision], N[(a + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.64999999999999985e103

   1. Initial program 64.5%

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

   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. lower-fma.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. div-sub N/A

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

      13. lower-/.f64 N/A

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in t around inf

      \[\leadsto a \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 48.9%

       \[\leadsto a \cdot 1 \]

   if -2.64999999999999985e103 < t

   1. Initial program 59.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 57.7%

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

Alternative 8: 51.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{+204}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -7.8e+204) (- a b) (+ a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -7.8e+204) {
		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 (t <= (-7.8d+204)) 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 (t <= -7.8e+204) {
		tmp = a - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -7.8e+204:
		tmp = a - b
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -7.8e+204)
		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 (t <= -7.8e+204)
		tmp = a - b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -7.8e+204], N[(a - b), $MachinePrecision], N[(a + z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.80000000000000033e204

   1. Initial program 42.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 22.4

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

   5. Applied rewrites 22.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 42.6%

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

   if -7.80000000000000033e204 < t

   1. Initial program 62.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 59.4

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 56.7%

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

Alternative 9: 51.7% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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 = a + z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 56.5

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

5. Applied rewrites 56.5%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 53.7%

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

Alternative 10: 13.6% 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 60.8%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 56.5

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

5. Applied rewrites 56.5%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 12.0%

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

Developer Target 1: 82.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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 2024220 
(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)))