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

Percentage Accurate: 60.8% → 89.1%

Time: 13.1s

Alternatives: 14

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 60.8% 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: 89.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{y + t}\\ t_2 := t + \left(x + y\right)\\ t_3 := \left(y + t\right) \cdot \left(y + t\right)\\ t_4 := \frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ t_5 := \left(x + y\right) \cdot \left(x + y\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, \frac{b}{t\_3}, t\_1\right) - \mathsf{fma}\left(y, \frac{z}{t\_3}, \frac{a}{y + t}\right), \mathsf{fma}\left(y, t\_1, a\right)\right) - y \cdot \frac{b}{y + t}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t\_2} + \frac{y}{t\_2}, \frac{\mathsf{fma}\left(a, y + t, y \cdot \left(-b\right)\right)}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{t\_5}, \frac{a}{x + y}\right) - \mathsf{fma}\left(a, \frac{y}{t\_5}, \frac{z}{x + y}\right), \mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\right) - y \cdot \frac{b}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ z (+ y t)))
        (t_2 (+ t (+ x y)))
        (t_3 (* (+ y t) (+ y t)))
        (t_4 (/ (- (+ (* z (+ x y)) (* (+ y t) a)) (* y b)) (+ y (+ x t))))
        (t_5 (* (+ x y) (+ x y))))
   (if (<= t_4 (- INFINITY))
     (-
      (fma
       x
       (- (fma y (/ b t_3) t_1) (fma y (/ z t_3) (/ a (+ y t))))
       (fma y t_1 a))
      (* y (/ b (+ y t))))
     (if (<= t_4 2e+307)
       (fma z (+ (/ x t_2) (/ y t_2)) (/ (fma a (+ y t) (* y (- b))) t_2))
       (-
        (fma
         t
         (- (fma y (/ b t_5) (/ a (+ x y))) (fma a (/ y t_5) (/ z (+ x y))))
         (fma a (/ y (+ x y)) z))
        (* y (/ b (+ x y))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z / (y + t);
	double t_2 = t + (x + y);
	double t_3 = (y + t) * (y + t);
	double t_4 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	double t_5 = (x + y) * (x + y);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = fma(x, (fma(y, (b / t_3), t_1) - fma(y, (z / t_3), (a / (y + t)))), fma(y, t_1, a)) - (y * (b / (y + t)));
	} else if (t_4 <= 2e+307) {
		tmp = fma(z, ((x / t_2) + (y / t_2)), (fma(a, (y + t), (y * -b)) / t_2));
	} else {
		tmp = fma(t, (fma(y, (b / t_5), (a / (x + y))) - fma(a, (y / t_5), (z / (x + y)))), fma(a, (y / (x + y)), z)) - (y * (b / (x + y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z / Float64(y + t))
	t_2 = Float64(t + Float64(x + y))
	t_3 = Float64(Float64(y + t) * Float64(y + t))
	t_4 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(Float64(y + t) * a)) - Float64(y * b)) / Float64(y + Float64(x + t)))
	t_5 = Float64(Float64(x + y) * Float64(x + y))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(fma(x, Float64(fma(y, Float64(b / t_3), t_1) - fma(y, Float64(z / t_3), Float64(a / Float64(y + t)))), fma(y, t_1, a)) - Float64(y * Float64(b / Float64(y + t))));
	elseif (t_4 <= 2e+307)
		tmp = fma(z, Float64(Float64(x / t_2) + Float64(y / t_2)), Float64(fma(a, Float64(y + t), Float64(y * Float64(-b))) / t_2));
	else
		tmp = Float64(fma(t, Float64(fma(y, Float64(b / t_5), Float64(a / Float64(x + y))) - fma(a, Float64(y / t_5), Float64(z / Float64(x + y)))), fma(a, Float64(y / Float64(x + y)), z)) - Float64(y * Float64(b / Float64(x + y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y + t), $MachinePrecision] * N[(y + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(N[(y + t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(x * N[(N[(y * N[(b / t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(y * N[(z / t$95$3), $MachinePrecision] + N[(a / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * t$95$1 + a), $MachinePrecision]), $MachinePrecision] - N[(y * N[(b / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+307], N[(z * N[(N[(x / t$95$2), $MachinePrecision] + N[(y / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(y + t), $MachinePrecision] + N[(y * (-b)), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(t * N[(N[(y * N[(b / t$95$5), $MachinePrecision] + N[(a / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(y / t$95$5), $MachinePrecision] + N[(z / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] - N[(y * N[(b / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.7%

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

   3. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 83.8%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.99999999999999997e307

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 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 98.0%

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

   if 1.99999999999999997e307 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 4.3%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 79.6%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(y, \frac{b}{\left(y + t\right) \cdot \left(y + t\right)}, \frac{z}{y + t}\right) - \mathsf{fma}\left(y, \frac{z}{\left(y + t\right) \cdot \left(y + t\right)}, \frac{a}{y + t}\right), \mathsf{fma}\left(y, \frac{z}{y + t}, a\right)\right) - y \cdot \frac{b}{y + t}\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}, \frac{\mathsf{fma}\left(a, y + t, y \cdot \left(-b\right)\right)}{t + \left(x + y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{\left(x + y\right) \cdot \left(x + y\right)}, \frac{a}{x + y}\right) - \mathsf{fma}\left(a, \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}, \frac{z}{x + y}\right), \mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\right) - y \cdot \frac{b}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ t_2 := t + \left(x + y\right)\\ t_3 := \left(x + y\right) \cdot \left(x + y\right)\\ t_4 := \mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{t\_3}, \frac{a}{x + y}\right) - \mathsf{fma}\left(a, \frac{y}{t\_3}, \frac{z}{x + y}\right), \mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\right) - y \cdot \frac{b}{x + y}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t\_2} + \frac{y}{t\_2}, \frac{\mathsf{fma}\left(a, y + t, y \cdot \left(-b\right)\right)}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* z (+ x y)) (* (+ y t) a)) (* y b)) (+ y (+ x t))))
        (t_2 (+ t (+ x y)))
        (t_3 (* (+ x y) (+ x y)))
        (t_4
         (-
          (fma
           t
           (- (fma y (/ b t_3) (/ a (+ x y))) (fma a (/ y t_3) (/ z (+ x y))))
           (fma a (/ y (+ x y)) z))
          (* y (/ b (+ x y))))))
   (if (<= t_1 (- INFINITY))
     t_4
     (if (<= t_1 2e+307)
       (fma z (+ (/ x t_2) (/ y t_2)) (/ (fma a (+ y t) (* y (- b))) t_2))
       t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	double t_2 = t + (x + y);
	double t_3 = (x + y) * (x + y);
	double t_4 = fma(t, (fma(y, (b / t_3), (a / (x + y))) - fma(a, (y / t_3), (z / (x + y)))), fma(a, (y / (x + y)), z)) - (y * (b / (x + y)));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_1 <= 2e+307) {
		tmp = fma(z, ((x / t_2) + (y / t_2)), (fma(a, (y + t), (y * -b)) / t_2));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(Float64(y + t) * a)) - Float64(y * b)) / Float64(y + Float64(x + t)))
	t_2 = Float64(t + Float64(x + y))
	t_3 = Float64(Float64(x + y) * Float64(x + y))
	t_4 = Float64(fma(t, Float64(fma(y, Float64(b / t_3), Float64(a / Float64(x + y))) - fma(a, Float64(y / t_3), Float64(z / Float64(x + y)))), fma(a, Float64(y / Float64(x + y)), z)) - Float64(y * Float64(b / Float64(x + y))))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_1 <= 2e+307)
		tmp = fma(z, Float64(Float64(x / t_2) + Float64(y / t_2)), Float64(fma(a, Float64(y + t), Float64(y * Float64(-b))) / 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[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(N[(y + t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t * N[(N[(y * N[(b / t$95$3), $MachinePrecision] + N[(a / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(y / t$95$3), $MachinePrecision] + N[(z / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] - N[(y * N[(b / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$4, If[LessEqual[t$95$1, 2e+307], N[(z * N[(N[(x / t$95$2), $MachinePrecision] + N[(y / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(y + t), $MachinePrecision] + N[(y * (-b)), $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 1.99999999999999997e307 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 4.9%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 81.4%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.99999999999999997e307

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 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 98.0%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{\left(x + y\right) \cdot \left(x + y\right)}, \frac{a}{x + y}\right) - \mathsf{fma}\left(a, \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}, \frac{z}{x + y}\right), \mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\right) - y \cdot \frac{b}{x + y}\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}, \frac{\mathsf{fma}\left(a, y + t, y \cdot \left(-b\right)\right)}{t + \left(x + y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{\left(x + y\right) \cdot \left(x + y\right)}, \frac{a}{x + y}\right) - \mathsf{fma}\left(a, \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}, \frac{z}{x + y}\right), \mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\right) - y \cdot \frac{b}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ t_2 := t + \left(x + y\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t\_2}, z - b\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t\_2} + \frac{y}{t\_2}, \frac{\mathsf{fma}\left(a, y + t, y \cdot \left(-b\right)\right)}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(y, \frac{z}{y + t}, -b \cdot \frac{y}{y + t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* z (+ x y)) (* (+ y t) a)) (* y b)) (+ y (+ x t))))
        (t_2 (+ t (+ x y))))
   (if (<= t_1 (- INFINITY))
     (fma (+ y t) (/ a t_2) (- z b))
     (if (<= t_1 2e+194)
       (fma z (+ (/ x t_2) (/ y t_2)) (/ (fma a (+ y t) (* y (- b))) t_2))
       (+ a (fma y (/ z (+ y t)) (- (* b (/ y (+ y t))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	double t_2 = t + (x + y);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((y + t), (a / t_2), (z - b));
	} else if (t_1 <= 2e+194) {
		tmp = fma(z, ((x / t_2) + (y / t_2)), (fma(a, (y + t), (y * -b)) / t_2));
	} else {
		tmp = a + fma(y, (z / (y + t)), -(b * (y / (y + t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(Float64(y + t) * a)) - Float64(y * b)) / Float64(y + Float64(x + t)))
	t_2 = Float64(t + Float64(x + y))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(y + t), Float64(a / t_2), Float64(z - b));
	elseif (t_1 <= 2e+194)
		tmp = fma(z, Float64(Float64(x / t_2) + Float64(y / t_2)), Float64(fma(a, Float64(y + t), Float64(y * Float64(-b))) / t_2));
	else
		tmp = Float64(a + fma(y, Float64(z / Float64(y + t)), Float64(-Float64(b * Float64(y / Float64(y + t))))));
	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[(N[(y + t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y + t), $MachinePrecision] * N[(a / t$95$2), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+194], N[(z * N[(N[(x / t$95$2), $MachinePrecision] + N[(y / t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(y + t), $MachinePrecision] + N[(y * (-b)), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(a + N[(y * N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision] + (-N[(b * N[(y / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 5.7%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 36.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 81.9%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.99999999999999989e194

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 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 97.9%

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

   if 1.99999999999999989e194 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 13.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 36.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 59.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 80.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t + \left(x + y\right)} + \frac{y}{t + \left(x + y\right)}, \frac{\mathsf{fma}\left(a, y + t, y \cdot \left(-b\right)\right)}{t + \left(x + y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(y, \frac{z}{y + t}, -b \cdot \frac{y}{y + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+194}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(y, \frac{z}{y + t}, -b \cdot \frac{y}{y + t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* z (+ x y)) (* (+ y t) a)) (* y b)) (+ y (+ x t)))))
   (if (<= t_1 -1e+291)
     (fma (+ y t) (/ a (+ t (+ x y))) (- z b))
     (if (<= t_1 2e+194)
       t_1
       (+ a (fma y (/ z (+ y t)) (- (* b (/ y (+ y t))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	double tmp;
	if (t_1 <= -1e+291) {
		tmp = fma((y + t), (a / (t + (x + y))), (z - b));
	} else if (t_1 <= 2e+194) {
		tmp = t_1;
	} else {
		tmp = a + fma(y, (z / (y + t)), -(b * (y / (y + t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(Float64(y + t) * a)) - Float64(y * b)) / Float64(y + Float64(x + t)))
	tmp = 0.0
	if (t_1 <= -1e+291)
		tmp = fma(Float64(y + t), Float64(a / Float64(t + Float64(x + y))), Float64(z - b));
	elseif (t_1 <= 2e+194)
		tmp = t_1;
	else
		tmp = Float64(a + fma(y, Float64(z / Float64(y + t)), Float64(-Float64(b * Float64(y / Float64(y + t))))));
	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[(N[(y + t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+291], N[(N[(y + t), $MachinePrecision] * N[(a / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+194], t$95$1, N[(a + N[(y * N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision] + (-N[(b * N[(y / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.9999999999999996e290

   1. Initial program 9.8%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 82.7%

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

   if -9.9999999999999996e290 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.99999999999999989e194

   1. Initial program 97.8%

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

   if 1.99999999999999989e194 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 13.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 36.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 59.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 80.5%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -1 \cdot 10^{+291}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(y, \frac{z}{y + t}, -b \cdot \frac{y}{y + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-138}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{x + t}, x \cdot \frac{z}{x + t}\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(y, \frac{z}{y + t}, -b \cdot \frac{y}{y + t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.6e-66)
   (fma (+ y t) (/ a (+ t (+ x y))) (- z b))
   (if (<= y 7.5e-138)
     (fma a (/ t (+ x t)) (* x (/ z (+ x t))))
     (+ a (fma y (/ z (+ y t)) (- (* b (/ y (+ y t)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.6e-66) {
		tmp = fma((y + t), (a / (t + (x + y))), (z - b));
	} else if (y <= 7.5e-138) {
		tmp = fma(a, (t / (x + t)), (x * (z / (x + t))));
	} else {
		tmp = a + fma(y, (z / (y + t)), -(b * (y / (y + t))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.6e-66)
		tmp = fma(Float64(y + t), Float64(a / Float64(t + Float64(x + y))), Float64(z - b));
	elseif (y <= 7.5e-138)
		tmp = fma(a, Float64(t / Float64(x + t)), Float64(x * Float64(z / Float64(x + t))));
	else
		tmp = Float64(a + fma(y, Float64(z / Float64(y + t)), Float64(-Float64(b * Float64(y / Float64(y + t))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.6e-66], N[(N[(y + t), $MachinePrecision] * N[(a / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e-138], N[(a * N[(t / N[(x + t), $MachinePrecision]), $MachinePrecision] + N[(x * N[(z / N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a + N[(y * N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision] + (-N[(b * N[(y / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.59999999999999991e-66

   1. Initial program 50.5%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 83.1%

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

   if -1.59999999999999991e-66 < y < 7.4999999999999995e-138

   1. Initial program 79.4%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 80.4%

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

   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 71.0%

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

   if 7.4999999999999995e-138 < y

   1. Initial program 58.5%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 67.7%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 77.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 79.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-138}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{x + t}, x \cdot \frac{z}{x + t}\right)\\ \mathbf{else}:\\ \;\;\;\;a + \mathsf{fma}\left(y, \frac{z}{y + t}, -b \cdot \frac{y}{y + t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{x + t}, x \cdot \frac{z}{x + t}\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;a + \left(z - b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= y -1.6e-66)
   (fma (+ y t) (/ a (+ t (+ x y))) (- z b))
   (if (<= y 2.9e-130)
     (fma a (/ t (+ x t)) (* x (/ z (+ x t))))
     (if (<= y 4.5e-6) (+ a (- z b)) (fma y (/ (- z b) (+ y t)) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (y <= -1.6e-66) {
		tmp = fma((y + t), (a / (t + (x + y))), (z - b));
	} else if (y <= 2.9e-130) {
		tmp = fma(a, (t / (x + t)), (x * (z / (x + t))));
	} else if (y <= 4.5e-6) {
		tmp = a + (z - b);
	} else {
		tmp = fma(y, ((z - b) / (y + t)), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (y <= -1.6e-66)
		tmp = fma(Float64(y + t), Float64(a / Float64(t + Float64(x + y))), Float64(z - b));
	elseif (y <= 2.9e-130)
		tmp = fma(a, Float64(t / Float64(x + t)), Float64(x * Float64(z / Float64(x + t))));
	elseif (y <= 4.5e-6)
		tmp = Float64(a + Float64(z - b));
	else
		tmp = fma(y, Float64(Float64(z - b) / Float64(y + t)), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[y, -1.6e-66], N[(N[(y + t), $MachinePrecision] * N[(a / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e-130], N[(a * N[(t / N[(x + t), $MachinePrecision]), $MachinePrecision] + N[(x * N[(z / N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e-6], N[(a + N[(z - b), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - b), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.59999999999999991e-66

   1. Initial program 50.5%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 83.1%

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

   if -1.59999999999999991e-66 < y < 2.9e-130

   1. Initial program 79.6%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 79.5%

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

   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 71.3%

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

   if 2.9e-130 < y < 4.50000000000000011e-6

   1. Initial program 82.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 4.50000000000000011e-6 < y

   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 b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 83.2%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{x + t}, x \cdot \frac{z}{x + t}\right)\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;a + \left(z - b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{if}\;t \leq -2.65 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z b) (+ y t)) a)))
   (if (<= t -2.65e-30)
     t_1
     (if (<= t 1.5e+31) (fma (+ y t) (/ a (+ t (+ x y))) (- z b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, ((z - b) / (y + t)), a);
	double tmp;
	if (t <= -2.65e-30) {
		tmp = t_1;
	} else if (t <= 1.5e+31) {
		tmp = fma((y + t), (a / (t + (x + y))), (z - b));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(Float64(z - b) / Float64(y + t)), a)
	tmp = 0.0
	if (t <= -2.65e-30)
		tmp = t_1;
	elseif (t <= 1.5e+31)
		tmp = fma(Float64(y + t), Float64(a / Float64(t + Float64(x + y))), Float64(z - b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(N[(z - b), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[t, -2.65e-30], t$95$1, If[LessEqual[t, 1.5e+31], N[(N[(y + t), $MachinePrecision] * N[(a / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.64999999999999987e-30 or 1.49999999999999995e31 < t

   1. Initial program 58.3%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.1%

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

   if -2.64999999999999987e-30 < t < 1.49999999999999995e31

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 76.6%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.65 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-a\right) \cdot \frac{\left(-y\right) - t}{t + \left(x + y\right)}\\ \mathbf{if}\;a \leq -2.7 \cdot 10^{+198}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (- a) (/ (- (- y) t) (+ t (+ x y))))))
   (if (<= a -2.7e+198)
     t_1
     (if (<= a 9.6e+144) (fma y (/ (- z b) (+ y t)) a) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -a * ((-y - t) / (t + (x + y)));
	double tmp;
	if (a <= -2.7e+198) {
		tmp = t_1;
	} else if (a <= 9.6e+144) {
		tmp = fma(y, ((z - b) / (y + t)), a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(-a) * Float64(Float64(Float64(-y) - t) / Float64(t + Float64(x + y))))
	tmp = 0.0
	if (a <= -2.7e+198)
		tmp = t_1;
	elseif (a <= 9.6e+144)
		tmp = fma(y, Float64(Float64(z - b) / Float64(y + t)), a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[((-a) * N[(N[((-y) - t), $MachinePrecision] / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.7e+198], t$95$1, If[LessEqual[a, 9.6e+144], N[(y * N[(N[(z - b), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.6999999999999999e198 or 9.6000000000000002e144 < a

   1. Initial program 51.9%

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

   3. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 78.6%

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

   if -2.6999999999999999e198 < a < 9.6000000000000002e144

   1. Initial program 65.4%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 69.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.4%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{+198}:\\ \;\;\;\;\left(-a\right) \cdot \frac{\left(-y\right) - t}{t + \left(x + y\right)}\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-a\right) \cdot \frac{\left(-y\right) - t}{t + \left(x + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{if}\;t \leq -7.4 \cdot 10^{-62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-32}:\\ \;\;\;\;a + \left(z - b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z b) (+ y t)) a)))
   (if (<= t -7.4e-62) t_1 (if (<= t 3.2e-32) (+ a (- z b)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, ((z - b) / (y + t)), a);
	double tmp;
	if (t <= -7.4e-62) {
		tmp = t_1;
	} else if (t <= 3.2e-32) {
		tmp = a + (z - b);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(Float64(z - b) / Float64(y + t)), a)
	tmp = 0.0
	if (t <= -7.4e-62)
		tmp = t_1;
	elseif (t <= 3.2e-32)
		tmp = Float64(a + Float64(z - b));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * N[(N[(z - b), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]}, If[LessEqual[t, -7.4e-62], t$95$1, If[LessEqual[t, 3.2e-32], N[(a + N[(z - b), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.3999999999999996e-62 or 3.2000000000000002e-32 < t

   1. Initial program 57.9%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Applied rewrites 69.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.6%

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

   if -7.3999999999999996e-62 < t < 3.2000000000000002e-32

   1. Initial program 67.8%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 68.9

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

   5. Applied rewrites 68.9%

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

Alternative 10: 60.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \left(z - b\right)\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.45 \cdot 10^{-130}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ a (- z b))))
   (if (<= y -1.4e-13) t_1 (if (<= y 3.45e-130) (+ z a) 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 <= -1.4e-13) {
		tmp = t_1;
	} else if (y <= 3.45e-130) {
		tmp = z + a;
	} 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 <= (-1.4d-13)) then
        tmp = t_1
    else if (y <= 3.45d-130) then
        tmp = z + a
    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 <= -1.4e-13) {
		tmp = t_1;
	} else if (y <= 3.45e-130) {
		tmp = z + a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a + (z - b)
	tmp = 0
	if y <= -1.4e-13:
		tmp = t_1
	elif y <= 3.45e-130:
		tmp = z + a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a + Float64(z - b))
	tmp = 0.0
	if (y <= -1.4e-13)
		tmp = t_1;
	elseif (y <= 3.45e-130)
		tmp = Float64(z + a);
	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 <= -1.4e-13)
		tmp = t_1;
	elseif (y <= 3.45e-130)
		tmp = z + a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a + N[(z - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.4e-13], t$95$1, If[LessEqual[y, 3.45e-130], N[(z + a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4000000000000001e-13 or 3.45000000000000018e-130 < y

   1. Initial program 52.3%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -1.4000000000000001e-13 < y < 3.45000000000000018e-130

   1. Initial program 79.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 33.5

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

   5. Applied rewrites 33.5%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 45.8%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-13}:\\ \;\;\;\;a + \left(z - b\right)\\ \mathbf{elif}\;y \leq 3.45 \cdot 10^{-130}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;a + \left(z - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+120}:\\ \;\;\;\;a - b\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{+174}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;z - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -6e+120) (- a b) (if (<= b 1.32e+174) (+ z a) (- z b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6e+120) {
		tmp = a - b;
	} else if (b <= 1.32e+174) {
		tmp = z + a;
	} else {
		tmp = z - b;
	}
	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 (b <= (-6d+120)) then
        tmp = a - b
    else if (b <= 1.32d+174) then
        tmp = z + a
    else
        tmp = z - b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6e+120) {
		tmp = a - b;
	} else if (b <= 1.32e+174) {
		tmp = z + a;
	} else {
		tmp = z - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -6e+120:
		tmp = a - b
	elif b <= 1.32e+174:
		tmp = z + a
	else:
		tmp = z - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6e+120)
		tmp = Float64(a - b);
	elseif (b <= 1.32e+174)
		tmp = Float64(z + a);
	else
		tmp = Float64(z - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -6e+120)
		tmp = a - b;
	elseif (b <= 1.32e+174)
		tmp = z + a;
	else
		tmp = z - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6e+120], N[(a - b), $MachinePrecision], If[LessEqual[b, 1.32e+174], N[(z + a), $MachinePrecision], N[(z - b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6e120

   1. Initial program 50.6%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 49.6

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

   5. Applied rewrites 49.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 46.2%

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

   if -6e120 < b < 1.31999999999999999e174

   1. Initial program 68.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 59.2%

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

   if 1.31999999999999999e174 < b

   1. Initial program 43.7%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 51.3%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+120}:\\ \;\;\;\;a - b\\ \mathbf{elif}\;b \leq 1.32 \cdot 10^{+174}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;z - b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+120}:\\ \;\;\;\;a - b\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+174}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;a - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -6e+120) (- a b) (if (<= b 1.15e+174) (+ z a) (- a b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6e+120) {
		tmp = a - b;
	} else if (b <= 1.15e+174) {
		tmp = z + a;
	} else {
		tmp = a - b;
	}
	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 (b <= (-6d+120)) then
        tmp = a - b
    else if (b <= 1.15d+174) then
        tmp = z + a
    else
        tmp = a - b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6e+120) {
		tmp = a - b;
	} else if (b <= 1.15e+174) {
		tmp = z + a;
	} else {
		tmp = a - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -6e+120:
		tmp = a - b
	elif b <= 1.15e+174:
		tmp = z + a
	else:
		tmp = a - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6e+120)
		tmp = Float64(a - b);
	elseif (b <= 1.15e+174)
		tmp = Float64(z + a);
	else
		tmp = Float64(a - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -6e+120)
		tmp = a - b;
	elseif (b <= 1.15e+174)
		tmp = z + a;
	else
		tmp = a - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6e+120], N[(a - b), $MachinePrecision], If[LessEqual[b, 1.15e+174], N[(z + a), $MachinePrecision], N[(a - b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < -6e120 or 1.1499999999999999e174 < b

   1. Initial program 47.2%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 54.3

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

   5. Applied rewrites 54.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 48.2%

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

   if -6e120 < b < 1.1499999999999999e174

   1. Initial program 68.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 60.3

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

   5. Applied rewrites 60.3%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 59.2%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6 \cdot 10^{+120}:\\ \;\;\;\;a - b\\ \mathbf{elif}\;b \leq 1.15 \cdot 10^{+174}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;a - b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+277}:\\ \;\;\;\;-b\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{+175}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;-b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= b -3.3e+277) (- b) (if (<= b 1.22e+175) (+ z a) (- b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.3e+277) {
		tmp = -b;
	} else if (b <= 1.22e+175) {
		tmp = z + a;
	} else {
		tmp = -b;
	}
	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 (b <= (-3.3d+277)) then
        tmp = -b
    else if (b <= 1.22d+175) then
        tmp = z + a
    else
        tmp = -b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.3e+277) {
		tmp = -b;
	} else if (b <= 1.22e+175) {
		tmp = z + a;
	} else {
		tmp = -b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -3.3e+277:
		tmp = -b
	elif b <= 1.22e+175:
		tmp = z + a
	else:
		tmp = -b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.3e+277)
		tmp = Float64(-b);
	elseif (b <= 1.22e+175)
		tmp = Float64(z + a);
	else
		tmp = Float64(-b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -3.3e+277)
		tmp = -b;
	elseif (b <= 1.22e+175)
		tmp = z + a;
	else
		tmp = -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.3e+277], (-b), If[LessEqual[b, 1.22e+175], N[(z + a), $MachinePrecision], (-b)]]

Derivation

1. Split input into 2 regimes

2. if b < -3.3000000000000001e277 or 1.22e175 < b

   1. Initial program 40.9%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 47.6%

      \[\leadsto -b \]

   if -3.3000000000000001e277 < b < 1.22e175

   1. Initial program 67.2%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 56.1%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{+277}:\\ \;\;\;\;-b\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{+175}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;-b\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 13.1% 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 62.7%

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

3. Taylor expanded in y around inf

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

   1. associate--l+ N/A

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

   2. lower-+.f64 N/A

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

   3. lower--.f64 58.6

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

5. Applied rewrites 58.6%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 18.8%

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

Developer Target 1: 82.3% 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 2024238 
(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)))