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

Percentage Accurate: 60.1% → 91.0%

Time: 13.2s

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: 60.1% 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: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ t_2 := t + \left(x + y\right)\\ t_3 := \frac{a}{t\_2}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y + t, t\_3, z \cdot \frac{x + y}{t\_2}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+281}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + t, t\_3, z - b\right)\\ \end{array} \end{array} \]

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)))
        (t_3 (/ a t_2)))
   (if (<= t_1 (- INFINITY))
     (fma (+ y t) t_3 (* z (/ (+ x y) t_2)))
     (if (<= t_1 2e+281) t_1 (fma (+ y t) t_3 (- z b))))))

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 = a / t_2;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((y + t), t_3, (z * ((x + y) / t_2)));
	} else if (t_1 <= 2e+281) {
		tmp = t_1;
	} else {
		tmp = fma((y + t), t_3, (z - b));
	}
	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(a / t_2)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(y + t), t_3, Float64(z * Float64(Float64(x + y) / t_2)));
	elseif (t_1 <= 2e+281)
		tmp = t_1;
	else
		tmp = fma(Float64(y + t), t_3, Float64(z - b));
	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[(a / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(y + t), $MachinePrecision] * t$95$3 + N[(z * N[(N[(x + y), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+281], t$95$1, N[(N[(y + t), $MachinePrecision] * t$95$3 + N[(z - b), $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 6.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 43.2%

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

   6. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. neg-mul-1 N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 86.2

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

   8. Applied rewrites 86.2%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e281

   1. Initial program 99.5%

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

   if 2.0000000000000001e281 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 6.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 40.5%

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

   6. Taylor expanded in y around inf

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

      1. lower--.f64 92.2

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

   8. Applied rewrites 92.2%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z \cdot \frac{x + y}{t + \left(x + y\right)}\right)\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ t_2 := \left(z + a\right) - b\\ \mathbf{if}\;t\_1 \leq -500000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-29}:\\ \;\;\;\;a \cdot \frac{t}{x + t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* z (+ x y)) (* (+ y t) a)) (* y b)) (+ y (+ x t))))
        (t_2 (- (+ z a) b)))
   (if (<= t_1 -500000000000.0)
     t_2
     (if (<= t_1 -2e-159)
       (fma a (/ y (+ x y)) z)
       (if (<= t_1 5e-29) (* a (/ t (+ x t))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	double t_2 = (z + a) - b;
	double tmp;
	if (t_1 <= -500000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -2e-159) {
		tmp = fma(a, (y / (x + y)), z);
	} else if (t_1 <= 5e-29) {
		tmp = a * (t / (x + t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(Float64(y + t) * a)) - Float64(y * b)) / Float64(y + Float64(x + t)))
	t_2 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (t_1 <= -500000000000.0)
		tmp = t_2;
	elseif (t_1 <= -2e-159)
		tmp = fma(a, Float64(y / Float64(x + y)), z);
	elseif (t_1 <= 5e-29)
		tmp = Float64(a * Float64(t / Float64(x + t)));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(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[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -500000000000.0], t$95$2, If[LessEqual[t$95$1, -2e-159], N[(a * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision], If[LessEqual[t$95$1, 5e-29], N[(a * N[(t / N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

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)) < -5e11 or 4.99999999999999986e-29 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 49.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 67.7

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

   5. Applied rewrites 67.7%

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

   if -5e11 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.99999999999999998e-159

   1. Initial program 99.6%

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

   3. Taylor expanded in b around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{b \cdot y}{t + \left(x + y\right)} + \left(\frac{a \cdot \left(t + y\right)}{t + \left(x + y\right)} + \frac{z \cdot \left(x + y\right)}{t + \left(x + y\right)}\right)} \]
   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 92.5%

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

   6. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. neg-mul-1 N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 82.0

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

   8. Applied rewrites 82.0%

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

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 64.8

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

   11. Applied rewrites 64.8%

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

   if -1.99999999999999998e-159 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999986e-29

   1. Initial program 98.8%

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

   3. Taylor expanded in 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 93.9%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 71.4

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

   8. Applied rewrites 71.4%

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

   9. Taylor expanded in y around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 71.4

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

   11. Applied rewrites 71.4%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -500000000000:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;a \cdot \frac{t}{x + t}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ t_2 := \left(z + a\right) - b\\ \mathbf{if}\;t\_1 \leq -50000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-139}:\\ \;\;\;\;a \cdot \frac{z}{a}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-29}:\\ \;\;\;\;a \cdot \frac{t}{x + t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* z (+ x y)) (* (+ y t) a)) (* y b)) (+ y (+ x t))))
        (t_2 (- (+ z a) b)))
   (if (<= t_1 -50000000000.0)
     t_2
     (if (<= t_1 -5e-139)
       (* a (/ z a))
       (if (<= t_1 5e-29) (* a (/ t (+ x t))) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	double t_2 = (z + a) - b;
	double tmp;
	if (t_1 <= -50000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -5e-139) {
		tmp = a * (z / a);
	} else if (t_1 <= 5e-29) {
		tmp = a * (t / (x + t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t))
    t_2 = (z + a) - b
    if (t_1 <= (-50000000000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-5d-139)) then
        tmp = a * (z / a)
    else if (t_1 <= 5d-29) then
        tmp = a * (t / (x + t))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	double t_2 = (z + a) - b;
	double tmp;
	if (t_1 <= -50000000000.0) {
		tmp = t_2;
	} else if (t_1 <= -5e-139) {
		tmp = a * (z / a);
	} else if (t_1 <= 5e-29) {
		tmp = a * (t / (x + t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t))
	t_2 = (z + a) - b
	tmp = 0
	if t_1 <= -50000000000.0:
		tmp = t_2
	elif t_1 <= -5e-139:
		tmp = a * (z / a)
	elif t_1 <= 5e-29:
		tmp = a * (t / (x + t))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(Float64(y + t) * a)) - Float64(y * b)) / Float64(y + Float64(x + t)))
	t_2 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (t_1 <= -50000000000.0)
		tmp = t_2;
	elseif (t_1 <= -5e-139)
		tmp = Float64(a * Float64(z / a));
	elseif (t_1 <= 5e-29)
		tmp = Float64(a * Float64(t / Float64(x + t)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	t_2 = (z + a) - b;
	tmp = 0.0;
	if (t_1 <= -50000000000.0)
		tmp = t_2;
	elseif (t_1 <= -5e-139)
		tmp = a * (z / a);
	elseif (t_1 <= 5e-29)
		tmp = a * (t / (x + t));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(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[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000000.0], t$95$2, If[LessEqual[t$95$1, -5e-139], N[(a * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-29], N[(a * N[(t / N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

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)) < -5e10 or 4.99999999999999986e-29 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 49.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 67.4

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

   5. Applied rewrites 67.4%

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

   if -5e10 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -5.00000000000000034e-139

   1. Initial program 99.6%

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

   3. Taylor expanded in a around -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 91.1%

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

   6. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \frac{\color{blue}{\mathsf{neg}\left(z\right)}}{a} \]

      4. lower-neg.f64 53.1

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

   8. Applied rewrites 53.1%

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

   if -5.00000000000000034e-139 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999986e-29

   1. Initial program 98.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 91.8%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 68.5

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

   8. Applied rewrites 68.5%

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

   9. Taylor expanded in y around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 68.4

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

   11. Applied rewrites 68.4%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -50000000000:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -5 \cdot 10^{-139}:\\ \;\;\;\;a \cdot \frac{z}{a}\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;a \cdot \frac{t}{x + t}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ t_2 := \mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+281}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* z (+ x y)) (* (+ y t) a)) (* y b)) (+ y (+ x t))))
        (t_2 (fma (+ y t) (/ a (+ t (+ x y))) (- z b))))
   (if (<= t_1 (- INFINITY)) t_2 (if (<= t_1 2e+281) t_1 t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	double t_2 = fma((y + t), (a / (t + (x + y))), (z - b));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_1 <= 2e+281) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(Float64(y + t) * a)) - Float64(y * b)) / Float64(y + Float64(x + t)))
	t_2 = fma(Float64(y + t), Float64(a / Float64(t + Float64(x + y))), Float64(z - b))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_1 <= 2e+281)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(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[(N[(y + t), $MachinePrecision] * N[(a / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], t$95$2, If[LessEqual[t$95$1, 2e+281], t$95$1, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 2.0000000000000001e281 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 6.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 41.6%

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

   6. Taylor expanded in y around inf

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

      1. lower--.f64 88.0

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

   8. Applied rewrites 88.0%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e281

   1. Initial program 99.5%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 2 \cdot 10^{+281}:\\ \;\;\;\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)}\\ t_2 := \left(z + a\right) - b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-145}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-29}:\\ \;\;\;\;a \cdot \frac{t}{x + t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* z (+ x y)) (* (+ y t) a)) (* y b)) (+ y (+ x t))))
        (t_2 (- (+ z a) b)))
   (if (<= t_1 -1e-145) t_2 (if (<= t_1 5e-29) (* a (/ t (+ x t))) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	double t_2 = (z + a) - b;
	double tmp;
	if (t_1 <= -1e-145) {
		tmp = t_2;
	} else if (t_1 <= 5e-29) {
		tmp = a * (t / (x + t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t))
    t_2 = (z + a) - b
    if (t_1 <= (-1d-145)) then
        tmp = t_2
    else if (t_1 <= 5d-29) then
        tmp = a * (t / (x + t))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	double t_2 = (z + a) - b;
	double tmp;
	if (t_1 <= -1e-145) {
		tmp = t_2;
	} else if (t_1 <= 5e-29) {
		tmp = a * (t / (x + t));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t))
	t_2 = (z + a) - b
	tmp = 0
	if t_1 <= -1e-145:
		tmp = t_2
	elif t_1 <= 5e-29:
		tmp = a * (t / (x + t))
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(Float64(y + t) * a)) - Float64(y * b)) / Float64(y + Float64(x + t)))
	t_2 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (t_1 <= -1e-145)
		tmp = t_2;
	elseif (t_1 <= 5e-29)
		tmp = Float64(a * Float64(t / Float64(x + t)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (((z * (x + y)) + ((y + t) * a)) - (y * b)) / (y + (x + t));
	t_2 = (z + a) - b;
	tmp = 0.0;
	if (t_1 <= -1e-145)
		tmp = t_2;
	elseif (t_1 <= 5e-29)
		tmp = a * (t / (x + t));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(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[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-145], t$95$2, If[LessEqual[t$95$1, 5e-29], N[(a * N[(t / N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.99999999999999915e-146 or 4.99999999999999986e-29 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 55.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 63.5

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

   5. Applied rewrites 63.5%

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

   if -9.99999999999999915e-146 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999986e-29

   1. Initial program 98.8%

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

   3. Taylor expanded in 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 91.5%

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

   6. Taylor expanded in a around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-+.f64 70.3

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

   8. Applied rewrites 70.3%

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

   9. Taylor expanded in y around 0

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

       1. associate-/l* N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 70.3

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

   11. Applied rewrites 70.3%

       \[\leadsto \color{blue}{a \cdot \frac{t}{x + t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq -1 \cdot 10^{-145}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + \left(y + t\right) \cdot a\right) - y \cdot b}{y + \left(x + t\right)} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;a \cdot \frac{t}{x + t}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{if}\;x \leq -9.2 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+268}:\\ \;\;\;\;z - \frac{\mathsf{fma}\left(y + t, z, -\mathsf{fma}\left(a, t, y \cdot \left(\left(z + a\right) - b\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (+ y t) (/ a (+ t (+ x y))) (- z b))))
   (if (<= x -9.2e+109)
     t_1
     (if (<= x 1.18e-57)
       (fma y (/ (- z b) (+ y t)) a)
       (if (<= x 3.7e+186)
         t_1
         (if (<= x 6.6e+268)
           (- z (/ (fma (+ y t) z (- (fma a t (* y (- (+ z a) b))))) x))
           (fma a (/ y (+ x y)) z)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y + t), (a / (t + (x + y))), (z - b));
	double tmp;
	if (x <= -9.2e+109) {
		tmp = t_1;
	} else if (x <= 1.18e-57) {
		tmp = fma(y, ((z - b) / (y + t)), a);
	} else if (x <= 3.7e+186) {
		tmp = t_1;
	} else if (x <= 6.6e+268) {
		tmp = z - (fma((y + t), z, -fma(a, t, (y * ((z + a) - b)))) / x);
	} else {
		tmp = fma(a, (y / (x + y)), z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(y + t), Float64(a / Float64(t + Float64(x + y))), Float64(z - b))
	tmp = 0.0
	if (x <= -9.2e+109)
		tmp = t_1;
	elseif (x <= 1.18e-57)
		tmp = fma(y, Float64(Float64(z - b) / Float64(y + t)), a);
	elseif (x <= 3.7e+186)
		tmp = t_1;
	elseif (x <= 6.6e+268)
		tmp = Float64(z - Float64(fma(Float64(y + t), z, Float64(-fma(a, t, Float64(y * Float64(Float64(z + a) - b))))) / x));
	else
		tmp = fma(a, Float64(y / Float64(x + y)), z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y + t), $MachinePrecision] * N[(a / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.2e+109], t$95$1, If[LessEqual[x, 1.18e-57], N[(y * N[(N[(z - b), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[x, 3.7e+186], t$95$1, If[LessEqual[x, 6.6e+268], N[(z - N[(N[(N[(y + t), $MachinePrecision] * z + (-N[(a * t + N[(y * N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(a * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.20000000000000042e109 or 1.18e-57 < x < 3.7e186

   1. Initial program 55.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 67.9%

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

   6. Taylor expanded in y around inf

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

      1. lower--.f64 71.9

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

   8. Applied rewrites 71.9%

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

   if -9.20000000000000042e109 < x < 1.18e-57

   1. Initial program 65.5%

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

   3. Taylor expanded in 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 68.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 85.4

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

   8. Applied rewrites 85.4%

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

   if 3.7e186 < x < 6.6000000000000002e268

   1. Initial program 72.3%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 72.8%

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

   if 6.6000000000000002e268 < x

   1. Initial program 31.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 38.9%

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

   6. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. neg-mul-1 N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 100.0

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

   11. Applied rewrites 100.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{elif}\;x \leq 1.18 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+186}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+268}:\\ \;\;\;\;z - \frac{\mathsf{fma}\left(y + t, z, -\mathsf{fma}\left(a, t, y \cdot \left(\left(z + a\right) - b\right)\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x + y\right)\\ \mathbf{if}\;y \leq -2.25 \cdot 10^{-38}:\\ \;\;\;\;\left(a + \left(z - b\right)\right) \cdot \frac{y}{t\_1}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-79}:\\ \;\;\;\;a \cdot \left(\frac{z}{a} + \frac{y + t}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ t (+ x y))))
   (if (<= y -2.25e-38)
     (* (+ a (- z b)) (/ y t_1))
     (if (<= y 1.55e-79)
       (* a (+ (/ z a) (/ (+ y t) t_1)))
       (fma y (/ (- z b) (+ y t)) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t + (x + y);
	double tmp;
	if (y <= -2.25e-38) {
		tmp = (a + (z - b)) * (y / t_1);
	} else if (y <= 1.55e-79) {
		tmp = a * ((z / a) + ((y + t) / t_1));
	} else {
		tmp = fma(y, ((z - b) / (y + t)), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t + Float64(x + y))
	tmp = 0.0
	if (y <= -2.25e-38)
		tmp = Float64(Float64(a + Float64(z - b)) * Float64(y / t_1));
	elseif (y <= 1.55e-79)
		tmp = Float64(a * Float64(Float64(z / a) + Float64(Float64(y + t) / t_1)));
	else
		tmp = fma(y, Float64(Float64(z - b) / Float64(y + t)), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.25e-38], N[(N[(a + N[(z - b), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e-79], N[(a * N[(N[(z / a), $MachinePrecision] + N[(N[(y + t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - b), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.25000000000000004e-38

   1. Initial program 51.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 44.2

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

   5. Applied rewrites 44.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-commutative N/A

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

      4. +-commutative N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate--l+ N/A

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

      14. lift--.f64 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-/.f64 81.9

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

   7. Applied rewrites 81.9%

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

   if -2.25000000000000004e-38 < y < 1.55e-79

   1. Initial program 77.6%

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

   3. Taylor expanded in a around -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 78.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-/.f64 74.1

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

   8. Applied rewrites 74.1%

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

   if 1.55e-79 < y

   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 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 58.6%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 86.4

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

   8. Applied rewrites 86.4%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-38}:\\ \;\;\;\;\left(a + \left(z - b\right)\right) \cdot \frac{y}{t + \left(x + y\right)}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-79}:\\ \;\;\;\;a \cdot \left(\frac{z}{a} + \frac{y + t}{t + \left(x + y\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z b) (+ y t)) a)))
   (if (<= t -1.15e-5)
     t_1
     (if (<= t 7.4e+143) (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 <= -1.15e-5) {
		tmp = t_1;
	} else if (t <= 7.4e+143) {
		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 <= -1.15e-5)
		tmp = t_1;
	elseif (t <= 7.4e+143)
		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, -1.15e-5], t$95$1, If[LessEqual[t, 7.4e+143], 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 < -1.15e-5 or 7.4000000000000003e143 < t

   1. Initial program 51.2%

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

   3. Taylor expanded in 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 63.1%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 85.1

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

   8. Applied rewrites 85.1%

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

   if -1.15e-5 < t < 7.4000000000000003e143

   1. Initial program 66.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 71.0%

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

   6. Taylor expanded in y around inf

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

      1. lower--.f64 74.0

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

   8. Applied rewrites 74.0%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{elif}\;t \leq 7.4 \cdot 10^{+143}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{t + \left(x + y\right)}, z - b\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a (/ y (+ x y)) z)))
   (if (<= x -1.4e+110)
     t_1
     (if (<= x 2.05e+102) (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 = fma(a, (y / (x + y)), z);
	double tmp;
	if (x <= -1.4e+110) {
		tmp = t_1;
	} else if (x <= 2.05e+102) {
		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 = fma(a, Float64(y / Float64(x + y)), z)
	tmp = 0.0
	if (x <= -1.4e+110)
		tmp = t_1;
	elseif (x <= 2.05e+102)
		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[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]}, If[LessEqual[x, -1.4e+110], t$95$1, If[LessEqual[x, 2.05e+102], 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 x < -1.39999999999999993e110 or 2.05e102 < x

   1. Initial program 44.1%

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

   3. Taylor expanded in 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 59.2%

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

   6. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. neg-mul-1 N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. +-commutative N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 83.4

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

   8. Applied rewrites 83.4%

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

   9. Taylor expanded in t around 0

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 64.3

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

   11. Applied rewrites 64.3%

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

   if -1.39999999999999993e110 < x < 2.05e102

   1. Initial program 67.5%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 71.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 79.9

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

   8. Applied rewrites 79.9%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+102}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-73}:\\ \;\;\;\;z + a\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+95}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;z + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -5e-73) (+ z a) (if (<= z 5.4e+95) (- a b) (+ z a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -5e-73) {
		tmp = z + a;
	} else if (z <= 5.4e+95) {
		tmp = a - b;
	} else {
		tmp = z + a;
	}
	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 (z <= (-5d-73)) then
        tmp = z + a
    else if (z <= 5.4d+95) then
        tmp = a - b
    else
        tmp = z + a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -5e-73) {
		tmp = z + a;
	} else if (z <= 5.4e+95) {
		tmp = a - b;
	} else {
		tmp = z + a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -5e-73:
		tmp = z + a
	elif z <= 5.4e+95:
		tmp = a - b
	else:
		tmp = z + a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -5e-73)
		tmp = Float64(z + a);
	elseif (z <= 5.4e+95)
		tmp = Float64(a - b);
	else
		tmp = Float64(z + a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -5e-73)
		tmp = z + a;
	elseif (z <= 5.4e+95)
		tmp = a - b;
	else
		tmp = z + a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -5e-73], N[(z + a), $MachinePrecision], If[LessEqual[z, 5.4e+95], N[(a - b), $MachinePrecision], N[(z + a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.9999999999999998e-73 or 5.4e95 < z

   1. Initial program 48.6%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 62.5

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

   5. Applied rewrites 62.5%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 58.8

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

   8. Applied rewrites 58.8%

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

   if -4.9999999999999998e-73 < z < 5.4e95

   1. Initial program 71.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in z around 0

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

      1. lower--.f64 53.9

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

   8. Applied rewrites 53.9%

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

Alternative 11: 57.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{+186}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -9.2e+186) a (- (+ z a) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -9.2e+186) {
		tmp = a;
	} else {
		tmp = (z + 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 (t <= (-9.2d+186)) then
        tmp = a
    else
        tmp = (z + 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 (t <= -9.2e+186) {
		tmp = a;
	} else {
		tmp = (z + a) - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -9.2e+186:
		tmp = a
	else:
		tmp = (z + a) - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -9.2e+186)
		tmp = a;
	else
		tmp = Float64(Float64(z + a) - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -9.2e+186)
		tmp = a;
	else
		tmp = (z + a) - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -9.2e+186], a, N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9.20000000000000054e186

   1. Initial program 59.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 16.2

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

   5. Applied rewrites 16.2%

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

   6. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-+.f64 3.8

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

   8. Applied rewrites 3.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 67.9%

       \[\leadsto a \cdot \color{blue}{1} \]

   if -9.20000000000000054e186 < t

   1. Initial program 61.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 61.0

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

   5. Applied rewrites 61.0%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{+186}:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;z - b\\ \mathbf{else}:\\ \;\;\;\;z + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= b -6.2e+18) (- z b) (+ z a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6.2e+18) {
		tmp = z - b;
	} else {
		tmp = z + a;
	}
	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 <= (-6.2d+18)) then
        tmp = z - b
    else
        tmp = z + a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -6.2e+18) {
		tmp = z - b;
	} else {
		tmp = z + a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -6.2e+18:
		tmp = z - b
	else:
		tmp = z + a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -6.2e+18)
		tmp = Float64(z - b);
	else
		tmp = Float64(z + a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -6.2e+18)
		tmp = z - b;
	else
		tmp = z + a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -6.2e+18], N[(z - b), $MachinePrecision], N[(z + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -6.2e18

   1. Initial program 60.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in a around 0

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

      1. lower--.f64 51.5

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

   8. Applied rewrites 51.5%

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

   if -6.2e18 < b

   1. Initial program 61.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 56.6

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

   8. Applied rewrites 56.6%

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

Alternative 13: 52.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+208}:\\ \;\;\;\;-b\\ \mathbf{else}:\\ \;\;\;\;z + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= b -3.2e+208) (- b) (+ z a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.2e+208) {
		tmp = -b;
	} else {
		tmp = z + a;
	}
	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 <= (-3.2d+208)) then
        tmp = -b
    else
        tmp = z + a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (b <= -3.2e+208) {
		tmp = -b;
	} else {
		tmp = z + a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if b <= -3.2e+208:
		tmp = -b
	else:
		tmp = z + a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (b <= -3.2e+208)
		tmp = Float64(-b);
	else
		tmp = Float64(z + a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (b <= -3.2e+208)
		tmp = -b;
	else
		tmp = z + a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[b, -3.2e+208], (-b), N[(z + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.2000000000000001e208

   1. Initial program 40.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 69.0

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

   5. Applied rewrites 69.0%

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

   6. Taylor expanded in b around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(b\right)} \]

      2. lower-neg.f64 62.1

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

   8. Applied rewrites 62.1%

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

   if -3.2000000000000001e208 < b

   1. Initial program 62.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 57.2

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 52.7

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

   8. Applied rewrites 52.7%

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

Alternative 14: 12.9% 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 61.1%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 58.1

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

5. Applied rewrites 58.1%

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

6. Taylor expanded in b around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(b\right)} \]

   2. lower-neg.f64 15.1

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

8. Applied rewrites 15.1%

   \[\leadsto \color{blue}{-b} \]
9. Add Preprocessing

Developer Target 1: 81.9% 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 2024219 
(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)))