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

Percentage Accurate: 59.9% → 89.7%

Time: 14.4s

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: 59.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(y + t\right)\\ t_2 := x + \left(y + t\right)\\ t_3 := \frac{\left(\left(x + y\right) \cdot z + t\_1\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x + y, z, t\_1\right)}{t\_2} - \frac{y \cdot b}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{\left(x + y\right) + t}, z - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (+ y t)))
        (t_2 (+ x (+ y t)))
        (t_3 (/ (- (+ (* (+ x y) z) t_1) (* y b)) (+ y (+ x t)))))
   (if (<= t_3 (- INFINITY))
     (fma y (/ (- z b) (+ y t)) a)
     (if (<= t_3 2e+259)
       (- (/ (fma (+ x y) z t_1) t_2) (/ (* y b) t_2))
       (fma (+ y t) (/ a (+ (+ x y) t)) (- z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (y + t);
	double t_2 = x + (y + t);
	double t_3 = ((((x + y) * z) + t_1) - (y * b)) / (y + (x + t));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = fma(y, ((z - b) / (y + t)), a);
	} else if (t_3 <= 2e+259) {
		tmp = (fma((x + y), z, t_1) / t_2) - ((y * b) / t_2);
	} else {
		tmp = fma((y + t), (a / ((x + y) + t)), (z - b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(y + t))
	t_2 = Float64(x + Float64(y + t))
	t_3 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + t_1) - Float64(y * b)) / Float64(y + Float64(x + t)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = fma(y, Float64(Float64(z - b) / Float64(y + t)), a);
	elseif (t_3 <= 2e+259)
		tmp = Float64(Float64(fma(Float64(x + y), z, t_1) / t_2) - Float64(Float64(y * b) / t_2));
	else
		tmp = fma(Float64(y + t), Float64(a / Float64(Float64(x + y) + t)), Float64(z - b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(y + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(y * N[(N[(z - b), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[t$95$3, 2e+259], N[(N[(N[(N[(x + y), $MachinePrecision] * z + t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision] - N[(N[(y * b), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(y + t), $MachinePrecision] * N[(a / N[(N[(x + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + 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 6.8%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \frac{y \cdot \left(z - b\right)}{t + y}} \]
   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 88.2

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

   8. Simplified 88.2%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2e259

   1. Initial program 98.4%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. div-sub N/A

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

      10. lower--.f64 N/A

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

   4. Applied egg-rr 98.4%

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

   if 2e259 < (/.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. Simplified 31.2%

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

   6. Taylor expanded in 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 80.6

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

   8. Simplified 80.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x + y, z, a \cdot \left(y + t\right)\right)}{x + \left(y + t\right)} - \frac{y \cdot b}{x + \left(y + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{\left(x + y\right) + t}, z - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(x + y\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+259}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{\left(x + y\right) + t}, z - b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ x y) z) (* a (+ y t))) (* y b)) (+ y (+ x t)))))
   (if (<= t_1 (- INFINITY))
     (fma y (/ (- z b) (+ y t)) a)
     (if (<= t_1 2e+259) t_1 (fma (+ y t) (/ a (+ (+ x y) t)) (- z b))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((x + y) * z) + (a * (y + t))) - (y * b)) / (y + (x + t));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(y, ((z - b) / (y + t)), a);
	} else if (t_1 <= 2e+259) {
		tmp = t_1;
	} else {
		tmp = fma((y + t), (a / ((x + y) + t)), (z - b));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(a * Float64(y + t))) - Float64(y * b)) / Float64(y + Float64(x + t)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(y, Float64(Float64(z - b) / Float64(y + t)), a);
	elseif (t_1 <= 2e+259)
		tmp = t_1;
	else
		tmp = fma(Float64(y + t), Float64(a / Float64(Float64(x + y) + t)), Float64(z - b));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(N[(z - b), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[t$95$1, 2e+259], t$95$1, N[(N[(y + t), $MachinePrecision] * N[(a / N[(N[(x + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision] + 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 6.8%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \frac{y \cdot \left(z - b\right)}{t + y}} \]
   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 88.2

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

   8. Simplified 88.2%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2e259

   1. Initial program 98.4%

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

   if 2e259 < (/.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. Simplified 31.2%

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

   6. Taylor expanded in 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 80.6

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

   8. Simplified 80.6%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x + y\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{elif}\;\frac{\left(\left(x + y\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 2 \cdot 10^{+259}:\\ \;\;\;\;\frac{\left(\left(x + y\right) \cdot z + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y + t, \frac{a}{\left(x + y\right) + t}, z - b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{y + t}, -b, a\right)\\ \mathbf{if}\;b \leq -8.9 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-260}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{y + t}, a\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+58}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (/ y (+ y t)) (- b) a)))
   (if (<= b -8.9e+52)
     t_1
     (if (<= b -3e-260)
       (fma y (/ z (+ y t)) a)
       (if (<= b 1.3e+58) (+ z a) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((y / (y + t)), -b, a);
	double tmp;
	if (b <= -8.9e+52) {
		tmp = t_1;
	} else if (b <= -3e-260) {
		tmp = fma(y, (z / (y + t)), a);
	} else if (b <= 1.3e+58) {
		tmp = z + a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(y / Float64(y + t)), Float64(-b), a)
	tmp = 0.0
	if (b <= -8.9e+52)
		tmp = t_1;
	elseif (b <= -3e-260)
		tmp = fma(y, Float64(z / Float64(y + t)), a);
	elseif (b <= 1.3e+58)
		tmp = Float64(z + a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y / N[(y + t), $MachinePrecision]), $MachinePrecision] * (-b) + a), $MachinePrecision]}, If[LessEqual[b, -8.9e+52], t$95$1, If[LessEqual[b, -3e-260], N[(y * N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[b, 1.3e+58], N[(z + a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.89999999999999994e52 or 1.29999999999999994e58 < b

   1. Initial program 57.4%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Simplified 63.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \frac{y \cdot \left(z - b\right)}{t + y}} \]
   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 62.5

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

   8. Simplified 62.5%

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

   9. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. *-commutative N/A

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

       4. associate-*l/ N/A

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

       5. distribute-lft-neg-out N/A

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

       6. mul-1-neg N/A

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

       7. *-commutative N/A

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

       8. associate-*l* N/A

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

       9. mul-1-neg N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. +-commutative N/A

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

       13. lower-+.f64 N/A

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

       14. lower-neg.f64 56.6

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

   11. Simplified 56.6%

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

   if -8.89999999999999994e52 < b < -3.0000000000000001e-260

   1. Initial program 58.8%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Simplified 76.5%

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

   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 82.2

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

   8. Simplified 82.2%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 76.1

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

   11. Simplified 76.1%

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

   if -3.0000000000000001e-260 < b < 1.29999999999999994e58

   1. Initial program 59.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 70.2

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

   5. Simplified 70.2%

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

   6. Taylor expanded in b around 0

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

      1. lower-+.f64 73.7

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

   8. Simplified 73.7%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.9 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + t}, -b, a\right)\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-260}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{y + t}, a\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+58}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{y + t}, -b, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 58.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, -\frac{b}{y + t}, a\right)\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{+27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-260}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{y + t}, a\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+58}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma y (- (/ b (+ y t))) a)))
   (if (<= b -6.5e+27)
     t_1
     (if (<= b -3e-260)
       (fma y (/ z (+ y t)) a)
       (if (<= b 1.3e+58) (+ z a) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma(y, -(b / (y + t)), a);
	double tmp;
	if (b <= -6.5e+27) {
		tmp = t_1;
	} else if (b <= -3e-260) {
		tmp = fma(y, (z / (y + t)), a);
	} else if (b <= 1.3e+58) {
		tmp = z + a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(y, Float64(-Float64(b / Float64(y + t))), a)
	tmp = 0.0
	if (b <= -6.5e+27)
		tmp = t_1;
	elseif (b <= -3e-260)
		tmp = fma(y, Float64(z / Float64(y + t)), a);
	elseif (b <= 1.3e+58)
		tmp = Float64(z + a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y * (-N[(b / N[(y + t), $MachinePrecision]), $MachinePrecision]) + a), $MachinePrecision]}, If[LessEqual[b, -6.5e+27], t$95$1, If[LessEqual[b, -3e-260], N[(y * N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[b, 1.3e+58], N[(z + a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.5000000000000005e27 or 1.29999999999999994e58 < b

   1. Initial program 57.8%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Simplified 64.3%

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

   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 62.6

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

   8. Simplified 62.6%

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

   9. Taylor expanded in z around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 55.7

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

   11. Simplified 55.7%

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

   if -6.5000000000000005e27 < b < -3.0000000000000001e-260

   1. Initial program 58.4%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Simplified 75.5%

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

   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 82.9

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

   8. Simplified 82.9%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 76.5

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

   11. Simplified 76.5%

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

   if -3.0000000000000001e-260 < b < 1.29999999999999994e58

   1. Initial program 59.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 70.2

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

   5. Simplified 70.2%

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

   6. Taylor expanded in b around 0

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

      1. lower-+.f64 73.7

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

   8. Simplified 73.7%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(y, -\frac{b}{y + t}, a\right)\\ \mathbf{elif}\;b \leq -3 \cdot 10^{-260}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{y + t}, a\right)\\ \mathbf{elif}\;b \leq 1.3 \cdot 10^{+58}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -\frac{b}{y + t}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{y + t}{y + \left(x + t\right)}, z\right)\\ \mathbf{if}\;x \leq -6.6 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma a (/ (+ y t) (+ y (+ x t))) z)))
   (if (<= x -6.6e+61)
     t_1
     (if (<= x 2.1e+94) (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 + t) / (y + (x + t))), z);
	double tmp;
	if (x <= -6.6e+61) {
		tmp = t_1;
	} else if (x <= 2.1e+94) {
		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(Float64(y + t) / Float64(y + Float64(x + t))), z)
	tmp = 0.0
	if (x <= -6.6e+61)
		tmp = t_1;
	elseif (x <= 2.1e+94)
		tmp = fma(y, Float64(Float64(z - b) / Float64(y + t)), a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(N[(y + t), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision]}, If[LessEqual[x, -6.6e+61], t$95$1, If[LessEqual[x, 2.1e+94], 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 < -6.5999999999999995e61 or 2.09999999999999989e94 < x

   1. Initial program 53.5%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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. Simplified 60.2%

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

   6. Taylor expanded in 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 61.3

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

   8. Simplified 61.3%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 N/A

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

       7. associate-+r+ N/A

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

       8. +-commutative N/A

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

       9. lower-+.f64 N/A

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

       10. +-commutative N/A

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

       11. lower-+.f64 80.9

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

   11. Simplified 80.9%

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

   if -6.5999999999999995e61 < x < 2.09999999999999989e94

   1. Initial program 60.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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. Simplified 75.2%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{a + \frac{y \cdot \left(z - b\right)}{t + y}} \]
   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 83.3

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

   8. Simplified 83.3%

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

Alternative 6: 70.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+116}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -1.02e+116) z (if (<= x 3.3e+192) (fma y (/ (- z b) (+ y t)) a) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1.02e+116) {
		tmp = z;
	} else if (x <= 3.3e+192) {
		tmp = fma(y, ((z - b) / (y + t)), a);
	} else {
		tmp = z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -1.02e+116)
		tmp = z;
	elseif (x <= 3.3e+192)
		tmp = fma(y, Float64(Float64(z - b) / Float64(y + t)), a);
	else
		tmp = z;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -1.02e+116], z, If[LessEqual[x, 3.3e+192], N[(y * N[(N[(z - b), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0199999999999999e116 or 3.3000000000000001e192 < x

   1. Initial program 50.0%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

   5. Simplified 73.1%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 62.1%

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

   if -1.0199999999999999e116 < x < 3.3000000000000001e192

   1. Initial program 60.4%

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

   3. Taylor expanded in b around 0

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

      1. mul-1-neg N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. sub-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

   5. Simplified 75.1%

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

   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 79.0

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

   8. Simplified 79.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+116}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+192}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{y + t}, a\right)\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + a\right) - b\\ \mathbf{if}\;y \leq -8.3 \cdot 10^{-128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-238}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t}, a\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ z a) b)))
   (if (<= y -8.3e-128) t_1 (if (<= y 8e-238) (fma z (/ x t) a) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + a) - b;
	double tmp;
	if (y <= -8.3e-128) {
		tmp = t_1;
	} else if (y <= 8e-238) {
		tmp = fma(z, (x / t), a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (y <= -8.3e-128)
		tmp = t_1;
	elseif (y <= 8e-238)
		tmp = fma(z, Float64(x / t), a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -8.3e-128], t$95$1, If[LessEqual[y, 8e-238], N[(z * N[(x / t), $MachinePrecision] + a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.30000000000000015e-128 or 7.9999999999999999e-238 < y

   1. Initial program 55.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 63.4

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

   5. Simplified 63.4%

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

   if -8.30000000000000015e-128 < y < 7.9999999999999999e-238

   1. Initial program 69.2%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 58.0%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 61.3

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

   8. Simplified 61.3%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 59.7

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

   11. Simplified 59.7%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.3 \cdot 10^{-128}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-238}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{x}{t}, a\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 6.3 \cdot 10^{+53}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{y + t}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 6.3e+53) (- (+ z a) b) (fma y (/ z (+ y t)) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 6.3e+53) {
		tmp = (z + a) - b;
	} else {
		tmp = fma(y, (z / (y + t)), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 6.3e+53)
		tmp = Float64(Float64(z + a) - b);
	else
		tmp = fma(y, Float64(z / Float64(y + t)), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 6.3e+53], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], N[(y * N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 6.29999999999999974e53

   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 57.8

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

   5. Simplified 57.8%

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

   if 6.29999999999999974e53 < t

   1. Initial program 47.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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. Simplified 60.4%

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

   6. Taylor expanded in 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 86.1

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

   8. Simplified 86.1%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lower-+.f64 78.5

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

   11. Simplified 78.5%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.3 \cdot 10^{+53}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{y + t}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{+77}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t 7e+77) (- (+ z a) b) (fma y (/ z t) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 7e+77) {
		tmp = (z + a) - b;
	} else {
		tmp = fma(y, (z / t), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 7e+77)
		tmp = Float64(Float64(z + a) - b);
	else
		tmp = fma(y, Float64(z / t), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 7e+77], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], N[(y * N[(z / t), $MachinePrecision] + a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.0000000000000003e77

   1. Initial program 63.0%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 58.1

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

   5. Simplified 58.1%

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

   if 7.0000000000000003e77 < t

   1. Initial program 45.9%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Simplified 51.9%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 71.7

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

   8. Simplified 71.7%

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

   9. Taylor expanded in b around 0

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

       1. +-commutative N/A

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

       2. associate-/l* N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-/.f64 66.7

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

   11. Simplified 66.7%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{+77}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+59}:\\ \;\;\;\;z + a\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+173}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -2.6e+59) (+ z a) (if (<= x 1.1e+173) (- (+ z a) b) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.6e+59) {
		tmp = z + a;
	} else if (x <= 1.1e+173) {
		tmp = (z + a) - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-2.6d+59)) then
        tmp = z + a
    else if (x <= 1.1d+173) then
        tmp = (z + a) - b
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.6e+59) {
		tmp = z + a;
	} else if (x <= 1.1e+173) {
		tmp = (z + a) - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -2.6e+59:
		tmp = z + a
	elif x <= 1.1e+173:
		tmp = (z + a) - b
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -2.6e+59)
		tmp = Float64(z + a);
	elseif (x <= 1.1e+173)
		tmp = Float64(Float64(z + a) - b);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -2.6e+59)
		tmp = z + a;
	elseif (x <= 1.1e+173)
		tmp = (z + a) - b;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -2.6e+59], N[(z + a), $MachinePrecision], If[LessEqual[x, 1.1e+173], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], z]]

Derivation

1. Split input into 3 regimes

2. if x < -2.59999999999999999e59

   1. Initial program 54.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 39.0

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

   5. Simplified 39.0%

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

   6. Taylor expanded in b around 0

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

      1. lower-+.f64 54.8

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

   8. Simplified 54.8%

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

   if -2.59999999999999999e59 < x < 1.1e173

   1. Initial program 59.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 62.6

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

   5. Simplified 62.6%

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

   if 1.1e173 < x

   1. Initial program 54.0%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

   5. Simplified 68.7%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 69.7%

      \[\leadsto \left(-z\right) \cdot \color{blue}{-1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+59}:\\ \;\;\;\;z + a\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+173}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-140}:\\ \;\;\;\;z + a\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-38}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;z + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2.1e-140) (+ z a) (if (<= z 5.2e-38) (- a b) (+ z a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2.1e-140) {
		tmp = z + a;
	} else if (z <= 5.2e-38) {
		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 <= (-2.1d-140)) then
        tmp = z + a
    else if (z <= 5.2d-38) 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 <= -2.1e-140) {
		tmp = z + a;
	} else if (z <= 5.2e-38) {
		tmp = a - b;
	} else {
		tmp = z + a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -2.1e-140:
		tmp = z + a
	elif z <= 5.2e-38:
		tmp = a - b
	else:
		tmp = z + a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2.1e-140)
		tmp = Float64(z + a);
	elseif (z <= 5.2e-38)
		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 <= -2.1e-140)
		tmp = z + a;
	elseif (z <= 5.2e-38)
		tmp = a - b;
	else
		tmp = z + a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2.1e-140], N[(z + a), $MachinePrecision], If[LessEqual[z, 5.2e-38], N[(a - b), $MachinePrecision], N[(z + a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.10000000000000017e-140 or 5.20000000000000022e-38 < z

   1. Initial program 57.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 55.7

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

   5. Simplified 55.7%

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

   6. Taylor expanded in b around 0

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

      1. lower-+.f64 58.2

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

   8. Simplified 58.2%

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

   if -2.10000000000000017e-140 < z < 5.20000000000000022e-38

   1. Initial program 60.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 60.4

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

   5. Simplified 60.4%

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

   6. Taylor expanded in z around 0

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

      1. lower--.f64 60.0

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

   8. Simplified 60.0%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-140}:\\ \;\;\;\;z + a\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-38}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;z + a\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+59}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -2.6e+59) (+ z a) (- (+ z a) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -2.6e+59) {
		tmp = z + 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 (x <= (-2.6d+59)) then
        tmp = z + 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 (x <= -2.6e+59) {
		tmp = z + a;
	} else {
		tmp = (z + a) - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -2.6e+59:
		tmp = z + a
	else:
		tmp = (z + a) - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -2.6e+59)
		tmp = Float64(z + 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 (x <= -2.6e+59)
		tmp = z + a;
	else
		tmp = (z + a) - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -2.6e+59], N[(z + a), $MachinePrecision], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.59999999999999999e59

   1. Initial program 54.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 39.0

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

   5. Simplified 39.0%

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

   6. Taylor expanded in b around 0

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

      1. lower-+.f64 54.8

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

   8. Simplified 54.8%

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

   if -2.59999999999999999e59 < x

   1. Initial program 59.4%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 60.9

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

   5. Simplified 60.9%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+59}:\\ \;\;\;\;z + a\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.3% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

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 = z + a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.6%

   \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
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 57.3

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

5. Simplified 57.3%

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

6. Taylor expanded in b around 0

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

   1. lower-+.f64 52.9

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

8. Simplified 52.9%

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

9. Final simplification 52.9%

   \[\leadsto z + a \]
10. Add Preprocessing

Alternative 14: 13.1% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.6%

   \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
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 57.3

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

5. Simplified 57.3%

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

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 12.3

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

8. Simplified 12.3%

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

Developer Target 1: 82.5% 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 2024208 
(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)))