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

Percentage Accurate: 60.8% → 88.2%

Time: 9.2s

Alternatives: 11

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ t_2 := \left(t + x\right) + y\\ t_3 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_2}\\ t_4 := t + \left(y + x\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+270}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, y, \mathsf{fma}\left(x, z, a \cdot t\right)\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{t}{t\_4} + \mathsf{fma}\left(\frac{z}{t\_4}, \frac{y + x}{a}, \frac{y}{t\_4}\right)\right) - \frac{y}{a} \cdot \frac{b}{t\_4}\right) \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b))
        (t_2 (+ (+ t x) y))
        (t_3 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) t_2))
        (t_4 (+ t (+ y x))))
   (if (<= t_3 -5e+270)
     t_1
     (if (<= t_3 4e+303)
       (/ (fma t_1 y (fma x z (* a t))) t_2)
       (*
        (-
         (+ (/ t t_4) (fma (/ z t_4) (/ (+ y x) a) (/ y t_4)))
         (* (/ y a) (/ b t_4)))
        a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double t_2 = (t + x) + y;
	double t_3 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / t_2;
	double t_4 = t + (y + x);
	double tmp;
	if (t_3 <= -5e+270) {
		tmp = t_1;
	} else if (t_3 <= 4e+303) {
		tmp = fma(t_1, y, fma(x, z, (a * t))) / t_2;
	} else {
		tmp = (((t / t_4) + fma((z / t_4), ((y + x) / a), (y / t_4))) - ((y / a) * (b / t_4))) * a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	t_2 = Float64(Float64(t + x) + y)
	t_3 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / t_2)
	t_4 = Float64(t + Float64(y + x))
	tmp = 0.0
	if (t_3 <= -5e+270)
		tmp = t_1;
	elseif (t_3 <= 4e+303)
		tmp = Float64(fma(t_1, y, fma(x, z, Float64(a * t))) / t_2);
	else
		tmp = Float64(Float64(Float64(Float64(t / t_4) + fma(Float64(z / t_4), Float64(Float64(y + x) / a), Float64(y / t_4))) - Float64(Float64(y / a) * Float64(b / t_4))) * a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+270], t$95$1, If[LessEqual[t$95$3, 4e+303], N[(N[(t$95$1 * y + N[(x * z + N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[(N[(t / t$95$4), $MachinePrecision] + N[(N[(z / t$95$4), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / a), $MachinePrecision] + N[(y / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y / a), $MachinePrecision] * N[(b / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $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)) < -4.99999999999999976e270

   1. Initial program 8.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 88.7

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

   5. Applied rewrites 88.7%

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

   if -4.99999999999999976e270 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4e303

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 4e303 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 4.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   8. Applied rewrites 77.5%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a + z\right) - b, y, \mathsf{fma}\left(x, z, a \cdot t\right)\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{t}{t + \left(y + x\right)} + \mathsf{fma}\left(\frac{z}{t + \left(y + x\right)}, \frac{y + x}{a}, \frac{y}{t + \left(y + x\right)}\right)\right) - \frac{y}{a} \cdot \frac{b}{t + \left(y + x\right)}\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -6 \cdot 10^{+258}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a, t\_3 \cdot y\right)}{t + y}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, z, a \cdot t\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, z, \left(z - b\right) \cdot y\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ t x) y))
        (t_2 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) t_1))
        (t_3 (- (+ a z) b)))
   (if (<= t_2 -6e+258)
     t_3
     (if (<= t_2 -1e+126)
       (/ (fma t a (* t_3 y)) (+ t y))
       (if (<= t_2 5e-111)
         (/ (fma x z (* a t)) (+ t x))
         (if (<= t_2 2e+159) (/ (fma x z (* (- z b) y)) t_1) t_3))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + x) + y;
	double t_2 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / t_1;
	double t_3 = (a + z) - b;
	double tmp;
	if (t_2 <= -6e+258) {
		tmp = t_3;
	} else if (t_2 <= -1e+126) {
		tmp = fma(t, a, (t_3 * y)) / (t + y);
	} else if (t_2 <= 5e-111) {
		tmp = fma(x, z, (a * t)) / (t + x);
	} else if (t_2 <= 2e+159) {
		tmp = fma(x, z, ((z - b) * y)) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + x) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / t_1)
	t_3 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_2 <= -6e+258)
		tmp = t_3;
	elseif (t_2 <= -1e+126)
		tmp = Float64(fma(t, a, Float64(t_3 * y)) / Float64(t + y));
	elseif (t_2 <= 5e-111)
		tmp = Float64(fma(x, z, Float64(a * t)) / Float64(t + x));
	elseif (t_2 <= 2e+159)
		tmp = Float64(fma(x, z, Float64(Float64(z - b) * y)) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$2, -6e+258], t$95$3, If[LessEqual[t$95$2, -1e+126], N[(N[(t * a + N[(t$95$3 * y), $MachinePrecision]), $MachinePrecision] / N[(t + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-111], N[(N[(x * z + N[(a * t), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e+159], N[(N[(x * z + N[(N[(z - b), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -5.9999999999999999e258 or 1.9999999999999999e159 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 16.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 76.5

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

   5. Applied rewrites 76.5%

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

   if -5.9999999999999999e258 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.99999999999999925e125

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -9.99999999999999925e125 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000003e-111

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if 5.0000000000000003e-111 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.9999999999999999e159

   1. Initial program 99.7%

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

   3. Taylor expanded in a around 0

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

      1. distribute-rgt-in N/A

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

      2. associate--l+ N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 73.9

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

   5. Applied rewrites 73.9%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -6 \cdot 10^{+258}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -1 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a, \left(\left(a + z\right) - b\right) \cdot y\right)}{t + y}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 5 \cdot 10^{-111}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, z, a \cdot t\right)}{t + x}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, z, \left(z - b\right) \cdot y\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := t + \left(y + x\right)\\ t_3 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_4 := \left(a + z\right) - b\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+270}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_4, y, \mathsf{fma}\left(x, z, a \cdot t\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y + x, \frac{z}{t\_2}, \frac{a}{t\_2} \cdot \left(t + y\right)\right)}{b} - \frac{y}{t\_2}\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ t x) y))
        (t_2 (+ t (+ y x)))
        (t_3 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) t_1))
        (t_4 (- (+ a z) b)))
   (if (<= t_3 -5e+270)
     t_4
     (if (<= t_3 4e+303)
       (/ (fma t_4 y (fma x z (* a t))) t_1)
       (*
        (- (/ (fma (+ y x) (/ z t_2) (* (/ a t_2) (+ t y))) b) (/ y t_2))
        b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + x) + y;
	double t_2 = t + (y + x);
	double t_3 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / t_1;
	double t_4 = (a + z) - b;
	double tmp;
	if (t_3 <= -5e+270) {
		tmp = t_4;
	} else if (t_3 <= 4e+303) {
		tmp = fma(t_4, y, fma(x, z, (a * t))) / t_1;
	} else {
		tmp = ((fma((y + x), (z / t_2), ((a / t_2) * (t + y))) / b) - (y / t_2)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + x) + y)
	t_2 = Float64(t + Float64(y + x))
	t_3 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / t_1)
	t_4 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_3 <= -5e+270)
		tmp = t_4;
	elseif (t_3 <= 4e+303)
		tmp = Float64(fma(t_4, y, fma(x, z, Float64(a * t))) / t_1);
	else
		tmp = Float64(Float64(Float64(fma(Float64(y + x), Float64(z / t_2), Float64(Float64(a / t_2) * Float64(t + y))) / b) - Float64(y / t_2)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(t + N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+270], t$95$4, If[LessEqual[t$95$3, 4e+303], N[(N[(t$95$4 * y + N[(x * z + N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[(N[(N[(y + x), $MachinePrecision] * N[(z / t$95$2), $MachinePrecision] + N[(N[(a / t$95$2), $MachinePrecision] * N[(t + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] - N[(y / t$95$2), $MachinePrecision]), $MachinePrecision] * b), $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)) < -4.99999999999999976e270

   1. Initial program 8.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 88.7

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

   5. Applied rewrites 88.7%

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

   if -4.99999999999999976e270 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4e303

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if 4e303 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 4.2%

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

   3. Taylor expanded in b around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

   5. Applied rewrites 72.0%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 4 \cdot 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a + z\right) - b, y, \mathsf{fma}\left(x, z, a \cdot t\right)\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\mathsf{fma}\left(y + x, \frac{z}{t + \left(y + x\right)}, \frac{a}{t + \left(y + x\right)} \cdot \left(t + y\right)\right)}{b} - \frac{y}{t + \left(y + x\right)}\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;t\_1 \leq -6 \cdot 10^{+258}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a, t\_2 \cdot y\right)}{t + y}\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, z, a \cdot t\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) (+ (+ t x) y)))
        (t_2 (- (+ a z) b)))
   (if (<= t_1 -6e+258)
     t_2
     (if (<= t_1 -1e+126)
       (/ (fma t a (* t_2 y)) (+ t y))
       (if (<= t_1 1.5e+214) (/ (fma x z (* a t)) (+ t x)) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / ((t + x) + y);
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -6e+258) {
		tmp = t_2;
	} else if (t_1 <= -1e+126) {
		tmp = fma(t, a, (t_2 * y)) / (t + y);
	} else if (t_1 <= 1.5e+214) {
		tmp = fma(x, z, (a * t)) / (t + x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / Float64(Float64(t + x) + y))
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_1 <= -6e+258)
		tmp = t_2;
	elseif (t_1 <= -1e+126)
		tmp = Float64(fma(t, a, Float64(t_2 * y)) / Float64(t + y));
	elseif (t_1 <= 1.5e+214)
		tmp = Float64(fma(x, z, Float64(a * t)) / Float64(t + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -6e+258], t$95$2, If[LessEqual[t$95$1, -1e+126], N[(N[(t * a + N[(t$95$2 * y), $MachinePrecision]), $MachinePrecision] / N[(t + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.5e+214], N[(N[(x * z + N[(a * t), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -5.9999999999999999e258 or 1.5000000000000001e214 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 14.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -5.9999999999999999e258 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.99999999999999925e125

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. associate--l+ N/A

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

      3. distribute-lft-in N/A

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

      4. associate-+l+ N/A

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

      5. *-commutative N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. associate--l+ N/A

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

      11. lower-fma.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower--.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if -9.99999999999999925e125 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.5000000000000001e214

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.7

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

   5. Applied rewrites 65.7%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -6 \cdot 10^{+258}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -1 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, a, \left(\left(a + z\right) - b\right) \cdot y\right)}{t + y}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 1.5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, z, a \cdot t\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+270}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 1.5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, y, \mathsf{fma}\left(x, z, a \cdot t\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ t x) y))
        (t_2 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) t_1))
        (t_3 (- (+ a z) b)))
   (if (<= t_2 -5e+270)
     t_3
     (if (<= t_2 1.5e+214) (/ (fma t_3 y (fma x z (* a t))) t_1) t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + x) + y;
	double t_2 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / t_1;
	double t_3 = (a + z) - b;
	double tmp;
	if (t_2 <= -5e+270) {
		tmp = t_3;
	} else if (t_2 <= 1.5e+214) {
		tmp = fma(t_3, y, fma(x, z, (a * t))) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + x) + y)
	t_2 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / t_1)
	t_3 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_2 <= -5e+270)
		tmp = t_3;
	elseif (t_2 <= 1.5e+214)
		tmp = Float64(fma(t_3, y, fma(x, z, Float64(a * t))) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+270], t$95$3, If[LessEqual[t$95$2, 1.5e+214], N[(N[(t$95$3 * y + N[(x * z + N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -4.99999999999999976e270 or 1.5000000000000001e214 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 12.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 77.3

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

   5. Applied rewrites 77.3%

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

   if -4.99999999999999976e270 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.5000000000000001e214

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.7

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

   5. Applied rewrites 99.7%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -5 \cdot 10^{+270}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 1.5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(a + z\right) - b, y, \mathsf{fma}\left(x, z, a \cdot t\right)\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+131}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, z, a \cdot t\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* (+ t y) a) (* z (+ y x))) (* b y)) (+ (+ t x) y)))
        (t_2 (- (+ a z) b)))
   (if (<= t_1 -1e+131)
     t_2
     (if (<= t_1 1.5e+214) (/ (fma x z (* a t)) (+ t x)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / ((t + x) + y);
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -1e+131) {
		tmp = t_2;
	} else if (t_1 <= 1.5e+214) {
		tmp = fma(x, z, (a * t)) / (t + x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(Float64(t + y) * a) + Float64(z * Float64(y + x))) - Float64(b * y)) / Float64(Float64(t + x) + y))
	t_2 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (t_1 <= -1e+131)
		tmp = t_2;
	elseif (t_1 <= 1.5e+214)
		tmp = Float64(fma(x, z, Float64(a * t)) / Float64(t + x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / N[(N[(t + x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+131], t$95$2, If[LessEqual[t$95$1, 1.5e+214], N[(N[(x * z + N[(a * t), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.9999999999999991e130 or 1.5000000000000001e214 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 27.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 75.5

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

   5. Applied rewrites 75.5%

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

   if -9.9999999999999991e130 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.5000000000000001e214

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 65.5

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

   5. Applied rewrites 65.5%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -1 \cdot 10^{+131}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 1.5 \cdot 10^{+214}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, z, a \cdot t\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -52000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-295}:\\ \;\;\;\;a + z\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{z}{t + \left(y + x\right)} \cdot \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -52000.0)
     t_1
     (if (<= y -2.9e-295)
       (+ a z)
       (if (<= y 4.5e-40) (* (/ z (+ t (+ y x))) (+ y x)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -52000.0) {
		tmp = t_1;
	} else if (y <= -2.9e-295) {
		tmp = a + z;
	} else if (y <= 4.5e-40) {
		tmp = (z / (t + (y + x))) * (y + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a + z) - b
    if (y <= (-52000.0d0)) then
        tmp = t_1
    else if (y <= (-2.9d-295)) then
        tmp = a + z
    else if (y <= 4.5d-40) then
        tmp = (z / (t + (y + x))) * (y + x)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -52000.0) {
		tmp = t_1;
	} else if (y <= -2.9e-295) {
		tmp = a + z;
	} else if (y <= 4.5e-40) {
		tmp = (z / (t + (y + x))) * (y + x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + z) - b
	tmp = 0
	if y <= -52000.0:
		tmp = t_1
	elif y <= -2.9e-295:
		tmp = a + z
	elif y <= 4.5e-40:
		tmp = (z / (t + (y + x))) * (y + x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -52000.0)
		tmp = t_1;
	elseif (y <= -2.9e-295)
		tmp = Float64(a + z);
	elseif (y <= 4.5e-40)
		tmp = Float64(Float64(z / Float64(t + Float64(y + x))) * Float64(y + x));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + z) - b;
	tmp = 0.0;
	if (y <= -52000.0)
		tmp = t_1;
	elseif (y <= -2.9e-295)
		tmp = a + z;
	elseif (y <= 4.5e-40)
		tmp = (z / (t + (y + x))) * (y + x);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -52000.0], t$95$1, If[LessEqual[y, -2.9e-295], N[(a + z), $MachinePrecision], If[LessEqual[y, 4.5e-40], N[(N[(z / N[(t + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -52000 or 4.5000000000000001e-40 < y

   1. Initial program 49.9%

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

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

      3. lower-+.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if -52000 < y < -2.90000000000000015e-295

   1. Initial program 76.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 37.1

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

   5. Applied rewrites 37.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 52.0%

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

   if -2.90000000000000015e-295 < y < 4.5000000000000001e-40

   1. Initial program 77.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-+.f64 56.0

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

   5. Applied rewrites 56.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -52000:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \leq -2.9 \cdot 10^{-295}:\\ \;\;\;\;a + z\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{z}{t + \left(y + x\right)} \cdot \left(y + x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -52000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-39}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -52000.0) t_1 (if (<= y 6e-39) (+ a z) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -52000.0) {
		tmp = t_1;
	} else if (y <= 6e-39) {
		tmp = a + z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a + z) - b
    if (y <= (-52000.0d0)) then
        tmp = t_1
    else if (y <= 6d-39) then
        tmp = a + z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a + z) - b;
	double tmp;
	if (y <= -52000.0) {
		tmp = t_1;
	} else if (y <= 6e-39) {
		tmp = a + z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a + z) - b
	tmp = 0
	if y <= -52000.0:
		tmp = t_1
	elif y <= 6e-39:
		tmp = a + z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a + z) - b)
	tmp = 0.0
	if (y <= -52000.0)
		tmp = t_1;
	elseif (y <= 6e-39)
		tmp = Float64(a + z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a + z) - b;
	tmp = 0.0;
	if (y <= -52000.0)
		tmp = t_1;
	elseif (y <= 6e-39)
		tmp = a + z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -52000.0], t$95$1, If[LessEqual[y, 6e-39], N[(a + z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -52000 or 6.00000000000000055e-39 < y

   1. Initial program 49.9%

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

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

      3. lower-+.f64 72.6

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

   5. Applied rewrites 72.6%

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

   if -52000 < y < 6.00000000000000055e-39

   1. Initial program 76.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 35.0

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

   5. Applied rewrites 35.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 46.6%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -52000:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-39}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-44}:\\ \;\;\;\;a + z\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-100}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.2e-44) (+ a z) (if (<= z 7.6e-100) (- a b) (+ a z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.2e-44) {
		tmp = a + z;
	} else if (z <= 7.6e-100) {
		tmp = a - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-1.2d-44)) then
        tmp = a + z
    else if (z <= 7.6d-100) then
        tmp = a - b
    else
        tmp = a + z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.2e-44) {
		tmp = a + z;
	} else if (z <= 7.6e-100) {
		tmp = a - b;
	} else {
		tmp = a + z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.2e-44:
		tmp = a + z
	elif z <= 7.6e-100:
		tmp = a - b
	else:
		tmp = a + z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.2e-44)
		tmp = Float64(a + z);
	elseif (z <= 7.6e-100)
		tmp = Float64(a - b);
	else
		tmp = Float64(a + z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.2e-44)
		tmp = a + z;
	elseif (z <= 7.6e-100)
		tmp = a - b;
	else
		tmp = a + z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.2e-44], N[(a + z), $MachinePrecision], If[LessEqual[z, 7.6e-100], N[(a - b), $MachinePrecision], N[(a + z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.20000000000000004e-44 or 7.59999999999999995e-100 < z

   1. Initial program 57.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 58.8

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 60.1%

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

   if -1.20000000000000004e-44 < z < 7.59999999999999995e-100

   1. Initial program 67.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 52.6%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-44}:\\ \;\;\;\;a + z\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{-100}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;a + z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.1% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 56.4

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

5. Applied rewrites 56.4%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 52.2%

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

9. Final simplification 52.2%

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

Alternative 11: 13.4% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-+.f64 56.4

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

5. Applied rewrites 56.4%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 13.0%

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

Developer Target 1: 82.6% 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 2024257 
(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)))