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

Percentage Accurate: 60.1% → 89.9%

Time: 8.9s

Alternatives: 18

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: 60.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(\frac{y + x}{t\_1} \cdot \frac{z}{a} - \mathsf{fma}\left(\frac{b}{a}, \frac{y}{t\_1}, \frac{-\left(t + y\right)}{t\_1}\right)\right) \cdot a\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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
         (*
          (-
           (* (/ (+ y x) t_1) (/ z a))
           (fma (/ b a) (/ y t_1) (/ (- (+ t y)) t_1)))
          a)))
   (if (<= t_2 (- INFINITY)) t_3 (if (<= t_2 5e+306) t_2 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 = ((((y + x) / t_1) * (z / a)) - fma((b / a), (y / t_1), (-(t + y) / t_1))) * a;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 5e+306) {
		tmp = t_2;
	} 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(Float64(Float64(Float64(y + x) / t_1) * Float64(z / a)) - fma(Float64(b / a), Float64(y / t_1), Float64(Float64(-Float64(t + y)) / t_1))) * a)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 5e+306)
		tmp = t_2;
	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[(N[(N[(N[(y + x), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision] - N[(N[(b / a), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision] + N[((-N[(t + y), $MachinePrecision]) / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 5e+306], t$95$2, 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)) < -inf.0 or 4.99999999999999993e306 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 5.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   7. Applied rewrites 79.6%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999993e306

   1. Initial program 99.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -\infty:\\ \;\;\;\;\left(\frac{y + x}{\left(t + x\right) + y} \cdot \frac{z}{a} - \mathsf{fma}\left(\frac{b}{a}, \frac{y}{\left(t + x\right) + y}, \frac{-\left(t + y\right)}{\left(t + x\right) + y}\right)\right) \cdot a\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y + x}{\left(t + x\right) + y} \cdot \frac{z}{a} - \mathsf{fma}\left(\frac{b}{a}, \frac{y}{\left(t + x\right) + y}, \frac{-\left(t + y\right)}{\left(t + x\right) + y}\right)\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(a + z\right) - b\\ t_4 := \frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{t\_1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+151}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+87}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - b, y, z \cdot x\right)}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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))
        (t_4 (/ (fma (- a b) y (* a t)) t_1)))
   (if (<= t_2 -5e+151)
     t_3
     (if (<= t_2 -1e+87)
       t_4
       (if (<= t_2 -5e-41)
         (/ (fma (- z b) y (* z x)) t_1)
         (if (<= t_2 2e-115)
           (/ (fma a t (* z x)) (+ t x))
           (if (<= t_2 4e+164) t_4 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 t_4 = fma((a - b), y, (a * t)) / t_1;
	double tmp;
	if (t_2 <= -5e+151) {
		tmp = t_3;
	} else if (t_2 <= -1e+87) {
		tmp = t_4;
	} else if (t_2 <= -5e-41) {
		tmp = fma((z - b), y, (z * x)) / t_1;
	} else if (t_2 <= 2e-115) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else if (t_2 <= 4e+164) {
		tmp = t_4;
	} 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)
	t_4 = Float64(fma(Float64(a - b), y, Float64(a * t)) / t_1)
	tmp = 0.0
	if (t_2 <= -5e+151)
		tmp = t_3;
	elseif (t_2 <= -1e+87)
		tmp = t_4;
	elseif (t_2 <= -5e-41)
		tmp = Float64(fma(Float64(z - b), y, Float64(z * x)) / t_1);
	elseif (t_2 <= 2e-115)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	elseif (t_2 <= 4e+164)
		tmp = t_4;
	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]}, Block[{t$95$4 = N[(N[(N[(a - b), $MachinePrecision] * y + N[(a * t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+151], t$95$3, If[LessEqual[t$95$2, -1e+87], t$95$4, If[LessEqual[t$95$2, -5e-41], N[(N[(N[(z - b), $MachinePrecision] * y + N[(z * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e-115], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+164], t$95$4, 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)) < -5.0000000000000002e151 or 4e164 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 24.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 71.3

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

   5. Applied rewrites 71.3%

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

   if -5.0000000000000002e151 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.9999999999999996e86 or 2.0000000000000001e-115 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4e164

   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 z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.9%

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

   if -9.9999999999999996e86 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -4.9999999999999996e-41

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   7. Applied rewrites 77.5%

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

   if -4.9999999999999996e-41 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e-115

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 74.6

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

   5. Applied rewrites 74.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -1 \cdot 10^{+87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{\left(t + x\right) + y}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z - b, y, z \cdot x\right)}{\left(t + x\right) + y}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(a + z\right) - b\\ t_4 := \frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{t\_1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+151}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{+87}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x, \left(z - b\right) \cdot y\right)}{t\_1}\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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))
        (t_4 (/ (fma (- a b) y (* a t)) t_1)))
   (if (<= t_2 -5e+151)
     t_3
     (if (<= t_2 -1e+87)
       t_4
       (if (<= t_2 -5e-41)
         (/ (fma z x (* (- z b) y)) t_1)
         (if (<= t_2 2e-115)
           (/ (fma a t (* z x)) (+ t x))
           (if (<= t_2 4e+164) t_4 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 t_4 = fma((a - b), y, (a * t)) / t_1;
	double tmp;
	if (t_2 <= -5e+151) {
		tmp = t_3;
	} else if (t_2 <= -1e+87) {
		tmp = t_4;
	} else if (t_2 <= -5e-41) {
		tmp = fma(z, x, ((z - b) * y)) / t_1;
	} else if (t_2 <= 2e-115) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else if (t_2 <= 4e+164) {
		tmp = t_4;
	} 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)
	t_4 = Float64(fma(Float64(a - b), y, Float64(a * t)) / t_1)
	tmp = 0.0
	if (t_2 <= -5e+151)
		tmp = t_3;
	elseif (t_2 <= -1e+87)
		tmp = t_4;
	elseif (t_2 <= -5e-41)
		tmp = Float64(fma(z, x, Float64(Float64(z - b) * y)) / t_1);
	elseif (t_2 <= 2e-115)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	elseif (t_2 <= 4e+164)
		tmp = t_4;
	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]}, Block[{t$95$4 = N[(N[(N[(a - b), $MachinePrecision] * y + N[(a * t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+151], t$95$3, If[LessEqual[t$95$2, -1e+87], t$95$4, If[LessEqual[t$95$2, -5e-41], N[(N[(z * x + N[(N[(z - b), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 2e-115], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+164], t$95$4, 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)) < -5.0000000000000002e151 or 4e164 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 24.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 71.3

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

   5. Applied rewrites 71.3%

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

   if -5.0000000000000002e151 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -9.9999999999999996e86 or 2.0000000000000001e-115 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4e164

   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 z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.9%

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

   if -9.9999999999999996e86 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -4.9999999999999996e-41

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -4.9999999999999996e-41 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e-115

   1. Initial program 99.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 74.6

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

   5. Applied rewrites 74.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -1 \cdot 10^{+87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{\left(t + x\right) + y}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x, \left(z - b\right) \cdot y\right)}{\left(t + x\right) + y}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+293}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(1, a, \frac{\mathsf{fma}\left(z, x, \left(z - b\right) \cdot y\right)}{t + \left(y + x\right)}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 -2e+293)
     t_3
     (if (<= t_2 -1e-195)
       (fma 1.0 a (/ (fma z x (* (- z b) y)) (+ t (+ y x))))
       (if (<= t_2 2e-115)
         (/ (fma a t (* z x)) (+ t x))
         (if (<= t_2 4e+164) (/ (fma (- a b) y (* 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 <= -2e+293) {
		tmp = t_3;
	} else if (t_2 <= -1e-195) {
		tmp = fma(1.0, a, (fma(z, x, ((z - b) * y)) / (t + (y + x))));
	} else if (t_2 <= 2e-115) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else if (t_2 <= 4e+164) {
		tmp = fma((a - b), y, (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 <= -2e+293)
		tmp = t_3;
	elseif (t_2 <= -1e-195)
		tmp = fma(1.0, a, Float64(fma(z, x, Float64(Float64(z - b) * y)) / Float64(t + Float64(y + x))));
	elseif (t_2 <= 2e-115)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	elseif (t_2 <= 4e+164)
		tmp = Float64(fma(Float64(a - b), y, 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, -2e+293], t$95$3, If[LessEqual[t$95$2, -1e-195], N[(1.0 * a + N[(N[(z * x + N[(N[(z - b), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / N[(t + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-115], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+164], N[(N[(N[(a - b), $MachinePrecision] * y + N[(a * t), $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)) < -1.9999999999999998e293 or 4e164 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 16.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if -1.9999999999999998e293 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.0000000000000001e-195

   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

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

      1. associate--l+ N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 87.4%

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

   if -1.0000000000000001e-195 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e-115

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 86.8

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

   5. Applied rewrites 86.8%

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

   if 2.0000000000000001e-115 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4e164

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 82.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -2 \cdot 10^{+293}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -1 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(1, a, \frac{\mathsf{fma}\left(z, x, \left(z - b\right) \cdot y\right)}{t + \left(y + x\right)}\right)\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(a + z\right) - b\\ t_4 := \frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{t\_1}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+151}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -8.5 \cdot 10^{+21}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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))
        (t_4 (/ (fma (- a b) y (* a t)) t_1)))
   (if (<= t_2 -5e+151)
     t_3
     (if (<= t_2 -8.5e+21)
       t_4
       (if (<= t_2 2e-115)
         (/ (fma a t (* z x)) (+ t x))
         (if (<= t_2 4e+164) t_4 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 t_4 = fma((a - b), y, (a * t)) / t_1;
	double tmp;
	if (t_2 <= -5e+151) {
		tmp = t_3;
	} else if (t_2 <= -8.5e+21) {
		tmp = t_4;
	} else if (t_2 <= 2e-115) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else if (t_2 <= 4e+164) {
		tmp = t_4;
	} 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)
	t_4 = Float64(fma(Float64(a - b), y, Float64(a * t)) / t_1)
	tmp = 0.0
	if (t_2 <= -5e+151)
		tmp = t_3;
	elseif (t_2 <= -8.5e+21)
		tmp = t_4;
	elseif (t_2 <= 2e-115)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	elseif (t_2 <= 4e+164)
		tmp = t_4;
	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]}, Block[{t$95$4 = N[(N[(N[(a - b), $MachinePrecision] * y + N[(a * t), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+151], t$95$3, If[LessEqual[t$95$2, -8.5e+21], t$95$4, If[LessEqual[t$95$2, 2e-115], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+164], t$95$4, t$95$3]]]]]]]]

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)) < -5.0000000000000002e151 or 4e164 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 24.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 71.3

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

   5. Applied rewrites 71.3%

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

   if -5.0000000000000002e151 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -8.5e21 or 2.0000000000000001e-115 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4e164

   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 z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 78.4

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

   5. Applied rewrites 78.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 78.4%

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

   if -8.5e21 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.0000000000000001e-115

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 67.4

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

   5. Applied rewrites 67.4%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -8.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{\left(t + x\right) + y}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+244}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t + y, a, \left(z - b\right) \cdot y\right)}{t + y}\\ \mathbf{elif}\;t\_2 \leq 100000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{t\_3 \cdot y}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 -2e+244)
     t_3
     (if (<= t_2 -2e+27)
       (/ (fma (+ t y) a (* (- z b) y)) (+ t y))
       (if (<= t_2 100000.0)
         (/ (fma a t (* z x)) (+ t x))
         (if (<= t_2 4e+164) (/ (* t_3 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 <= -2e+244) {
		tmp = t_3;
	} else if (t_2 <= -2e+27) {
		tmp = fma((t + y), a, ((z - b) * y)) / (t + y);
	} else if (t_2 <= 100000.0) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else if (t_2 <= 4e+164) {
		tmp = (t_3 * 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 <= -2e+244)
		tmp = t_3;
	elseif (t_2 <= -2e+27)
		tmp = Float64(fma(Float64(t + y), a, Float64(Float64(z - b) * y)) / Float64(t + y));
	elseif (t_2 <= 100000.0)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	elseif (t_2 <= 4e+164)
		tmp = Float64(Float64(t_3 * 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, -2e+244], t$95$3, If[LessEqual[t$95$2, -2e+27], N[(N[(N[(t + y), $MachinePrecision] * a + N[(N[(z - b), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision] / N[(t + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 100000.0], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+164], N[(N[(t$95$3 * y), $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)) < -2.00000000000000015e244 or 4e164 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 20.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if -2.00000000000000015e244 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -2e27

   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 74.0

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

   5. Applied rewrites 74.0%

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

   if -2e27 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1e5

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 67.9

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

   5. Applied rewrites 67.9%

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

   if 1e5 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4e164

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 65.0

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

   5. Applied rewrites 65.0%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -2 \cdot 10^{+244}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t + y, a, \left(z - b\right) \cdot y\right)}{t + y}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 100000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\left(\left(a + z\right) - b\right) \cdot y}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(a + z\right) - b\\ t_4 := \frac{t\_3 \cdot y}{t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+125}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -8.5 \cdot 10^{+21}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_2 \leq 100000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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))
        (t_4 (/ (* t_3 y) t_1)))
   (if (<= t_2 -2e+125)
     t_3
     (if (<= t_2 -8.5e+21)
       t_4
       (if (<= t_2 100000.0)
         (/ (fma a t (* z x)) (+ t x))
         (if (<= t_2 4e+164) t_4 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 t_4 = (t_3 * y) / t_1;
	double tmp;
	if (t_2 <= -2e+125) {
		tmp = t_3;
	} else if (t_2 <= -8.5e+21) {
		tmp = t_4;
	} else if (t_2 <= 100000.0) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else if (t_2 <= 4e+164) {
		tmp = t_4;
	} 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)
	t_4 = Float64(Float64(t_3 * y) / t_1)
	tmp = 0.0
	if (t_2 <= -2e+125)
		tmp = t_3;
	elseif (t_2 <= -8.5e+21)
		tmp = t_4;
	elseif (t_2 <= 100000.0)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	elseif (t_2 <= 4e+164)
		tmp = t_4;
	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]}, Block[{t$95$4 = N[(N[(t$95$3 * y), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+125], t$95$3, If[LessEqual[t$95$2, -8.5e+21], t$95$4, If[LessEqual[t$95$2, 100000.0], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+164], t$95$4, t$95$3]]]]]]]]

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)) < -1.9999999999999998e125 or 4e164 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 26.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 70.7

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

   5. Applied rewrites 70.7%

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

   if -1.9999999999999998e125 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -8.5e21 or 1e5 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4e164

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-+.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if -8.5e21 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1e5

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 68.8

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

   5. Applied rewrites 68.8%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -2 \cdot 10^{+125}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -8.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\left(\left(a + z\right) - b\right) \cdot y}{\left(t + x\right) + y}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 100000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\left(\left(a + z\right) - b\right) \cdot y}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+84}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -5e+84)
     t_2
     (if (<= t_1 1e-101)
       (/ (fma a t (* z x)) (+ t x))
       (if (<= t_1 4e+164) (/ (fma (- a b) y (* 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 <= -5e+84) {
		tmp = t_2;
	} else if (t_1 <= 1e-101) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else if (t_1 <= 4e+164) {
		tmp = fma((a - b), y, (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 <= -5e+84)
		tmp = t_2;
	elseif (t_1 <= 1e-101)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	elseif (t_1 <= 4e+164)
		tmp = Float64(fma(Float64(a - b), y, 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, -5e+84], t$95$2, If[LessEqual[t$95$1, 1e-101], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e+164], N[(N[(N[(a - b), $MachinePrecision] * y + 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)) < -5.0000000000000001e84 or 4e164 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 33.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 68.3

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

   5. Applied rewrites 68.3%

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

   if -5.0000000000000001e84 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.00000000000000005e-101

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if 1.00000000000000005e-101 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4e164

   1. Initial program 99.6%

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

   3. Taylor expanded in z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.8%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. lower-+.f64 66.9

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

   11. Applied rewrites 66.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 10^{-101}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a - b, y, a \cdot t\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ t_3 := \left(a + z\right) - b\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+84}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 100000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\left(a - b\right) \cdot y}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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+84)
     t_3
     (if (<= t_2 100000.0)
       (/ (fma a t (* z x)) (+ t x))
       (if (<= t_2 4e+164) (/ (* (- a 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 <= -5e+84) {
		tmp = t_3;
	} else if (t_2 <= 100000.0) {
		tmp = fma(a, t, (z * x)) / (t + x);
	} else if (t_2 <= 4e+164) {
		tmp = ((a - 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 <= -5e+84)
		tmp = t_3;
	elseif (t_2 <= 100000.0)
		tmp = Float64(fma(a, t, Float64(z * x)) / Float64(t + x));
	elseif (t_2 <= 4e+164)
		tmp = Float64(Float64(Float64(a - 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, -5e+84], t$95$3, If[LessEqual[t$95$2, 100000.0], N[(N[(a * t + N[(z * x), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+164], N[(N[(N[(a - b), $MachinePrecision] * y), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]]

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)) < -5.0000000000000001e84 or 4e164 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 33.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 68.3

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

   5. Applied rewrites 68.3%

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

   if -5.0000000000000001e84 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1e5

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if 1e5 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4e164

   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 z around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 82.0

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 57.7%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 100000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 4 \cdot 10^{+164}:\\ \;\;\;\;\frac{\left(a - b\right) \cdot y}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + x\right) + y\\ t_2 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{t\_1}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+293}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y + x}{t\_1} \cdot \frac{z}{a} - \left(\frac{b}{a} - 1\right)\right) \cdot a\\ \end{array} \end{array} \]

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)))
   (if (<= t_2 -2e+293)
     (- (+ a z) b)
     (if (<= t_2 5e+306)
       t_2
       (* (- (* (/ (+ y x) t_1) (/ z a)) (- (/ b a) 1.0)) a)))))

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 tmp;
	if (t_2 <= -2e+293) {
		tmp = (a + z) - b;
	} else if (t_2 <= 5e+306) {
		tmp = t_2;
	} else {
		tmp = ((((y + x) / t_1) * (z / a)) - ((b / a) - 1.0)) * 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (t + x) + y
    t_2 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / t_1
    if (t_2 <= (-2d+293)) then
        tmp = (a + z) - b
    else if (t_2 <= 5d+306) then
        tmp = t_2
    else
        tmp = ((((y + x) / t_1) * (z / a)) - ((b / a) - 1.0d0)) * 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 t_1 = (t + x) + y;
	double t_2 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / t_1;
	double tmp;
	if (t_2 <= -2e+293) {
		tmp = (a + z) - b;
	} else if (t_2 <= 5e+306) {
		tmp = t_2;
	} else {
		tmp = ((((y + x) / t_1) * (z / a)) - ((b / a) - 1.0)) * a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t + x) + y
	t_2 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / t_1
	tmp = 0
	if t_2 <= -2e+293:
		tmp = (a + z) - b
	elif t_2 <= 5e+306:
		tmp = t_2
	else:
		tmp = ((((y + x) / t_1) * (z / a)) - ((b / a) - 1.0)) * a
	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)
	tmp = 0.0
	if (t_2 <= -2e+293)
		tmp = Float64(Float64(a + z) - b);
	elseif (t_2 <= 5e+306)
		tmp = t_2;
	else
		tmp = Float64(Float64(Float64(Float64(Float64(y + x) / t_1) * Float64(z / a)) - Float64(Float64(b / a) - 1.0)) * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + x) + y;
	t_2 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / t_1;
	tmp = 0.0;
	if (t_2 <= -2e+293)
		tmp = (a + z) - b;
	elseif (t_2 <= 5e+306)
		tmp = t_2;
	else
		tmp = ((((y + x) / t_1) * (z / a)) - ((b / a) - 1.0)) * a;
	end
	tmp_2 = 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]}, If[LessEqual[t$95$2, -2e+293], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], If[LessEqual[t$95$2, 5e+306], t$95$2, N[(N[(N[(N[(N[(y + x), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(z / a), $MachinePrecision]), $MachinePrecision] - N[(N[(b / a), $MachinePrecision] - 1.0), $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)) < -1.9999999999999998e293

   1. Initial program 8.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 70.6

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

   5. Applied rewrites 70.6%

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

   if -1.9999999999999998e293 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999993e306

   1. Initial program 99.6%

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

   if 4.99999999999999993e306 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 4.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 5.7%

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

   5. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 N/A

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 74.6%

       \[\leadsto \left(-a\right) \cdot \left(\left(\frac{b}{a} - 1\right) - \color{blue}{\frac{z}{a}} \cdot \frac{x + y}{\left(x + t\right) + y}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -2 \cdot 10^{+293}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y + x}{\left(t + x\right) + y} \cdot \frac{z}{a} - \left(\frac{b}{a} - 1\right)\right) \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+293}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+254}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -2e+293) t_2 (if (<= t_1 1e+254) t_1 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 <= -2e+293) {
		tmp = t_2;
	} else if (t_1 <= 1e+254) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / ((t + x) + y)
    t_2 = (a + z) - b
    if (t_1 <= (-2d+293)) then
        tmp = t_2
    else if (t_1 <= 1d+254) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / ((t + x) + y);
	double t_2 = (a + z) - b;
	double tmp;
	if (t_1 <= -2e+293) {
		tmp = t_2;
	} else if (t_1 <= 1e+254) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / ((t + x) + y)
	t_2 = (a + z) - b
	tmp = 0
	if t_1 <= -2e+293:
		tmp = t_2
	elif t_1 <= 1e+254:
		tmp = t_1
	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 <= -2e+293)
		tmp = t_2;
	elseif (t_1 <= 1e+254)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((((t + y) * a) + (z * (y + x))) - (b * y)) / ((t + x) + y);
	t_2 = (a + z) - b;
	tmp = 0.0;
	if (t_1 <= -2e+293)
		tmp = t_2;
	elseif (t_1 <= 1e+254)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(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, -2e+293], t$95$2, If[LessEqual[t$95$1, 1e+254], t$95$1, t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.9999999999999998e293 or 9.9999999999999994e253 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 9.0%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -1.9999999999999998e293 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999994e253

   1. Initial program 99.6%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -2 \cdot 10^{+293}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 10^{+254}:\\ \;\;\;\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y}\\ t_2 := \left(a + z\right) - b\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+84}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -5e+84)
     t_2
     (if (<= t_1 1e+46) (/ (fma a t (* z x)) (+ 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 <= -5e+84) {
		tmp = t_2;
	} else if (t_1 <= 1e+46) {
		tmp = fma(a, t, (z * x)) / (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 <= -5e+84)
		tmp = t_2;
	elseif (t_1 <= 1e+46)
		tmp = Float64(fma(a, t, Float64(z * x)) / 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, -5e+84], t$95$2, If[LessEqual[t$95$1, 1e+46], N[(N[(a * t + N[(z * x), $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)) < -5.0000000000000001e84 or 9.9999999999999999e45 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 41.1%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 65.3

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

   5. Applied rewrites 65.3%

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

   if -5.0000000000000001e84 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999999e45

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 62.0

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

   5. Applied rewrites 62.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;\frac{\left(\left(t + y\right) \cdot a + z \cdot \left(y + x\right)\right) - b \cdot y}{\left(t + x\right) + y} \leq 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, z \cdot x\right)}{t + x}\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{t + x} \cdot a\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -290000000:\\ \;\;\;\;\frac{z}{t + \left(y + x\right)} \cdot \left(y + x\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+161}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ t (+ t x)) a)))
   (if (<= t -6.2e+122)
     t_1
     (if (<= t -290000000.0)
       (* (/ z (+ t (+ y x))) (+ y x))
       (if (<= t 6e+161) (- (+ a z) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t / (t + x)) * a;
	double tmp;
	if (t <= -6.2e+122) {
		tmp = t_1;
	} else if (t <= -290000000.0) {
		tmp = (z / (t + (y + x))) * (y + x);
	} else if (t <= 6e+161) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / (t + x)) * a
    if (t <= (-6.2d+122)) then
        tmp = t_1
    else if (t <= (-290000000.0d0)) then
        tmp = (z / (t + (y + x))) * (y + x)
    else if (t <= 6d+161) then
        tmp = (a + z) - b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t / (t + x)) * a;
	double tmp;
	if (t <= -6.2e+122) {
		tmp = t_1;
	} else if (t <= -290000000.0) {
		tmp = (z / (t + (y + x))) * (y + x);
	} else if (t <= 6e+161) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t / (t + x)) * a
	tmp = 0
	if t <= -6.2e+122:
		tmp = t_1
	elif t <= -290000000.0:
		tmp = (z / (t + (y + x))) * (y + x)
	elif t <= 6e+161:
		tmp = (a + z) - b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t / Float64(t + x)) * a)
	tmp = 0.0
	if (t <= -6.2e+122)
		tmp = t_1;
	elseif (t <= -290000000.0)
		tmp = Float64(Float64(z / Float64(t + Float64(y + x))) * Float64(y + x));
	elseif (t <= 6e+161)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t / (t + x)) * a;
	tmp = 0.0;
	if (t <= -6.2e+122)
		tmp = t_1;
	elseif (t <= -290000000.0)
		tmp = (z / (t + (y + x))) * (y + x);
	elseif (t <= 6e+161)
		tmp = (a + z) - b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t / N[(t + x), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t, -6.2e+122], t$95$1, If[LessEqual[t, -290000000.0], N[(N[(z / N[(t + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e+161], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.19999999999999998e122 or 6.00000000000000023e161 < t

   1. Initial program 53.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 51.9

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.4%

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

   if -6.19999999999999998e122 < t < -2.9e8

   1. Initial program 67.6%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 63.7

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

   5. Applied rewrites 63.7%

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

   if -2.9e8 < t < 6.00000000000000023e161

   1. Initial program 66.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 63.2

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

   5. Applied rewrites 63.2%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{+122}:\\ \;\;\;\;\frac{t}{t + x} \cdot a\\ \mathbf{elif}\;t \leq -290000000:\\ \;\;\;\;\frac{z}{t + \left(y + x\right)} \cdot \left(y + x\right)\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+161}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{t + x} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{t + x} \cdot a\\ \mathbf{if}\;t \leq -6.7 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+161}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (/ t (+ t x)) a)))
   (if (<= t -6.7e+115) t_1 (if (<= t 6e+161) (- (+ a z) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t / (t + x)) * a;
	double tmp;
	if (t <= -6.7e+115) {
		tmp = t_1;
	} else if (t <= 6e+161) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t / (t + x)) * a
    if (t <= (-6.7d+115)) then
        tmp = t_1
    else if (t <= 6d+161) then
        tmp = (a + z) - b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t / (t + x)) * a;
	double tmp;
	if (t <= -6.7e+115) {
		tmp = t_1;
	} else if (t <= 6e+161) {
		tmp = (a + z) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t / (t + x)) * a
	tmp = 0
	if t <= -6.7e+115:
		tmp = t_1
	elif t <= 6e+161:
		tmp = (a + z) - b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t / Float64(t + x)) * a)
	tmp = 0.0
	if (t <= -6.7e+115)
		tmp = t_1;
	elseif (t <= 6e+161)
		tmp = Float64(Float64(a + z) - b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t / (t + x)) * a;
	tmp = 0.0;
	if (t <= -6.7e+115)
		tmp = t_1;
	elseif (t <= 6e+161)
		tmp = (a + z) - b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t / N[(t + x), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t, -6.7e+115], t$95$1, If[LessEqual[t, 6e+161], N[(N[(a + z), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.6999999999999997e115 or 6.00000000000000023e161 < t

   1. Initial program 54.5%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 57.0%

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

   if -6.6999999999999997e115 < t < 6.00000000000000023e161

   1. Initial program 65.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 60.8

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

   5. Applied rewrites 60.8%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.7 \cdot 10^{+115}:\\ \;\;\;\;\frac{t}{t + x} \cdot a\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+161}:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{t + x} \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + z\right) - b\\ \mathbf{if}\;y \leq -32000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 215000000000:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ a z) b)))
   (if (<= y -32000.0) t_1 (if (<= y 215000000000.0) (+ 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 <= -32000.0) {
		tmp = t_1;
	} else if (y <= 215000000000.0) {
		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 <= (-32000.0d0)) then
        tmp = t_1
    else if (y <= 215000000000.0d0) 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 <= -32000.0) {
		tmp = t_1;
	} else if (y <= 215000000000.0) {
		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 <= -32000.0:
		tmp = t_1
	elif y <= 215000000000.0:
		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 <= -32000.0)
		tmp = t_1;
	elseif (y <= 215000000000.0)
		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 <= -32000.0)
		tmp = t_1;
	elseif (y <= 215000000000.0)
		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, -32000.0], t$95$1, If[LessEqual[y, 215000000000.0], N[(a + z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -32000 or 2.15e11 < y

   1. Initial program 45.2%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 66.8

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

   5. Applied rewrites 66.8%

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

   if -32000 < y < 2.15e11

   1. Initial program 76.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 40.0

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 4.9%

      \[\leadsto -b \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 50.3%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -32000:\\ \;\;\;\;\left(a + z\right) - b\\ \mathbf{elif}\;y \leq 215000000000:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;\left(a + z\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+119}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;a - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (if (<= t 3.3e+119) (+ a z) (- a b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 3.3e+119) {
		tmp = a + z;
	} else {
		tmp = a - b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= 3.3d+119) then
        tmp = a + z
    else
        tmp = a - b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= 3.3e+119) {
		tmp = a + z;
	} else {
		tmp = a - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= 3.3e+119:
		tmp = a + z
	else:
		tmp = a - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= 3.3e+119)
		tmp = Float64(a + z);
	else
		tmp = Float64(a - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= 3.3e+119)
		tmp = a + z;
	else
		tmp = a - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, 3.3e+119], N[(a + z), $MachinePrecision], N[(a - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.3000000000000002e119

   1. Initial program 64.4%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 10.9%

      \[\leadsto -b \]

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 54.5%

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

   if 3.3000000000000002e119 < t

   1. Initial program 50.8%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 43.8%

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

4. Final simplification 52.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+119}:\\ \;\;\;\;a + z\\ \mathbf{else}:\\ \;\;\;\;a - b\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.8% 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 62.1%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 52.4

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 11.1%

   \[\leadsto -b \]

9. Taylor expanded in b around 0

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

11. Applied rewrites 50.5%

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

12. Final simplification 50.5%

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

Alternative 18: 13.1% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in y around inf

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

   1. lower--.f64 N/A

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

   2. lower-+.f64 52.4

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 11.1%

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

Developer Target 1: 81.8% 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 2024295 
(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)))