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

Percentage Accurate: 59.6% → 88.2%

Time: 11.7s

Alternatives: 16

Speedup: 1.4×

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: 59.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{x + y}\\ t_2 := y + \left(x + t\right)\\ t_3 := \frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{t\_2}\\ t_4 := \left(x + y\right) \cdot \left(x + y\right)\\ t_5 := \mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{t\_4}, \frac{a}{x + y}\right) - \mathsf{fma}\left(a, \frac{y}{t\_4}, \frac{z}{x + y}\right), \mathsf{fma}\left(a, t\_1, z\right)\right) - b \cdot t\_1\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq 10^{+298}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, \left(z + a\right) - b, x \cdot z\right)\right)}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ y (+ x y)))
        (t_2 (+ y (+ x t)))
        (t_3 (/ (- (+ (* z (+ x y)) (* a (+ y t))) (* y b)) t_2))
        (t_4 (* (+ x y) (+ x y)))
        (t_5
         (-
          (fma
           t
           (- (fma y (/ b t_4) (/ a (+ x y))) (fma a (/ y t_4) (/ z (+ x y))))
           (fma a t_1 z))
          (* b t_1))))
   (if (<= t_3 (- INFINITY))
     t_5
     (if (<= t_3 1e+298)
       (/ (fma a t (fma y (- (+ z a) b) (* x z))) t_2)
       t_5))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y / (x + y);
	double t_2 = y + (x + t);
	double t_3 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / t_2;
	double t_4 = (x + y) * (x + y);
	double t_5 = fma(t, (fma(y, (b / t_4), (a / (x + y))) - fma(a, (y / t_4), (z / (x + y)))), fma(a, t_1, z)) - (b * t_1);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_3 <= 1e+298) {
		tmp = fma(a, t, fma(y, ((z + a) - b), (x * z))) / t_2;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y / Float64(x + y))
	t_2 = Float64(y + Float64(x + t))
	t_3 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(a * Float64(y + t))) - Float64(y * b)) / t_2)
	t_4 = Float64(Float64(x + y) * Float64(x + y))
	t_5 = Float64(fma(t, Float64(fma(y, Float64(b / t_4), Float64(a / Float64(x + y))) - fma(a, Float64(y / t_4), Float64(z / Float64(x + y)))), fma(a, t_1, z)) - Float64(b * t_1))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_3 <= 1e+298)
		tmp = Float64(fma(a, t, fma(y, Float64(Float64(z + a) - b), Float64(x * z))) / t_2);
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t * N[(N[(y * N[(b / t$95$4), $MachinePrecision] + N[(a / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(y / t$95$4), $MachinePrecision] + N[(z / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a * t$95$1 + z), $MachinePrecision]), $MachinePrecision] - N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$5, If[LessEqual[t$95$3, 1e+298], N[(N[(a * t + N[(y * N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], t$95$5]]]]]]]

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 9.9999999999999996e297 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 4.9%

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

   3. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

   5. Applied rewrites 71.1%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999996e297

   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

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{\left(x + y\right) \cdot \left(x + y\right)}, \frac{a}{x + y}\right) - \mathsf{fma}\left(a, \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}, \frac{z}{x + y}\right), \mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\right) - b \cdot \frac{y}{x + y}\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 10^{+298}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, \left(z + a\right) - b, x \cdot z\right)\right)}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \mathsf{fma}\left(y, \frac{b}{\left(x + y\right) \cdot \left(x + y\right)}, \frac{a}{x + y}\right) - \mathsf{fma}\left(a, \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}, \frac{z}{x + y}\right), \mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\right) - b \cdot \frac{y}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 66.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{t\_1}\\ t_3 := \left(z + a\right) - b\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+235}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{x + t}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x, y \cdot \left(z - b\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t)))
        (t_2 (/ (- (+ (* z (+ x y)) (* a (+ y t))) (* y b)) t_1))
        (t_3 (- (+ z a) b)))
   (if (<= t_2 -1e+235)
     t_3
     (if (<= t_2 2000000000.0)
       (/ (fma a t (* x z)) (+ x t))
       (if (<= t_2 5e+97) (/ (fma z x (* y (- z b))) t_1) t_3)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / t_1;
	double t_3 = (z + a) - b;
	double tmp;
	if (t_2 <= -1e+235) {
		tmp = t_3;
	} else if (t_2 <= 2000000000.0) {
		tmp = fma(a, t, (x * z)) / (x + t);
	} else if (t_2 <= 5e+97) {
		tmp = fma(z, x, (y * (z - b))) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	t_2 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(a * Float64(y + t))) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (t_2 <= -1e+235)
		tmp = t_3;
	elseif (t_2 <= 2000000000.0)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(x + t));
	elseif (t_2 <= 5e+97)
		tmp = Float64(fma(z, x, Float64(y * Float64(z - b))) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+235], t$95$3, If[LessEqual[t$95$2, 2000000000.0], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(x + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+97], N[(N[(z * x + N[(y * N[(z - b), $MachinePrecision]), $MachinePrecision]), $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)) < -1.0000000000000001e235 or 4.99999999999999999e97 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 20.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. 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 -1.0000000000000001e235 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2e9

   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 70.8

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

   5. Applied rewrites 70.8%

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

   if 2e9 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 4.99999999999999999e97

   1. Initial program 99.6%

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

   3. Taylor expanded in a around 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 74.8

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

   5. Applied rewrites 74.8%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -1 \cdot 10^{+235}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 2000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{x + t}\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 5 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(z, x, y \cdot \left(z - b\right)\right)}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{t\_1}\\ t_3 := \left(z + a\right) - b\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, \mathsf{fma}\left(y, t\_3, x \cdot z\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t)))
        (t_2 (/ (- (+ (* z (+ x y)) (* a (+ y t))) (* y b)) t_1))
        (t_3 (- (+ z a) b)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 5e+268) (/ (fma a t (fma y t_3 (* x z))) t_1) t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / t_1;
	double t_3 = (z + a) - b;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= 5e+268) {
		tmp = fma(a, t, fma(y, t_3, (x * z))) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	t_2 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(a * Float64(y + t))) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= 5e+268)
		tmp = Float64(fma(a, t, fma(y, t_3, Float64(x * z))) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, 5e+268], N[(N[(a * t + N[(y * t$95$3 + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 5.0000000000000002e268 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 5.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 70.0

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

   5. Applied rewrites 70.0%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 5.0000000000000002e268

   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

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

      1. lower-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 87.7%

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

Alternative 4: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{t\_1}\\ t_3 := \left(z + a\right) - b\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+235}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 1.3 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right) - y \cdot b}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t)))
        (t_2 (/ (- (+ (* z (+ x y)) (* a (+ y t))) (* y b)) t_1))
        (t_3 (- (+ z a) b)))
   (if (<= t_2 -1e+235)
     t_3
     (if (<= t_2 1.3e+90) (/ (- (fma a t (* x z)) (* y b)) t_1) t_3))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double t_2 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / t_1;
	double t_3 = (z + a) - b;
	double tmp;
	if (t_2 <= -1e+235) {
		tmp = t_3;
	} else if (t_2 <= 1.3e+90) {
		tmp = (fma(a, t, (x * z)) - (y * b)) / t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	t_2 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(a * Float64(y + t))) - Float64(y * b)) / t_1)
	t_3 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (t_2 <= -1e+235)
		tmp = t_3;
	elseif (t_2 <= 1.3e+90)
		tmp = Float64(Float64(fma(a, t, Float64(x * z)) - Float64(y * b)) / t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+235], t$95$3, If[LessEqual[t$95$2, 1.3e+90], N[(N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -1.0000000000000001e235 or 1.2999999999999999e90 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 21.7%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 71.4

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

   5. Applied rewrites 71.4%

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

   if -1.0000000000000001e235 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 1.2999999999999999e90

   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

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -1 \cdot 10^{+235}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 1.3 \cdot 10^{+90}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)}\\ t_2 := \left(z + a\right) - b\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+235}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{x + t}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* z (+ x y)) (* a (+ y t))) (* y b)) (+ y (+ x t))))
        (t_2 (- (+ z a) b)))
   (if (<= t_1 -1e+235)
     t_2
     (if (<= t_1 1e+38) (/ (fma a t (* x z)) (+ x t)) t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / (y + (x + t));
	double t_2 = (z + a) - b;
	double tmp;
	if (t_1 <= -1e+235) {
		tmp = t_2;
	} else if (t_1 <= 1e+38) {
		tmp = fma(a, t, (x * z)) / (x + t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + Float64(a * Float64(y + t))) - Float64(y * b)) / Float64(y + Float64(x + t)))
	t_2 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (t_1 <= -1e+235)
		tmp = t_2;
	elseif (t_1 <= 1e+38)
		tmp = Float64(fma(a, t, Float64(x * z)) / Float64(x + t));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+235], t$95$2, If[LessEqual[t$95$1, 1e+38], N[(N[(a * t + N[(x * z), $MachinePrecision]), $MachinePrecision] / N[(x + t), $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)) < -1.0000000000000001e235 or 9.99999999999999977e37 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 27.5%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -1.0000000000000001e235 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.99999999999999977e37

   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 69.3

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

   5. Applied rewrites 69.3%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -1 \cdot 10^{+235}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a, t, x \cdot z\right)}{x + t}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t}, a\right)\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;a - z \cdot \frac{\left(-x\right) - y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -4.2e+116)
   (fma y (/ (- z b) t) a)
   (if (<= t 2.85e+54) (fma a (/ y (+ x y)) z) (- a (* z (/ (- (- x) y) t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -4.2e+116) {
		tmp = fma(y, ((z - b) / t), a);
	} else if (t <= 2.85e+54) {
		tmp = fma(a, (y / (x + y)), z);
	} else {
		tmp = a - (z * ((-x - y) / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -4.2e+116)
		tmp = fma(y, Float64(Float64(z - b) / t), a);
	elseif (t <= 2.85e+54)
		tmp = fma(a, Float64(y / Float64(x + y)), z);
	else
		tmp = Float64(a - Float64(z * Float64(Float64(Float64(-x) - y) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -4.2e+116], N[(y * N[(N[(z - b), $MachinePrecision] / t), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[t, 2.85e+54], N[(a * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision], N[(a - N[(z * N[(N[((-x) - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.2000000000000002e116

   1. Initial program 54.4%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.9%

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

   if -4.2000000000000002e116 < t < 2.8499999999999998e54

   1. Initial program 67.7%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in a around -inf

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

   8. Applied rewrites 18.1%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 61.8%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 67.7%

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

   if 2.8499999999999998e54 < t

   1. Initial program 51.4%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 62.5%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t}, a\right)\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;a - z \cdot \frac{\left(-x\right) - y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - b}{t}, a\right)\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{y + t}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -4.2e+116)
   (fma y (/ (- z b) t) a)
   (if (<= t 3.1e+47) (fma a (/ y (+ x y)) z) (fma y (/ z (+ y t)) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -4.2e+116) {
		tmp = fma(y, ((z - b) / t), a);
	} else if (t <= 3.1e+47) {
		tmp = fma(a, (y / (x + y)), z);
	} else {
		tmp = fma(y, (z / (y + t)), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -4.2e+116)
		tmp = fma(y, Float64(Float64(z - b) / t), a);
	elseif (t <= 3.1e+47)
		tmp = fma(a, Float64(y / Float64(x + y)), z);
	else
		tmp = fma(y, Float64(z / Float64(y + t)), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -4.2e+116], N[(y * N[(N[(z - b), $MachinePrecision] / t), $MachinePrecision] + a), $MachinePrecision], If[LessEqual[t, 3.1e+47], N[(a * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision], N[(y * N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.2000000000000002e116

   1. Initial program 54.4%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 71.9%

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

   if -4.2000000000000002e116 < t < 3.1000000000000001e47

   1. Initial program 67.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in a around -inf

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

   8. Applied rewrites 18.2%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 61.5%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 67.5%

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

   if 3.1000000000000001e47 < t

   1. Initial program 52.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in a around -inf

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

   8. Applied rewrites 25.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 71.3%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 60.8%

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

Alternative 8: 61.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+116}:\\ \;\;\;\;a - \frac{y \cdot b}{t}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{y + t}, a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -7.5e+116)
   (- a (/ (* y b) t))
   (if (<= t 3.1e+47) (fma a (/ y (+ x y)) z) (fma y (/ z (+ y t)) a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -7.5e+116) {
		tmp = a - ((y * b) / t);
	} else if (t <= 3.1e+47) {
		tmp = fma(a, (y / (x + y)), z);
	} else {
		tmp = fma(y, (z / (y + t)), a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -7.5e+116)
		tmp = Float64(a - Float64(Float64(y * b) / t));
	elseif (t <= 3.1e+47)
		tmp = fma(a, Float64(y / Float64(x + y)), z);
	else
		tmp = fma(y, Float64(z / Float64(y + t)), a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -7.5e+116], N[(a - N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.1e+47], N[(a * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision], N[(y * N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.5e116

   1. Initial program 54.4%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in b around inf

      \[\leadsto a - \frac{b \cdot y}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.9%

      \[\leadsto a - \frac{y \cdot b}{t} \]

   if -7.5e116 < t < 3.1000000000000001e47

   1. Initial program 67.5%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 73.4%

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

   6. Taylor expanded in a around -inf

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

   8. Applied rewrites 18.2%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 61.5%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 67.5%

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

   if 3.1000000000000001e47 < t

   1. Initial program 52.3%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in a around -inf

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

   8. Applied rewrites 25.5%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 71.3%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 60.8%

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

Alternative 9: 59.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a - \frac{y \cdot b}{t}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+193}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{x + y}, z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- a (/ (* y b) t))))
   (if (<= t -7.5e+116)
     t_1
     (if (<= t 1.15e+193) (fma a (/ y (+ x y)) z) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a - ((y * b) / t);
	double tmp;
	if (t <= -7.5e+116) {
		tmp = t_1;
	} else if (t <= 1.15e+193) {
		tmp = fma(a, (y / (x + y)), z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a - Float64(Float64(y * b) / t))
	tmp = 0.0
	if (t <= -7.5e+116)
		tmp = t_1;
	elseif (t <= 1.15e+193)
		tmp = fma(a, Float64(y / Float64(x + y)), z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a - N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.5e+116], t$95$1, If[LessEqual[t, 1.15e+193], N[(a * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] + z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.5e116 or 1.15000000000000007e193 < t

   1. Initial program 49.6%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in b around inf

      \[\leadsto a - \frac{b \cdot y}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.6%

      \[\leadsto a - \frac{y \cdot b}{t} \]

   if -7.5e116 < t < 1.15000000000000007e193

   1. Initial program 66.9%

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

   3. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in a around -inf

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

   8. Applied rewrites 19.2%

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

   9. Taylor expanded in b around 0

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

   11. Applied rewrites 62.3%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 63.2%

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

Alternative 10: 59.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a - \frac{y \cdot b}{t}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+116}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+185}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- a (/ (* y b) t))))
   (if (<= t -7.5e+116) t_1 (if (<= t 2.2e+185) (- (+ z a) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a - ((y * b) / t);
	double tmp;
	if (t <= -7.5e+116) {
		tmp = t_1;
	} else if (t <= 2.2e+185) {
		tmp = (z + a) - 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 = a - ((y * b) / t)
    if (t <= (-7.5d+116)) then
        tmp = t_1
    else if (t <= 2.2d+185) then
        tmp = (z + a) - 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 = a - ((y * b) / t);
	double tmp;
	if (t <= -7.5e+116) {
		tmp = t_1;
	} else if (t <= 2.2e+185) {
		tmp = (z + a) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a - ((y * b) / t)
	tmp = 0
	if t <= -7.5e+116:
		tmp = t_1
	elif t <= 2.2e+185:
		tmp = (z + a) - b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a - Float64(Float64(y * b) / t))
	tmp = 0.0
	if (t <= -7.5e+116)
		tmp = t_1;
	elseif (t <= 2.2e+185)
		tmp = Float64(Float64(z + a) - b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a - ((y * b) / t);
	tmp = 0.0;
	if (t <= -7.5e+116)
		tmp = t_1;
	elseif (t <= 2.2e+185)
		tmp = (z + a) - b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a - N[(N[(y * b), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.5e+116], t$95$1, If[LessEqual[t, 2.2e+185], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.5e116 or 2.2000000000000001e185 < t

   1. Initial program 49.6%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 58.2%

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

   6. Taylor expanded in b around inf

      \[\leadsto a - \frac{b \cdot y}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.6%

      \[\leadsto a - \frac{y \cdot b}{t} \]

   if -7.5e116 < t < 2.2000000000000001e185

   1. Initial program 66.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+116}:\\ \;\;\;\;a - \frac{y \cdot b}{t}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+185}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;a - \frac{y \cdot b}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a - a \cdot \frac{x}{t}\\ \mathbf{if}\;t \leq -5.3 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- a (* a (/ x t)))))
   (if (<= t -5.3e+119) t_1 (if (<= t 5e+193) (- (+ z a) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a - (a * (x / t));
	double tmp;
	if (t <= -5.3e+119) {
		tmp = t_1;
	} else if (t <= 5e+193) {
		tmp = (z + a) - 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 = a - (a * (x / t))
    if (t <= (-5.3d+119)) then
        tmp = t_1
    else if (t <= 5d+193) then
        tmp = (z + a) - 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 = a - (a * (x / t));
	double tmp;
	if (t <= -5.3e+119) {
		tmp = t_1;
	} else if (t <= 5e+193) {
		tmp = (z + a) - b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a - (a * (x / t))
	tmp = 0
	if t <= -5.3e+119:
		tmp = t_1
	elif t <= 5e+193:
		tmp = (z + a) - b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a - Float64(a * Float64(x / t)))
	tmp = 0.0
	if (t <= -5.3e+119)
		tmp = t_1;
	elseif (t <= 5e+193)
		tmp = Float64(Float64(z + a) - b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a - (a * (x / t));
	tmp = 0.0;
	if (t <= -5.3e+119)
		tmp = t_1;
	elseif (t <= 5e+193)
		tmp = (z + a) - b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a - N[(a * N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.3e+119], t$95$1, If[LessEqual[t, 5e+193], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.29999999999999972e119 or 4.99999999999999972e193 < t

   1. Initial program 49.0%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 53.0%

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

   if -5.29999999999999972e119 < t < 4.99999999999999972e193

   1. Initial program 67.1%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 60.5

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

   5. Applied rewrites 60.5%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.3 \cdot 10^{+119}:\\ \;\;\;\;a - a \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+193}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;a - a \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+201}:\\ \;\;\;\;z \cdot 1\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+164}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;z \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -1.4e+201) (* z 1.0) (if (<= x 1.05e+164) (- (+ z a) b) (* z 1.0))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1.4e+201) {
		tmp = z * 1.0;
	} else if (x <= 1.05e+164) {
		tmp = (z + a) - b;
	} else {
		tmp = z * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-1.4d+201)) then
        tmp = z * 1.0d0
    else if (x <= 1.05d+164) then
        tmp = (z + a) - b
    else
        tmp = z * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -1.4e+201) {
		tmp = z * 1.0;
	} else if (x <= 1.05e+164) {
		tmp = (z + a) - b;
	} else {
		tmp = z * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -1.4e+201:
		tmp = z * 1.0
	elif x <= 1.05e+164:
		tmp = (z + a) - b
	else:
		tmp = z * 1.0
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -1.4e+201)
		tmp = Float64(z * 1.0);
	elseif (x <= 1.05e+164)
		tmp = Float64(Float64(z + a) - b);
	else
		tmp = Float64(z * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -1.4e+201)
		tmp = z * 1.0;
	elseif (x <= 1.05e+164)
		tmp = (z + a) - b;
	else
		tmp = z * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -1.4e+201], N[(z * 1.0), $MachinePrecision], If[LessEqual[x, 1.05e+164], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], N[(z * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.40000000000000003e201 or 1.04999999999999995e164 < x

   1. Initial program 51.2%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

      2. lower-+.f64 N/A

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

      3. lower--.f64 N/A

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

      4. associate-+r+ N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

   5. Applied rewrites 10.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 28.7%

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

   9. Taylor expanded in t around 0

      \[\leadsto z \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 60.0%

       \[\leadsto z \cdot 1 \]

   if -1.40000000000000003e201 < x < 1.04999999999999995e164

   1. Initial program 64.3%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 55.0

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

   5. Applied rewrites 55.0%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+201}:\\ \;\;\;\;z \cdot 1\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+164}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;z \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+94}:\\ \;\;\;\;z + a\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-55}:\\ \;\;\;\;z - b\\ \mathbf{else}:\\ \;\;\;\;z + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= a -3.4e+94) (+ z a) (if (<= a 1.35e-55) (- z b) (+ z a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -3.4e+94) {
		tmp = z + a;
	} else if (a <= 1.35e-55) {
		tmp = z - b;
	} else {
		tmp = z + a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-3.4d+94)) then
        tmp = z + a
    else if (a <= 1.35d-55) then
        tmp = z - b
    else
        tmp = z + a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (a <= -3.4e+94) {
		tmp = z + a;
	} else if (a <= 1.35e-55) {
		tmp = z - b;
	} else {
		tmp = z + a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if a <= -3.4e+94:
		tmp = z + a
	elif a <= 1.35e-55:
		tmp = z - b
	else:
		tmp = z + a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (a <= -3.4e+94)
		tmp = Float64(z + a);
	elseif (a <= 1.35e-55)
		tmp = Float64(z - b);
	else
		tmp = Float64(z + a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (a <= -3.4e+94)
		tmp = z + a;
	elseif (a <= 1.35e-55)
		tmp = z - b;
	else
		tmp = z + a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[a, -3.4e+94], N[(z + a), $MachinePrecision], If[LessEqual[a, 1.35e-55], N[(z - b), $MachinePrecision], N[(z + a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4000000000000002e94 or 1.35000000000000002e-55 < a

   1. Initial program 54.1%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 58.6%

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

   if -3.4000000000000002e94 < a < 1.35000000000000002e-55

   1. Initial program 68.3%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 49.9

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

   5. Applied rewrites 49.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 49.1%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+94}:\\ \;\;\;\;z + a\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{-55}:\\ \;\;\;\;z - b\\ \mathbf{else}:\\ \;\;\;\;z + a\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.15 \cdot 10^{-124}:\\ \;\;\;\;z + a\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-129}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;z + a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.15e-124) (+ z a) (if (<= z 1.05e-129) (- a b) (+ z a))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.15e-124) {
		tmp = z + a;
	} else if (z <= 1.05e-129) {
		tmp = a - b;
	} else {
		tmp = z + a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-3.15d-124)) then
        tmp = z + a
    else if (z <= 1.05d-129) then
        tmp = a - b
    else
        tmp = z + a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.15e-124) {
		tmp = z + a;
	} else if (z <= 1.05e-129) {
		tmp = a - b;
	} else {
		tmp = z + a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -3.15e-124:
		tmp = z + a
	elif z <= 1.05e-129:
		tmp = a - b
	else:
		tmp = z + a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.15e-124)
		tmp = Float64(z + a);
	elseif (z <= 1.05e-129)
		tmp = Float64(a - b);
	else
		tmp = Float64(z + a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -3.15e-124)
		tmp = z + a;
	elseif (z <= 1.05e-129)
		tmp = a - b;
	else
		tmp = z + a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.15e-124], N[(z + a), $MachinePrecision], If[LessEqual[z, 1.05e-129], N[(a - b), $MachinePrecision], N[(z + a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.15000000000000014e-124 or 1.05e-129 < z

   1. Initial program 59.9%

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

   3. Taylor expanded in y around inf

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

      1. lower--.f64 N/A

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

      2. lower-+.f64 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 51.6%

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

   if -3.15000000000000014e-124 < z < 1.05e-129

   1. Initial program 67.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 53.0%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.15 \cdot 10^{-124}:\\ \;\;\;\;z + a\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-129}:\\ \;\;\;\;a - b\\ \mathbf{else}:\\ \;\;\;\;z + a\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.4% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.9%

   \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
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 51.6

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

5. Applied rewrites 51.6%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 47.7%

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

9. Final simplification 47.7%

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

Alternative 16: 13.3% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.9%

   \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
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 51.6

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

5. Applied rewrites 51.6%

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

6. Taylor expanded in b around inf

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

8. Applied rewrites 12.6%

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

Developer Target 1: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\\ t_3 := \frac{t\_2}{t\_1}\\ t_4 := \left(z + a\right) - b\\ \mathbf{if}\;t\_3 < -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 < 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{t\_1}{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))
        (t_3 (/ t_2 t_1))
        (t_4 (- (+ z a) b)))
   (if (< t_3 -3.5813117084150564e+153)
     t_4
     (if (< t_3 1.2285964308315609e+82) (/ 1.0 (/ t_1 t_2)) t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	double t_3 = t_2 / t_1;
	double t_4 = (z + a) - b;
	double tmp;
	if (t_3 < -3.5813117084150564e+153) {
		tmp = t_4;
	} else if (t_3 < 1.2285964308315609e+82) {
		tmp = 1.0 / (t_1 / t_2);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (x + t) + y
    t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b)
    t_3 = t_2 / t_1
    t_4 = (z + a) - b
    if (t_3 < (-3.5813117084150564d+153)) then
        tmp = t_4
    else if (t_3 < 1.2285964308315609d+82) then
        tmp = 1.0d0 / (t_1 / t_2)
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	double t_3 = t_2 / t_1;
	double t_4 = (z + a) - b;
	double tmp;
	if (t_3 < -3.5813117084150564e+153) {
		tmp = t_4;
	} else if (t_3 < 1.2285964308315609e+82) {
		tmp = 1.0 / (t_1 / t_2);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + t) + y
	t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b)
	t_3 = t_2 / t_1
	t_4 = (z + a) - b
	tmp = 0
	if t_3 < -3.5813117084150564e+153:
		tmp = t_4
	elif t_3 < 1.2285964308315609e+82:
		tmp = 1.0 / (t_1 / t_2)
	else:
		tmp = t_4
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b))
	t_3 = Float64(t_2 / t_1)
	t_4 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (t_3 < -3.5813117084150564e+153)
		tmp = t_4;
	elseif (t_3 < 1.2285964308315609e+82)
		tmp = Float64(1.0 / Float64(t_1 / t_2));
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + t) + y;
	t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	t_3 = t_2 / t_1;
	t_4 = (z + a) - b;
	tmp = 0.0;
	if (t_3 < -3.5813117084150564e+153)
		tmp = t_4;
	elseif (t_3 < 1.2285964308315609e+82)
		tmp = 1.0 / (t_1 / t_2);
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[Less[t$95$3, -3.5813117084150564e+153], t$95$4, If[Less[t$95$3, 1.2285964308315609e+82], N[(1.0 / N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Reproduce

herbie shell --seed 2024226 
(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)))