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

Percentage Accurate: 61.0% → 96.2%

Time: 16.6s

Alternatives: 17

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.0% 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: 96.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ \mathsf{fma}\left(z, \frac{x}{t_1} + \frac{y}{t_1}, \frac{y}{\frac{t_1}{a - b}} + \frac{a}{\frac{t_1}{t}}\right) \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t))))
   (fma z (+ (/ x t_1) (/ y t_1)) (+ (/ y (/ t_1 (- a b))) (/ a (/ t_1 t))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	return fma(z, ((x / t_1) + (y / t_1)), ((y / (t_1 / (a - b))) + (a / (t_1 / t))));
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	return fma(z, Float64(Float64(x / t_1) + Float64(y / t_1)), Float64(Float64(y / Float64(t_1 / Float64(a - b))) + Float64(a / Float64(t_1 / t))))
end

Wolfram

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

Derivation

1. Initial program 57.0%

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

3. Simplified 57.4%

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

4. Taylor expanded in z around 0 70.1%

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

   1. *-commutative 70.1%

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

   2. fma-def 70.1%

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

   3. associate-/l* 83.3%

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

   4. associate-/l* 97.3%

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

6. Simplified 97.3%

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

7. Final simplification 97.3%

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

Alternative 2: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(y + t\right)\\ t_2 := y + \left(x + t\right)\\ t_3 := \frac{\left(z \cdot \left(x + y\right) + t_1\right) - y \cdot b}{t_2}\\ \mathbf{if}\;t_3 \leq -\infty \lor \neg \left(t_3 \leq 4 \cdot 10^{+202}\right):\\ \;\;\;\;z + \left(\frac{y}{\frac{t_2}{a - b}} + \frac{a}{\frac{t_2}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{x}{t_2} + \frac{y}{t_2}\right) + \frac{t_1 - y \cdot b}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (+ y t)))
        (t_2 (+ y (+ x t)))
        (t_3 (/ (- (+ (* z (+ x y)) t_1) (* y b)) t_2)))
   (if (or (<= t_3 (- INFINITY)) (not (<= t_3 4e+202)))
     (+ z (+ (/ y (/ t_2 (- a b))) (/ a (/ t_2 t))))
     (+ (* z (+ (/ x t_2) (/ y t_2))) (/ (- t_1 (* y b)) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (y + t);
	double t_2 = y + (x + t);
	double t_3 = (((z * (x + y)) + t_1) - (y * b)) / t_2;
	double tmp;
	if ((t_3 <= -((double) INFINITY)) || !(t_3 <= 4e+202)) {
		tmp = z + ((y / (t_2 / (a - b))) + (a / (t_2 / t)));
	} else {
		tmp = (z * ((x / t_2) + (y / t_2))) + ((t_1 - (y * b)) / t_2);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (y + t);
	double t_2 = y + (x + t);
	double t_3 = (((z * (x + y)) + t_1) - (y * b)) / t_2;
	double tmp;
	if ((t_3 <= -Double.POSITIVE_INFINITY) || !(t_3 <= 4e+202)) {
		tmp = z + ((y / (t_2 / (a - b))) + (a / (t_2 / t)));
	} else {
		tmp = (z * ((x / t_2) + (y / t_2))) + ((t_1 - (y * b)) / t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (y + t)
	t_2 = y + (x + t)
	t_3 = (((z * (x + y)) + t_1) - (y * b)) / t_2
	tmp = 0
	if (t_3 <= -math.inf) or not (t_3 <= 4e+202):
		tmp = z + ((y / (t_2 / (a - b))) + (a / (t_2 / t)))
	else:
		tmp = (z * ((x / t_2) + (y / t_2))) + ((t_1 - (y * b)) / t_2)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(y + t))
	t_2 = Float64(y + Float64(x + t))
	t_3 = Float64(Float64(Float64(Float64(z * Float64(x + y)) + t_1) - Float64(y * b)) / t_2)
	tmp = 0.0
	if ((t_3 <= Float64(-Inf)) || !(t_3 <= 4e+202))
		tmp = Float64(z + Float64(Float64(y / Float64(t_2 / Float64(a - b))) + Float64(a / Float64(t_2 / t))));
	else
		tmp = Float64(Float64(z * Float64(Float64(x / t_2) + Float64(y / t_2))) + Float64(Float64(t_1 - Float64(y * b)) / t_2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (y + t);
	t_2 = y + (x + t);
	t_3 = (((z * (x + y)) + t_1) - (y * b)) / t_2;
	tmp = 0.0;
	if ((t_3 <= -Inf) || ~((t_3 <= 4e+202)))
		tmp = z + ((y / (t_2 / (a - b))) + (a / (t_2 / t)));
	else
		tmp = (z * ((x / t_2) + (y / t_2))) + ((t_1 - (y * b)) / t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, If[Or[LessEqual[t$95$3, (-Infinity)], N[Not[LessEqual[t$95$3, 4e+202]], $MachinePrecision]], N[(z + N[(N[(y / N[(t$95$2 / N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(t$95$2 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(N[(x / t$95$2), $MachinePrecision] + N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 - N[(y * b), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]

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 3.9999999999999996e202 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 9.2%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in z around inf 9.1%

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

      1. *-commutative 9.1%

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

      2. associate-/l* 33.2%

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

      3. associate-/l* 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in y around inf 92.8%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 3.9999999999999996e202

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.8%

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

      1. associate--l+ 99.8%

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

      2. *-commutative 99.8%

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

      3. div-sub 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 4 \cdot 10^{+202}\right):\\ \;\;\;\;z + \left(\frac{y}{\frac{y + \left(x + t\right)}{a - b}} + \frac{a}{\frac{y + \left(x + t\right)}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(\frac{x}{y + \left(x + t\right)} + \frac{y}{y + \left(x + t\right)}\right) + \frac{a \cdot \left(y + t\right) - y \cdot b}{y + \left(x + t\right)}\\ \end{array} \]

Alternative 3: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ t_2 := \frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{t_1}\\ \mathbf{if}\;t_2 \leq -\infty \lor \neg \left(t_2 \leq 4 \cdot 10^{+202}\right):\\ \;\;\;\;z + \left(\frac{y}{\frac{t_1}{a - b}} + \frac{a}{\frac{t_1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

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)))
   (if (or (<= t_2 (- INFINITY)) (not (<= t_2 4e+202)))
     (+ z (+ (/ y (/ t_1 (- a b))) (/ a (/ t_1 t))))
     t_2)))

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 tmp;
	if ((t_2 <= -((double) INFINITY)) || !(t_2 <= 4e+202)) {
		tmp = z + ((y / (t_1 / (a - b))) + (a / (t_1 / t)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static 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 tmp;
	if ((t_2 <= -Double.POSITIVE_INFINITY) || !(t_2 <= 4e+202)) {
		tmp = z + ((y / (t_1 / (a - b))) + (a / (t_1 / t)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (x + t)
	t_2 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / t_1
	tmp = 0
	if (t_2 <= -math.inf) or not (t_2 <= 4e+202):
		tmp = z + ((y / (t_1 / (a - b))) + (a / (t_1 / t)))
	else:
		tmp = t_2
	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)
	tmp = 0.0
	if ((t_2 <= Float64(-Inf)) || !(t_2 <= 4e+202))
		tmp = Float64(z + Float64(Float64(y / Float64(t_1 / Float64(a - b))) + Float64(a / Float64(t_1 / t))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (x + t);
	t_2 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / t_1;
	tmp = 0.0;
	if ((t_2 <= -Inf) || ~((t_2 <= 4e+202)))
		tmp = z + ((y / (t_1 / (a - b))) + (a / (t_1 / t)));
	else
		tmp = t_2;
	end
	tmp_2 = 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]}, If[Or[LessEqual[t$95$2, (-Infinity)], N[Not[LessEqual[t$95$2, 4e+202]], $MachinePrecision]], N[(z + N[(N[(y / N[(t$95$1 / N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0 or 3.9999999999999996e202 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 9.2%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in z around inf 9.1%

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

      1. *-commutative 9.1%

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

      2. associate-/l* 33.2%

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

      3. associate-/l* 56.0%

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

   6. Simplified 56.0%

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

   7. Taylor expanded in y around inf 92.8%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 3.9999999999999996e202

   1. Initial program 99.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 4 \cdot 10^{+202}\right):\\ \;\;\;\;z + \left(\frac{y}{\frac{y + \left(x + t\right)}{a - b}} + \frac{a}{\frac{y + \left(x + t\right)}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)}\\ \end{array} \]

Alternative 4: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 4 \cdot 10^{+202}\right):\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}} + \frac{y + t}{\frac{y + t}{a}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* z (+ x y)) (* a (+ y t))) (* y b)) (+ y (+ x t)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 4e+202)))
     (+ (/ (- z b) (/ (+ y t) y)) (/ (+ y t) (/ (+ y t) a)))
     t_1)))

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 tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 4e+202)) {
		tmp = ((z - b) / ((y + t) / y)) + ((y + t) / ((y + t) / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static 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 tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 4e+202)) {
		tmp = ((z - b) / ((y + t) / y)) + ((y + t) / ((y + t) / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / (y + (x + t))
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 4e+202):
		tmp = ((z - b) / ((y + t) / y)) + ((y + t) / ((y + t) / a))
	else:
		tmp = t_1
	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)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 4e+202))
		tmp = Float64(Float64(Float64(z - b) / Float64(Float64(y + t) / y)) + Float64(Float64(y + t) / Float64(Float64(y + t) / a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (((z * (x + y)) + (a * (y + t))) - (y * b)) / (y + (x + t));
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 4e+202)))
		tmp = ((z - b) / ((y + t) / y)) + ((y + t) / ((y + t) / a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision] + N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 4e+202]], $MachinePrecision]], N[(N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(N[(y + t), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]

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 3.9999999999999996e202 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 9.2%

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

   3. Simplified 10.1%

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

   4. Taylor expanded in x around 0 16.5%

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

   5. Taylor expanded in a around -inf 16.3%

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

      1. mul-1-neg 16.3%

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

      2. unsub-neg 16.3%

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

      3. associate-/l* 40.6%

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

      4. associate-/l* 76.5%

         \[\leadsto \frac{z - b}{\frac{y + t}{y}} - \color{blue}{\frac{-1 \cdot t + -1 \cdot y}{\frac{y + t}{a}}} \]

      5. mul-1-neg 76.5%

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

      6. unsub-neg 76.5%

         \[\leadsto \frac{z - b}{\frac{y + t}{y}} - \frac{\color{blue}{-1 \cdot t - y}}{\frac{y + t}{a}} \]

      7. mul-1-neg 76.5%

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

   7. Simplified 76.5%

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

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 3.9999999999999996e202

   1. Initial program 99.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq -\infty \lor \neg \left(\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)} \leq 4 \cdot 10^{+202}\right):\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}} + \frac{y + t}{\frac{y + t}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(z \cdot \left(x + y\right) + a \cdot \left(y + t\right)\right) - y \cdot b}{y + \left(x + t\right)}\\ \end{array} \]

Alternative 5: 95.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(x + t\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{+64} \lor \neg \left(x \leq 9 \cdot 10^{+55}\right):\\ \;\;\;\;z + \left(\frac{y}{\frac{t_1}{a - b}} + \frac{a}{\frac{t_1}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - b}{\frac{t_1}{y}} + \left(\frac{a}{\frac{t_1}{y + t}} + \frac{z \cdot x}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ x t))))
   (if (or (<= x -4e+64) (not (<= x 9e+55)))
     (+ z (+ (/ y (/ t_1 (- a b))) (/ a (/ t_1 t))))
     (+ (/ (- z b) (/ t_1 y)) (+ (/ a (/ t_1 (+ y t))) (/ (* z x) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (x + t);
	double tmp;
	if ((x <= -4e+64) || !(x <= 9e+55)) {
		tmp = z + ((y / (t_1 / (a - b))) + (a / (t_1 / t)));
	} else {
		tmp = ((z - b) / (t_1 / y)) + ((a / (t_1 / (y + t))) + ((z * x) / 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 = y + (x + t)
    if ((x <= (-4d+64)) .or. (.not. (x <= 9d+55))) then
        tmp = z + ((y / (t_1 / (a - b))) + (a / (t_1 / t)))
    else
        tmp = ((z - b) / (t_1 / y)) + ((a / (t_1 / (y + t))) + ((z * x) / 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 = y + (x + t);
	double tmp;
	if ((x <= -4e+64) || !(x <= 9e+55)) {
		tmp = z + ((y / (t_1 / (a - b))) + (a / (t_1 / t)));
	} else {
		tmp = ((z - b) / (t_1 / y)) + ((a / (t_1 / (y + t))) + ((z * x) / t_1));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (x + t)
	tmp = 0
	if (x <= -4e+64) or not (x <= 9e+55):
		tmp = z + ((y / (t_1 / (a - b))) + (a / (t_1 / t)))
	else:
		tmp = ((z - b) / (t_1 / y)) + ((a / (t_1 / (y + t))) + ((z * x) / t_1))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(x + t))
	tmp = 0.0
	if ((x <= -4e+64) || !(x <= 9e+55))
		tmp = Float64(z + Float64(Float64(y / Float64(t_1 / Float64(a - b))) + Float64(a / Float64(t_1 / t))));
	else
		tmp = Float64(Float64(Float64(z - b) / Float64(t_1 / y)) + Float64(Float64(a / Float64(t_1 / Float64(y + t))) + Float64(Float64(z * x) / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (x + t);
	tmp = 0.0;
	if ((x <= -4e+64) || ~((x <= 9e+55)))
		tmp = z + ((y / (t_1 / (a - b))) + (a / (t_1 / t)));
	else
		tmp = ((z - b) / (t_1 / y)) + ((a / (t_1 / (y + t))) + ((z * x) / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, -4e+64], N[Not[LessEqual[x, 9e+55]], $MachinePrecision]], N[(z + N[(N[(y / N[(t$95$1 / N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a / N[(t$95$1 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z - b), $MachinePrecision] / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] + N[(N[(a / N[(t$95$1 / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(z * x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000009e64 or 8.99999999999999996e55 < x

   1. Initial program 44.3%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in z around inf 44.3%

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

      1. *-commutative 44.3%

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

      2. associate-/l* 57.7%

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

      3. associate-/l* 67.5%

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

   6. Simplified 67.5%

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

   7. Taylor expanded in y around inf 93.9%

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

   if -4.00000000000000009e64 < x < 8.99999999999999996e55

   1. Initial program 64.9%

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

   3. Simplified 65.4%

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

   4. Taylor expanded in a around inf 65.2%

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

      1. associate-/l* 78.5%

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

      2. +-commutative 78.5%

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

      3. associate-/l* 98.1%

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

   6. Simplified 98.1%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+64} \lor \neg \left(x \leq 9 \cdot 10^{+55}\right):\\ \;\;\;\;z + \left(\frac{y}{\frac{y + \left(x + t\right)}{a - b}} + \frac{a}{\frac{y + \left(x + t\right)}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - b}{\frac{y + \left(x + t\right)}{y}} + \left(\frac{a}{\frac{y + \left(x + t\right)}{y + t}} + \frac{z \cdot x}{y + \left(x + t\right)}\right)\\ \end{array} \]

Alternative 6: 58.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + t\right) \cdot \frac{a}{t + \left(x + y\right)}\\ t_2 := \frac{z - b}{\frac{y + t}{y}}\\ t_3 := \left(z + a\right) - b\\ t_4 := z \cdot \frac{x + y}{y + \left(x + t\right)}\\ \mathbf{if}\;a \leq -7.8 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.2 \cdot 10^{+71}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -700000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -7.6 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -8.8 \cdot 10^{-151}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-242}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-64}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+185}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ y t) (/ a (+ t (+ x y)))))
        (t_2 (/ (- z b) (/ (+ y t) y)))
        (t_3 (- (+ z a) b))
        (t_4 (* z (/ (+ x y) (+ y (+ x t))))))
   (if (<= a -7.8e+152)
     t_1
     (if (<= a -3.2e+71)
       t_3
       (if (<= a -700000.0)
         t_1
         (if (<= a -7.6e-62)
           t_2
           (if (<= a -8.8e-151)
             t_4
             (if (<= a 1.5e-242)
               t_2
               (if (<= a 5.2e-64) t_4 (if (<= a 8e+185) t_3 t_1))))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (y + t) * (a / (t + (x + y)));
	double t_2 = (z - b) / ((y + t) / y);
	double t_3 = (z + a) - b;
	double t_4 = z * ((x + y) / (y + (x + t)));
	double tmp;
	if (a <= -7.8e+152) {
		tmp = t_1;
	} else if (a <= -3.2e+71) {
		tmp = t_3;
	} else if (a <= -700000.0) {
		tmp = t_1;
	} else if (a <= -7.6e-62) {
		tmp = t_2;
	} else if (a <= -8.8e-151) {
		tmp = t_4;
	} else if (a <= 1.5e-242) {
		tmp = t_2;
	} else if (a <= 5.2e-64) {
		tmp = t_4;
	} else if (a <= 8e+185) {
		tmp = t_3;
	} 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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (y + t) * (a / (t + (x + y)))
    t_2 = (z - b) / ((y + t) / y)
    t_3 = (z + a) - b
    t_4 = z * ((x + y) / (y + (x + t)))
    if (a <= (-7.8d+152)) then
        tmp = t_1
    else if (a <= (-3.2d+71)) then
        tmp = t_3
    else if (a <= (-700000.0d0)) then
        tmp = t_1
    else if (a <= (-7.6d-62)) then
        tmp = t_2
    else if (a <= (-8.8d-151)) then
        tmp = t_4
    else if (a <= 1.5d-242) then
        tmp = t_2
    else if (a <= 5.2d-64) then
        tmp = t_4
    else if (a <= 8d+185) then
        tmp = t_3
    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 = (y + t) * (a / (t + (x + y)));
	double t_2 = (z - b) / ((y + t) / y);
	double t_3 = (z + a) - b;
	double t_4 = z * ((x + y) / (y + (x + t)));
	double tmp;
	if (a <= -7.8e+152) {
		tmp = t_1;
	} else if (a <= -3.2e+71) {
		tmp = t_3;
	} else if (a <= -700000.0) {
		tmp = t_1;
	} else if (a <= -7.6e-62) {
		tmp = t_2;
	} else if (a <= -8.8e-151) {
		tmp = t_4;
	} else if (a <= 1.5e-242) {
		tmp = t_2;
	} else if (a <= 5.2e-64) {
		tmp = t_4;
	} else if (a <= 8e+185) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (y + t) * (a / (t + (x + y)))
	t_2 = (z - b) / ((y + t) / y)
	t_3 = (z + a) - b
	t_4 = z * ((x + y) / (y + (x + t)))
	tmp = 0
	if a <= -7.8e+152:
		tmp = t_1
	elif a <= -3.2e+71:
		tmp = t_3
	elif a <= -700000.0:
		tmp = t_1
	elif a <= -7.6e-62:
		tmp = t_2
	elif a <= -8.8e-151:
		tmp = t_4
	elif a <= 1.5e-242:
		tmp = t_2
	elif a <= 5.2e-64:
		tmp = t_4
	elif a <= 8e+185:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(y + t) * Float64(a / Float64(t + Float64(x + y))))
	t_2 = Float64(Float64(z - b) / Float64(Float64(y + t) / y))
	t_3 = Float64(Float64(z + a) - b)
	t_4 = Float64(z * Float64(Float64(x + y) / Float64(y + Float64(x + t))))
	tmp = 0.0
	if (a <= -7.8e+152)
		tmp = t_1;
	elseif (a <= -3.2e+71)
		tmp = t_3;
	elseif (a <= -700000.0)
		tmp = t_1;
	elseif (a <= -7.6e-62)
		tmp = t_2;
	elseif (a <= -8.8e-151)
		tmp = t_4;
	elseif (a <= 1.5e-242)
		tmp = t_2;
	elseif (a <= 5.2e-64)
		tmp = t_4;
	elseif (a <= 8e+185)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (y + t) * (a / (t + (x + y)));
	t_2 = (z - b) / ((y + t) / y);
	t_3 = (z + a) - b;
	t_4 = z * ((x + y) / (y + (x + t)));
	tmp = 0.0;
	if (a <= -7.8e+152)
		tmp = t_1;
	elseif (a <= -3.2e+71)
		tmp = t_3;
	elseif (a <= -700000.0)
		tmp = t_1;
	elseif (a <= -7.6e-62)
		tmp = t_2;
	elseif (a <= -8.8e-151)
		tmp = t_4;
	elseif (a <= 1.5e-242)
		tmp = t_2;
	elseif (a <= 5.2e-64)
		tmp = t_4;
	elseif (a <= 8e+185)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(y + t), $MachinePrecision] * N[(a / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(x + y), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -7.8e+152], t$95$1, If[LessEqual[a, -3.2e+71], t$95$3, If[LessEqual[a, -700000.0], t$95$1, If[LessEqual[a, -7.6e-62], t$95$2, If[LessEqual[a, -8.8e-151], t$95$4, If[LessEqual[a, 1.5e-242], t$95$2, If[LessEqual[a, 5.2e-64], t$95$4, If[LessEqual[a, 8e+185], t$95$3, t$95$1]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -7.80000000000000022e152 or -3.20000000000000023e71 < a < -7e5 or 7.9999999999999998e185 < a

   1. Initial program 42.8%

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

   2. Taylor expanded in a around inf 38.2%

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

      1. expm1-log1p-u 15.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right)\right)} \]

      2. expm1-udef 10.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right)} - 1} \]

      3. associate-/l* 29.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}}\right)} - 1 \]

      4. +-commutative 29.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\frac{y + \color{blue}{\left(x + t\right)}}{y + t}}\right)} - 1 \]

   4. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{\frac{y + \left(x + t\right)}{y + t}}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 34.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\frac{y + \left(x + t\right)}{y + t}}\right)\right)} \]

      2. expm1-log1p 84.8%

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

      3. associate-/r/ 83.6%

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

      4. *-commutative 83.6%

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

      5. *-rgt-identity 83.6%

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

      6. associate-*r/ 83.4%

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

      7. associate-*r/ 83.6%

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

      8. *-rgt-identity 83.6%

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

      9. associate-+r+ 83.6%

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

   6. Simplified 83.6%

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

   if -7.80000000000000022e152 < a < -3.20000000000000023e71 or 5.2e-64 < a < 7.9999999999999998e185

   1. Initial program 55.2%

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

   2. Taylor expanded in y around inf 68.4%

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

      1. +-commutative 68.4%

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

   4. Simplified 68.4%

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

   if -7e5 < a < -7.60000000000000013e-62 or -8.7999999999999997e-151 < a < 1.5e-242

   1. Initial program 68.2%

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

   3. Simplified 68.7%

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

   4. Taylor expanded in x around 0 54.9%

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

   5. Taylor expanded in a around 0 46.9%

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

      1. associate-/l* 69.3%

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

   7. Simplified 69.3%

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

   if -7.60000000000000013e-62 < a < -8.7999999999999997e-151 or 1.5e-242 < a < 5.2e-64

   1. Initial program 70.0%

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

   2. Taylor expanded in z around inf 39.9%

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

      1. *-un-lft-identity 39.9%

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

      2. associate-/l* 60.5%

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

   4. Applied egg-rr 60.5%

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

      1. *-lft-identity 60.5%

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

      2. associate-/r/ 63.7%

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

   6. Simplified 63.7%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{+152}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{a}{t + \left(x + y\right)}\\ \mathbf{elif}\;a \leq -3.2 \cdot 10^{+71}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;a \leq -700000:\\ \;\;\;\;\left(y + t\right) \cdot \frac{a}{t + \left(x + y\right)}\\ \mathbf{elif}\;a \leq -7.6 \cdot 10^{-62}:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;a \leq -8.8 \cdot 10^{-151}:\\ \;\;\;\;z \cdot \frac{x + y}{y + \left(x + t\right)}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-242}:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-64}:\\ \;\;\;\;z \cdot \frac{x + y}{y + \left(x + t\right)}\\ \mathbf{elif}\;a \leq 8 \cdot 10^{+185}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{a}{t + \left(x + y\right)}\\ \end{array} \]

Alternative 7: 63.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + a\right) - b\\ t_2 := \frac{t \cdot a + y \cdot t_1}{y + t}\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{+14}:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{z \cdot x - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-112}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{-157}:\\ \;\;\;\;\frac{z \cdot x + t \cdot a}{x + t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ z a) b)) (t_2 (/ (+ (* t a) (* y t_1)) (+ y t))))
   (if (<= y -3.3e+62)
     t_1
     (if (<= y -1.55e+14)
       (/ (- z b) (/ (+ y t) y))
       (if (<= y -1.15e-33)
         t_2
         (if (<= y -1.5e-55)
           (/ (- (* z x) (* y b)) (+ y (+ x t)))
           (if (<= y -4.1e-112)
             t_2
             (if (<= y 1.36e-157) (/ (+ (* z x) (* t a)) (+ x t)) t_1))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + a) - b;
	double t_2 = ((t * a) + (y * t_1)) / (y + t);
	double tmp;
	if (y <= -3.3e+62) {
		tmp = t_1;
	} else if (y <= -1.55e+14) {
		tmp = (z - b) / ((y + t) / y);
	} else if (y <= -1.15e-33) {
		tmp = t_2;
	} else if (y <= -1.5e-55) {
		tmp = ((z * x) - (y * b)) / (y + (x + t));
	} else if (y <= -4.1e-112) {
		tmp = t_2;
	} else if (y <= 1.36e-157) {
		tmp = ((z * x) + (t * a)) / (x + t);
	} 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) :: t_2
    real(8) :: tmp
    t_1 = (z + a) - b
    t_2 = ((t * a) + (y * t_1)) / (y + t)
    if (y <= (-3.3d+62)) then
        tmp = t_1
    else if (y <= (-1.55d+14)) then
        tmp = (z - b) / ((y + t) / y)
    else if (y <= (-1.15d-33)) then
        tmp = t_2
    else if (y <= (-1.5d-55)) then
        tmp = ((z * x) - (y * b)) / (y + (x + t))
    else if (y <= (-4.1d-112)) then
        tmp = t_2
    else if (y <= 1.36d-157) then
        tmp = ((z * x) + (t * a)) / (x + t)
    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 = (z + a) - b;
	double t_2 = ((t * a) + (y * t_1)) / (y + t);
	double tmp;
	if (y <= -3.3e+62) {
		tmp = t_1;
	} else if (y <= -1.55e+14) {
		tmp = (z - b) / ((y + t) / y);
	} else if (y <= -1.15e-33) {
		tmp = t_2;
	} else if (y <= -1.5e-55) {
		tmp = ((z * x) - (y * b)) / (y + (x + t));
	} else if (y <= -4.1e-112) {
		tmp = t_2;
	} else if (y <= 1.36e-157) {
		tmp = ((z * x) + (t * a)) / (x + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z + a) - b
	t_2 = ((t * a) + (y * t_1)) / (y + t)
	tmp = 0
	if y <= -3.3e+62:
		tmp = t_1
	elif y <= -1.55e+14:
		tmp = (z - b) / ((y + t) / y)
	elif y <= -1.15e-33:
		tmp = t_2
	elif y <= -1.5e-55:
		tmp = ((z * x) - (y * b)) / (y + (x + t))
	elif y <= -4.1e-112:
		tmp = t_2
	elif y <= 1.36e-157:
		tmp = ((z * x) + (t * a)) / (x + t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + a) - b)
	t_2 = Float64(Float64(Float64(t * a) + Float64(y * t_1)) / Float64(y + t))
	tmp = 0.0
	if (y <= -3.3e+62)
		tmp = t_1;
	elseif (y <= -1.55e+14)
		tmp = Float64(Float64(z - b) / Float64(Float64(y + t) / y));
	elseif (y <= -1.15e-33)
		tmp = t_2;
	elseif (y <= -1.5e-55)
		tmp = Float64(Float64(Float64(z * x) - Float64(y * b)) / Float64(y + Float64(x + t)));
	elseif (y <= -4.1e-112)
		tmp = t_2;
	elseif (y <= 1.36e-157)
		tmp = Float64(Float64(Float64(z * x) + Float64(t * a)) / Float64(x + t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z + a) - b;
	t_2 = ((t * a) + (y * t_1)) / (y + t);
	tmp = 0.0;
	if (y <= -3.3e+62)
		tmp = t_1;
	elseif (y <= -1.55e+14)
		tmp = (z - b) / ((y + t) / y);
	elseif (y <= -1.15e-33)
		tmp = t_2;
	elseif (y <= -1.5e-55)
		tmp = ((z * x) - (y * b)) / (y + (x + t));
	elseif (y <= -4.1e-112)
		tmp = t_2;
	elseif (y <= 1.36e-157)
		tmp = ((z * x) + (t * a)) / (x + t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t * a), $MachinePrecision] + N[(y * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.3e+62], t$95$1, If[LessEqual[y, -1.55e+14], N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.15e-33], t$95$2, If[LessEqual[y, -1.5e-55], N[(N[(N[(z * x), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4.1e-112], t$95$2, If[LessEqual[y, 1.36e-157], N[(N[(N[(z * x), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] / N[(x + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -3.3e62 or 1.36e-157 < y

   1. Initial program 44.7%

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

   2. Taylor expanded in y around inf 70.7%

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

      1. +-commutative 70.7%

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

   4. Simplified 70.7%

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

   if -3.3e62 < y < -1.55e14

   1. Initial program 52.8%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in x around 0 46.5%

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

   5. Taylor expanded in a around 0 47.9%

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

      1. associate-/l* 76.1%

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

   7. Simplified 76.1%

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

   if -1.55e14 < y < -1.14999999999999993e-33 or -1.50000000000000008e-55 < y < -4.09999999999999996e-112

   1. Initial program 75.2%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in x around 0 71.5%

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

   if -1.14999999999999993e-33 < y < -1.50000000000000008e-55

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 78.2%

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

   if -4.09999999999999996e-112 < y < 1.36e-157

   1. Initial program 77.6%

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

   2. Taylor expanded in y around 0 68.5%

      \[\leadsto \color{blue}{\frac{z \cdot x + a \cdot t}{t + x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+62}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{+14}:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;\frac{t \cdot a + y \cdot \left(\left(z + a\right) - b\right)}{y + t}\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{z \cdot x - y \cdot b}{y + \left(x + t\right)}\\ \mathbf{elif}\;y \leq -4.1 \cdot 10^{-112}:\\ \;\;\;\;\frac{t \cdot a + y \cdot \left(\left(z + a\right) - b\right)}{y + t}\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{-157}:\\ \;\;\;\;\frac{z \cdot x + t \cdot a}{x + t}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Alternative 8: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-90} \lor \neg \left(y \leq 2.82 \cdot 10^{-169}\right):\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}} + \frac{y + t}{\frac{y + t}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x + t \cdot a}{x + t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.1e-90) (not (<= y 2.82e-169)))
   (+ (/ (- z b) (/ (+ y t) y)) (/ (+ y t) (/ (+ y t) a)))
   (/ (+ (* z x) (* t a)) (+ x t))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.1e-90) || !(y <= 2.82e-169)) {
		tmp = ((z - b) / ((y + t) / y)) + ((y + t) / ((y + t) / a));
	} else {
		tmp = ((z * x) + (t * a)) / (x + t);
	}
	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 ((y <= (-1.1d-90)) .or. (.not. (y <= 2.82d-169))) then
        tmp = ((z - b) / ((y + t) / y)) + ((y + t) / ((y + t) / a))
    else
        tmp = ((z * x) + (t * a)) / (x + t)
    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 ((y <= -1.1e-90) || !(y <= 2.82e-169)) {
		tmp = ((z - b) / ((y + t) / y)) + ((y + t) / ((y + t) / a));
	} else {
		tmp = ((z * x) + (t * a)) / (x + t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.1e-90) or not (y <= 2.82e-169):
		tmp = ((z - b) / ((y + t) / y)) + ((y + t) / ((y + t) / a))
	else:
		tmp = ((z * x) + (t * a)) / (x + t)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.1e-90) || !(y <= 2.82e-169))
		tmp = Float64(Float64(Float64(z - b) / Float64(Float64(y + t) / y)) + Float64(Float64(y + t) / Float64(Float64(y + t) / a)));
	else
		tmp = Float64(Float64(Float64(z * x) + Float64(t * a)) / Float64(x + t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.1e-90) || ~((y <= 2.82e-169)))
		tmp = ((z - b) / ((y + t) / y)) + ((y + t) / ((y + t) / a));
	else
		tmp = ((z * x) + (t * a)) / (x + t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.1e-90], N[Not[LessEqual[y, 2.82e-169]], $MachinePrecision]], N[(N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(N[(y + t), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * x), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] / N[(x + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.09999999999999993e-90 or 2.8199999999999998e-169 < y

   1. Initial program 50.0%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in x around 0 42.6%

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

   5. Taylor expanded in a around -inf 42.5%

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

      1. mul-1-neg 42.5%

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

      2. unsub-neg 42.5%

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

      3. associate-/l* 57.8%

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

      4. associate-/l* 72.7%

         \[\leadsto \frac{z - b}{\frac{y + t}{y}} - \color{blue}{\frac{-1 \cdot t + -1 \cdot y}{\frac{y + t}{a}}} \]

      5. mul-1-neg 72.7%

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

      6. unsub-neg 72.7%

         \[\leadsto \frac{z - b}{\frac{y + t}{y}} - \frac{\color{blue}{-1 \cdot t - y}}{\frac{y + t}{a}} \]

      7. mul-1-neg 72.7%

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

   7. Simplified 72.7%

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

   if -1.09999999999999993e-90 < y < 2.8199999999999998e-169

   1. Initial program 78.0%

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

   2. Taylor expanded in y around 0 68.9%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-90} \lor \neg \left(y \leq 2.82 \cdot 10^{-169}\right):\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}} + \frac{y + t}{\frac{y + t}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x + t \cdot a}{x + t}\\ \end{array} \]

Alternative 9: 57.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + a\right) - b\\ t_2 := \left(y + t\right) \cdot \frac{a}{t + \left(x + y\right)}\\ \mathbf{if}\;a \leq -1.12 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{+70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -430000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{-300}:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ z a) b)) (t_2 (* (+ y t) (/ a (+ t (+ x y))))))
   (if (<= a -1.12e+153)
     t_2
     (if (<= a -1.45e+70)
       t_1
       (if (<= a -430000.0)
         t_2
         (if (<= a 2.25e-300)
           (/ (- z b) (/ (+ y t) y))
           (if (<= a 1.2e+185) t_1 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + a) - b;
	double t_2 = (y + t) * (a / (t + (x + y)));
	double tmp;
	if (a <= -1.12e+153) {
		tmp = t_2;
	} else if (a <= -1.45e+70) {
		tmp = t_1;
	} else if (a <= -430000.0) {
		tmp = t_2;
	} else if (a <= 2.25e-300) {
		tmp = (z - b) / ((y + t) / y);
	} else if (a <= 1.2e+185) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z + a) - b
    t_2 = (y + t) * (a / (t + (x + y)))
    if (a <= (-1.12d+153)) then
        tmp = t_2
    else if (a <= (-1.45d+70)) then
        tmp = t_1
    else if (a <= (-430000.0d0)) then
        tmp = t_2
    else if (a <= 2.25d-300) then
        tmp = (z - b) / ((y + t) / y)
    else if (a <= 1.2d+185) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + a) - b;
	double t_2 = (y + t) * (a / (t + (x + y)));
	double tmp;
	if (a <= -1.12e+153) {
		tmp = t_2;
	} else if (a <= -1.45e+70) {
		tmp = t_1;
	} else if (a <= -430000.0) {
		tmp = t_2;
	} else if (a <= 2.25e-300) {
		tmp = (z - b) / ((y + t) / y);
	} else if (a <= 1.2e+185) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z + a) - b
	t_2 = (y + t) * (a / (t + (x + y)))
	tmp = 0
	if a <= -1.12e+153:
		tmp = t_2
	elif a <= -1.45e+70:
		tmp = t_1
	elif a <= -430000.0:
		tmp = t_2
	elif a <= 2.25e-300:
		tmp = (z - b) / ((y + t) / y)
	elif a <= 1.2e+185:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + a) - b)
	t_2 = Float64(Float64(y + t) * Float64(a / Float64(t + Float64(x + y))))
	tmp = 0.0
	if (a <= -1.12e+153)
		tmp = t_2;
	elseif (a <= -1.45e+70)
		tmp = t_1;
	elseif (a <= -430000.0)
		tmp = t_2;
	elseif (a <= 2.25e-300)
		tmp = Float64(Float64(z - b) / Float64(Float64(y + t) / y));
	elseif (a <= 1.2e+185)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z + a) - b;
	t_2 = (y + t) * (a / (t + (x + y)));
	tmp = 0.0;
	if (a <= -1.12e+153)
		tmp = t_2;
	elseif (a <= -1.45e+70)
		tmp = t_1;
	elseif (a <= -430000.0)
		tmp = t_2;
	elseif (a <= 2.25e-300)
		tmp = (z - b) / ((y + t) / y);
	elseif (a <= 1.2e+185)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y + t), $MachinePrecision] * N[(a / N[(t + N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.12e+153], t$95$2, If[LessEqual[a, -1.45e+70], t$95$1, If[LessEqual[a, -430000.0], t$95$2, If[LessEqual[a, 2.25e-300], N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.2e+185], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.1200000000000001e153 or -1.4499999999999999e70 < a < -4.3e5 or 1.19999999999999995e185 < a

   1. Initial program 42.8%

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

   2. Taylor expanded in a around inf 38.2%

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

      1. expm1-log1p-u 15.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right)\right)} \]

      2. expm1-udef 10.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right)} - 1} \]

      3. associate-/l* 29.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}}\right)} - 1 \]

      4. +-commutative 29.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{a}{\frac{y + \color{blue}{\left(x + t\right)}}{y + t}}\right)} - 1 \]

   4. Applied egg-rr 29.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{a}{\frac{y + \left(x + t\right)}{y + t}}\right)} - 1} \]
   5. Step-by-step derivation

      1. expm1-def 34.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a}{\frac{y + \left(x + t\right)}{y + t}}\right)\right)} \]

      2. expm1-log1p 84.8%

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

      3. associate-/r/ 83.6%

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

      4. *-commutative 83.6%

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

      5. *-rgt-identity 83.6%

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

      6. associate-*r/ 83.4%

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

      7. associate-*r/ 83.6%

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

      8. *-rgt-identity 83.6%

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

      9. associate-+r+ 83.6%

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

   6. Simplified 83.6%

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

   if -1.1200000000000001e153 < a < -1.4499999999999999e70 or 2.25e-300 < a < 1.19999999999999995e185

   1. Initial program 62.3%

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

   2. Taylor expanded in y around inf 61.4%

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

      1. +-commutative 61.4%

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

   4. Simplified 61.4%

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

   if -4.3e5 < a < 2.25e-300

   1. Initial program 67.2%

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

   3. Simplified 67.3%

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

   4. Taylor expanded in x around 0 57.8%

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

   5. Taylor expanded in a around 0 42.0%

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

      1. associate-/l* 62.0%

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

   7. Simplified 62.0%

      \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.12 \cdot 10^{+153}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{a}{t + \left(x + y\right)}\\ \mathbf{elif}\;a \leq -1.45 \cdot 10^{+70}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;a \leq -430000:\\ \;\;\;\;\left(y + t\right) \cdot \frac{a}{t + \left(x + y\right)}\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{-300}:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+185}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;\left(y + t\right) \cdot \frac{a}{t + \left(x + y\right)}\\ \end{array} \]

Alternative 10: 61.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + y \cdot \left(\frac{z}{t} - \frac{b}{t}\right)\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+41}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+138} \lor \neg \left(t \leq 1.6 \cdot 10^{+168}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x + y}{y + \left(x + t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ a (* y (- (/ z t) (/ b t))))))
   (if (<= t -1.5e+177)
     t_1
     (if (<= t 1.42e+41)
       (- (+ z a) b)
       (if (or (<= t 3.5e+138) (not (<= t 1.6e+168)))
         t_1
         (* z (/ (+ x y) (+ y (+ x t)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a + (y * ((z / t) - (b / t)));
	double tmp;
	if (t <= -1.5e+177) {
		tmp = t_1;
	} else if (t <= 1.42e+41) {
		tmp = (z + a) - b;
	} else if ((t <= 3.5e+138) || !(t <= 1.6e+168)) {
		tmp = t_1;
	} else {
		tmp = z * ((x + y) / (y + (x + t)));
	}
	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 * ((z / t) - (b / t)))
    if (t <= (-1.5d+177)) then
        tmp = t_1
    else if (t <= 1.42d+41) then
        tmp = (z + a) - b
    else if ((t <= 3.5d+138) .or. (.not. (t <= 1.6d+168))) then
        tmp = t_1
    else
        tmp = z * ((x + y) / (y + (x + t)))
    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 * ((z / t) - (b / t)));
	double tmp;
	if (t <= -1.5e+177) {
		tmp = t_1;
	} else if (t <= 1.42e+41) {
		tmp = (z + a) - b;
	} else if ((t <= 3.5e+138) || !(t <= 1.6e+168)) {
		tmp = t_1;
	} else {
		tmp = z * ((x + y) / (y + (x + t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a + (y * ((z / t) - (b / t)))
	tmp = 0
	if t <= -1.5e+177:
		tmp = t_1
	elif t <= 1.42e+41:
		tmp = (z + a) - b
	elif (t <= 3.5e+138) or not (t <= 1.6e+168):
		tmp = t_1
	else:
		tmp = z * ((x + y) / (y + (x + t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a + Float64(y * Float64(Float64(z / t) - Float64(b / t))))
	tmp = 0.0
	if (t <= -1.5e+177)
		tmp = t_1;
	elseif (t <= 1.42e+41)
		tmp = Float64(Float64(z + a) - b);
	elseif ((t <= 3.5e+138) || !(t <= 1.6e+168))
		tmp = t_1;
	else
		tmp = Float64(z * Float64(Float64(x + y) / Float64(y + Float64(x + t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a + (y * ((z / t) - (b / t)));
	tmp = 0.0;
	if (t <= -1.5e+177)
		tmp = t_1;
	elseif (t <= 1.42e+41)
		tmp = (z + a) - b;
	elseif ((t <= 3.5e+138) || ~((t <= 1.6e+168)))
		tmp = t_1;
	else
		tmp = z * ((x + y) / (y + (x + t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a + N[(y * N[(N[(z / t), $MachinePrecision] - N[(b / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.5e+177], t$95$1, If[LessEqual[t, 1.42e+41], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], If[Or[LessEqual[t, 3.5e+138], N[Not[LessEqual[t, 1.6e+168]], $MachinePrecision]], t$95$1, N[(z * N[(N[(x + y), $MachinePrecision] / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.5e177 or 1.42000000000000007e41 < t < 3.4999999999999998e138 or 1.6000000000000001e168 < t

   1. Initial program 50.0%

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

   3. Simplified 50.6%

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

   4. Taylor expanded in x around 0 43.0%

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

   5. Taylor expanded in y around 0 76.4%

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

   if -1.5e177 < t < 1.42000000000000007e41

   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. Taylor expanded in y around inf 64.1%

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

      1. +-commutative 64.1%

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

   4. Simplified 64.1%

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

   if 3.4999999999999998e138 < t < 1.6000000000000001e168

   1. Initial program 28.9%

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

   2. Taylor expanded in z around inf 21.4%

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

      1. *-un-lft-identity 21.4%

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

      2. associate-/l* 79.8%

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

   4. Applied egg-rr 79.8%

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

      1. *-lft-identity 79.8%

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

      2. associate-/r/ 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{+177}:\\ \;\;\;\;a + y \cdot \left(\frac{z}{t} - \frac{b}{t}\right)\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+41}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+138} \lor \neg \left(t \leq 1.6 \cdot 10^{+168}\right):\\ \;\;\;\;a + y \cdot \left(\frac{z}{t} - \frac{b}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x + y}{y + \left(x + t\right)}\\ \end{array} \]

Alternative 11: 55.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + a\right) - b\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;y \leq -1.06 \cdot 10^{-117}:\\ \;\;\;\;a\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{z}{\frac{x + t}{x}}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-162}:\\ \;\;\;\;\frac{t}{\frac{x + t}{a}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ z a) b)))
   (if (<= y -1.85e+62)
     t_1
     (if (<= y -8.5e-68)
       (/ (- z b) (/ (+ y t) y))
       (if (<= y -1.06e-117)
         a
         (if (<= y -1.2e-281)
           (/ z (/ (+ x t) x))
           (if (<= y 4.4e-162) (/ t (/ (+ x t) a)) t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + a) - b;
	double tmp;
	if (y <= -1.85e+62) {
		tmp = t_1;
	} else if (y <= -8.5e-68) {
		tmp = (z - b) / ((y + t) / y);
	} else if (y <= -1.06e-117) {
		tmp = a;
	} else if (y <= -1.2e-281) {
		tmp = z / ((x + t) / x);
	} else if (y <= 4.4e-162) {
		tmp = t / ((x + t) / a);
	} 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 = (z + a) - b
    if (y <= (-1.85d+62)) then
        tmp = t_1
    else if (y <= (-8.5d-68)) then
        tmp = (z - b) / ((y + t) / y)
    else if (y <= (-1.06d-117)) then
        tmp = a
    else if (y <= (-1.2d-281)) then
        tmp = z / ((x + t) / x)
    else if (y <= 4.4d-162) then
        tmp = t / ((x + t) / a)
    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 = (z + a) - b;
	double tmp;
	if (y <= -1.85e+62) {
		tmp = t_1;
	} else if (y <= -8.5e-68) {
		tmp = (z - b) / ((y + t) / y);
	} else if (y <= -1.06e-117) {
		tmp = a;
	} else if (y <= -1.2e-281) {
		tmp = z / ((x + t) / x);
	} else if (y <= 4.4e-162) {
		tmp = t / ((x + t) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z + a) - b
	tmp = 0
	if y <= -1.85e+62:
		tmp = t_1
	elif y <= -8.5e-68:
		tmp = (z - b) / ((y + t) / y)
	elif y <= -1.06e-117:
		tmp = a
	elif y <= -1.2e-281:
		tmp = z / ((x + t) / x)
	elif y <= 4.4e-162:
		tmp = t / ((x + t) / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (y <= -1.85e+62)
		tmp = t_1;
	elseif (y <= -8.5e-68)
		tmp = Float64(Float64(z - b) / Float64(Float64(y + t) / y));
	elseif (y <= -1.06e-117)
		tmp = a;
	elseif (y <= -1.2e-281)
		tmp = Float64(z / Float64(Float64(x + t) / x));
	elseif (y <= 4.4e-162)
		tmp = Float64(t / Float64(Float64(x + t) / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z + a) - b;
	tmp = 0.0;
	if (y <= -1.85e+62)
		tmp = t_1;
	elseif (y <= -8.5e-68)
		tmp = (z - b) / ((y + t) / y);
	elseif (y <= -1.06e-117)
		tmp = a;
	elseif (y <= -1.2e-281)
		tmp = z / ((x + t) / x);
	elseif (y <= 4.4e-162)
		tmp = t / ((x + t) / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -1.85e+62], t$95$1, If[LessEqual[y, -8.5e-68], N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.06e-117], a, If[LessEqual[y, -1.2e-281], N[(z / N[(N[(x + t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e-162], N[(t / N[(N[(x + t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.85000000000000007e62 or 4.3999999999999998e-162 < y

   1. Initial program 44.7%

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

   2. Taylor expanded in y around inf 70.7%

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

      1. +-commutative 70.7%

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

   4. Simplified 70.7%

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

   if -1.85000000000000007e62 < y < -8.50000000000000026e-68

   1. Initial program 72.9%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in x around 0 60.4%

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

   5. Taylor expanded in a around 0 47.0%

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

      1. associate-/l* 57.1%

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

   7. Simplified 57.1%

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

   if -8.50000000000000026e-68 < y < -1.06000000000000008e-117

   1. Initial program 74.2%

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

   2. Taylor expanded in t around inf 56.3%

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

   if -1.06000000000000008e-117 < y < -1.2e-281

   1. Initial program 81.7%

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

   2. Taylor expanded in z around inf 52.0%

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

   3. Taylor expanded in y around 0 49.2%

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

      1. associate-/l* 54.3%

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

   5. Simplified 54.3%

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

   if -1.2e-281 < y < 4.3999999999999998e-162

   1. Initial program 71.1%

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

   2. Taylor expanded in a around inf 41.7%

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

   3. Taylor expanded in y around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-/l* 59.2%

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

      3. +-commutative 59.2%

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

   5. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{t}{\frac{x + t}{a}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+62}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;y \leq -1.06 \cdot 10^{-117}:\\ \;\;\;\;a\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{z}{\frac{x + t}{x}}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-162}:\\ \;\;\;\;\frac{t}{\frac{x + t}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Alternative 12: 62.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + a\right) - b\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -64:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-150}:\\ \;\;\;\;\frac{z \cdot x + t \cdot a}{x + t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ z a) b)))
   (if (<= y -1.05e+62)
     t_1
     (if (<= y -64.0)
       (/ (- z b) (/ (+ y t) y))
       (if (<= y 5.8e-150) (/ (+ (* z x) (* t a)) (+ x t)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + a) - b;
	double tmp;
	if (y <= -1.05e+62) {
		tmp = t_1;
	} else if (y <= -64.0) {
		tmp = (z - b) / ((y + t) / y);
	} else if (y <= 5.8e-150) {
		tmp = ((z * x) + (t * a)) / (x + t);
	} 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 = (z + a) - b
    if (y <= (-1.05d+62)) then
        tmp = t_1
    else if (y <= (-64.0d0)) then
        tmp = (z - b) / ((y + t) / y)
    else if (y <= 5.8d-150) then
        tmp = ((z * x) + (t * a)) / (x + t)
    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 = (z + a) - b;
	double tmp;
	if (y <= -1.05e+62) {
		tmp = t_1;
	} else if (y <= -64.0) {
		tmp = (z - b) / ((y + t) / y);
	} else if (y <= 5.8e-150) {
		tmp = ((z * x) + (t * a)) / (x + t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z + a) - b
	tmp = 0
	if y <= -1.05e+62:
		tmp = t_1
	elif y <= -64.0:
		tmp = (z - b) / ((y + t) / y)
	elif y <= 5.8e-150:
		tmp = ((z * x) + (t * a)) / (x + t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (y <= -1.05e+62)
		tmp = t_1;
	elseif (y <= -64.0)
		tmp = Float64(Float64(z - b) / Float64(Float64(y + t) / y));
	elseif (y <= 5.8e-150)
		tmp = Float64(Float64(Float64(z * x) + Float64(t * a)) / Float64(x + t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z + a) - b;
	tmp = 0.0;
	if (y <= -1.05e+62)
		tmp = t_1;
	elseif (y <= -64.0)
		tmp = (z - b) / ((y + t) / y);
	elseif (y <= 5.8e-150)
		tmp = ((z * x) + (t * a)) / (x + t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -1.05e+62], t$95$1, If[LessEqual[y, -64.0], N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e-150], N[(N[(N[(z * x), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] / N[(x + t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e62 or 5.7999999999999996e-150 < y

   1. Initial program 44.7%

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

   2. Taylor expanded in y around inf 70.7%

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

      1. +-commutative 70.7%

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

   4. Simplified 70.7%

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

   if -1.05e62 < y < -64

   1. Initial program 52.9%

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

   3. Simplified 52.9%

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

   4. Taylor expanded in x around 0 47.7%

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

   5. Taylor expanded in a around 0 48.8%

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

      1. associate-/l* 72.3%

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

   7. Simplified 72.3%

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

   if -64 < y < 5.7999999999999996e-150

   1. Initial program 78.8%

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

   2. Taylor expanded in y around 0 63.6%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+62}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;y \leq -64:\\ \;\;\;\;\frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-150}:\\ \;\;\;\;\frac{z \cdot x + t \cdot a}{x + t}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Alternative 13: 53.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + a\right) - b\\ \mathbf{if}\;b \leq 8 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{\frac{x + y}{a}}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{-y}{y + \left(x + t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ z a) b)))
   (if (<= b 8e-112)
     t_1
     (if (<= b 3.4e-37)
       (/ y (/ (+ x y) a))
       (if (<= b 8.6e+136) t_1 (* b (/ (- y) (+ y (+ x t)))))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + a) - b;
	double tmp;
	if (b <= 8e-112) {
		tmp = t_1;
	} else if (b <= 3.4e-37) {
		tmp = y / ((x + y) / a);
	} else if (b <= 8.6e+136) {
		tmp = t_1;
	} else {
		tmp = b * (-y / (y + (x + t)));
	}
	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 = (z + a) - b
    if (b <= 8d-112) then
        tmp = t_1
    else if (b <= 3.4d-37) then
        tmp = y / ((x + y) / a)
    else if (b <= 8.6d+136) then
        tmp = t_1
    else
        tmp = b * (-y / (y + (x + t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + a) - b;
	double tmp;
	if (b <= 8e-112) {
		tmp = t_1;
	} else if (b <= 3.4e-37) {
		tmp = y / ((x + y) / a);
	} else if (b <= 8.6e+136) {
		tmp = t_1;
	} else {
		tmp = b * (-y / (y + (x + t)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z + a) - b
	tmp = 0
	if b <= 8e-112:
		tmp = t_1
	elif b <= 3.4e-37:
		tmp = y / ((x + y) / a)
	elif b <= 8.6e+136:
		tmp = t_1
	else:
		tmp = b * (-y / (y + (x + t)))
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (b <= 8e-112)
		tmp = t_1;
	elseif (b <= 3.4e-37)
		tmp = Float64(y / Float64(Float64(x + y) / a));
	elseif (b <= 8.6e+136)
		tmp = t_1;
	else
		tmp = Float64(b * Float64(Float64(-y) / Float64(y + Float64(x + t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z + a) - b;
	tmp = 0.0;
	if (b <= 8e-112)
		tmp = t_1;
	elseif (b <= 3.4e-37)
		tmp = y / ((x + y) / a);
	elseif (b <= 8.6e+136)
		tmp = t_1;
	else
		tmp = b * (-y / (y + (x + t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[b, 8e-112], t$95$1, If[LessEqual[b, 3.4e-37], N[(y / N[(N[(x + y), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.6e+136], t$95$1, N[(b * N[((-y) / N[(y + N[(x + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if b < 7.9999999999999996e-112 or 3.40000000000000018e-37 < b < 8.5999999999999997e136

   1. Initial program 55.3%

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

   2. Taylor expanded in y around inf 65.4%

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

      1. +-commutative 65.4%

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

   4. Simplified 65.4%

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

   if 7.9999999999999996e-112 < b < 3.40000000000000018e-37

   1. Initial program 77.0%

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

   2. Taylor expanded in a around inf 54.0%

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

   3. Taylor expanded in t around 0 49.0%

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

      1. *-commutative 49.0%

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

      2. associate-/l* 62.7%

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

   5. Simplified 62.7%

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

   if 8.5999999999999997e136 < b

   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. Taylor expanded in b around inf 27.4%

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

      1. neg-mul-1 27.4%

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

      2. distribute-rgt-neg-in 27.4%

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

   4. Simplified 27.4%

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

      1. expm1-log1p-u 11.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y \cdot \left(-b\right)}{\left(x + t\right) + y}\right)\right)} \]

      2. expm1-udef 5.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y \cdot \left(-b\right)}{\left(x + t\right) + y}\right)} - 1} \]

      3. associate-/l* 15.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{y}{\frac{\left(x + t\right) + y}{-b}}}\right)} - 1 \]

      4. +-commutative 15.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{y}{\frac{\color{blue}{y + \left(x + t\right)}}{-b}}\right)} - 1 \]

   6. Applied egg-rr 15.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{\frac{y + \left(x + t\right)}{-b}}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 20.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{\frac{y + \left(x + t\right)}{-b}}\right)\right)} \]

      2. expm1-log1p 47.9%

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

      3. associate-/r/ 52.4%

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

   8. Simplified 52.4%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-112}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;b \leq 3.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{y}{\frac{x + y}{a}}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+136}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{-y}{y + \left(x + t\right)}\\ \end{array} \]

Alternative 14: 56.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z + a\right) - b\\ \mathbf{if}\;y \leq -5.1 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-284}:\\ \;\;\;\;\frac{z}{\frac{x + t}{x}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{t}{\frac{x + t}{a}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (+ z a) b)))
   (if (<= y -5.1e-120)
     t_1
     (if (<= y -1.25e-284)
       (/ z (/ (+ x t) x))
       (if (<= y 4.5e-165) (/ t (/ (+ x t) a)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (z + a) - b;
	double tmp;
	if (y <= -5.1e-120) {
		tmp = t_1;
	} else if (y <= -1.25e-284) {
		tmp = z / ((x + t) / x);
	} else if (y <= 4.5e-165) {
		tmp = t / ((x + t) / a);
	} 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 = (z + a) - b
    if (y <= (-5.1d-120)) then
        tmp = t_1
    else if (y <= (-1.25d-284)) then
        tmp = z / ((x + t) / x)
    else if (y <= 4.5d-165) then
        tmp = t / ((x + t) / a)
    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 = (z + a) - b;
	double tmp;
	if (y <= -5.1e-120) {
		tmp = t_1;
	} else if (y <= -1.25e-284) {
		tmp = z / ((x + t) / x);
	} else if (y <= 4.5e-165) {
		tmp = t / ((x + t) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (z + a) - b
	tmp = 0
	if y <= -5.1e-120:
		tmp = t_1
	elif y <= -1.25e-284:
		tmp = z / ((x + t) / x)
	elif y <= 4.5e-165:
		tmp = t / ((x + t) / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (y <= -5.1e-120)
		tmp = t_1;
	elseif (y <= -1.25e-284)
		tmp = Float64(z / Float64(Float64(x + t) / x));
	elseif (y <= 4.5e-165)
		tmp = Float64(t / Float64(Float64(x + t) / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z + a) - b;
	tmp = 0.0;
	if (y <= -5.1e-120)
		tmp = t_1;
	elseif (y <= -1.25e-284)
		tmp = z / ((x + t) / x);
	elseif (y <= 4.5e-165)
		tmp = t / ((x + t) / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[y, -5.1e-120], t$95$1, If[LessEqual[y, -1.25e-284], N[(z / N[(N[(x + t), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e-165], N[(t / N[(N[(x + t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.0999999999999998e-120 or 4.49999999999999992e-165 < y

   1. Initial program 50.5%

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

   2. Taylor expanded in y around inf 63.9%

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

      1. +-commutative 63.9%

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

   4. Simplified 63.9%

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

   if -5.0999999999999998e-120 < y < -1.24999999999999993e-284

   1. Initial program 81.7%

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

   2. Taylor expanded in z around inf 52.0%

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

   3. Taylor expanded in y around 0 49.2%

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

      1. associate-/l* 54.3%

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

   5. Simplified 54.3%

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

   if -1.24999999999999993e-284 < y < 4.49999999999999992e-165

   1. Initial program 71.1%

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

   2. Taylor expanded in a around inf 41.7%

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

   3. Taylor expanded in y around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. associate-/l* 59.2%

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

      3. +-commutative 59.2%

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

   5. Simplified 59.2%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.1 \cdot 10^{-120}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{-284}:\\ \;\;\;\;\frac{z}{\frac{x + t}{x}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{t}{\frac{x + t}{a}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Alternative 15: 59.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+99}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+219}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= x -3.2e+99) z (if (<= x 9.6e+219) (- (+ z a) b) z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -3.2e+99) {
		tmp = z;
	} else if (x <= 9.6e+219) {
		tmp = (z + a) - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (x <= (-3.2d+99)) then
        tmp = z
    else if (x <= 9.6d+219) then
        tmp = (z + a) - b
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (x <= -3.2e+99) {
		tmp = z;
	} else if (x <= 9.6e+219) {
		tmp = (z + a) - b;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if x <= -3.2e+99:
		tmp = z
	elif x <= 9.6e+219:
		tmp = (z + a) - b
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (x <= -3.2e+99)
		tmp = z;
	elseif (x <= 9.6e+219)
		tmp = Float64(Float64(z + a) - b);
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (x <= -3.2e+99)
		tmp = z;
	elseif (x <= 9.6e+219)
		tmp = (z + a) - b;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[x, -3.2e+99], z, If[LessEqual[x, 9.6e+219], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], z]]

Derivation

1. Split input into 2 regimes

2. if x < -3.19999999999999999e99 or 9.6000000000000002e219 < x

   1. Initial program 46.5%

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

   2. Taylor expanded in x around inf 59.1%

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

   if -3.19999999999999999e99 < x < 9.6000000000000002e219

   1. Initial program 60.4%

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

   2. Taylor expanded in y around inf 62.0%

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

      1. +-commutative 62.0%

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

   4. Simplified 62.0%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+99}:\\ \;\;\;\;z\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{+219}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 16: 43.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-68}:\\ \;\;\;\;a\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+77}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -5e-68) a (if (<= t 4.3e+77) z a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -5e-68) {
		tmp = a;
	} else if (t <= 4.3e+77) {
		tmp = z;
	} else {
		tmp = 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 (t <= (-5d-68)) then
        tmp = a
    else if (t <= 4.3d+77) then
        tmp = z
    else
        tmp = 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 (t <= -5e-68) {
		tmp = a;
	} else if (t <= 4.3e+77) {
		tmp = z;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -5e-68:
		tmp = a
	elif t <= 4.3e+77:
		tmp = z
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -5e-68)
		tmp = a;
	elseif (t <= 4.3e+77)
		tmp = z;
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -5e-68)
		tmp = a;
	elseif (t <= 4.3e+77)
		tmp = z;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -5e-68], a, If[LessEqual[t, 4.3e+77], z, a]]

Derivation

1. Split input into 2 regimes

2. if t < -4.99999999999999971e-68 or 4.29999999999999991e77 < t

   1. Initial program 49.9%

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

   2. Taylor expanded in t around inf 53.5%

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

   if -4.99999999999999971e-68 < t < 4.29999999999999991e77

   1. Initial program 63.8%

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

   2. Taylor expanded in x around inf 47.5%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{-68}:\\ \;\;\;\;a\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+77}:\\ \;\;\;\;z\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 17: 32.5% accurate, 21.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return 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 = a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := a

Derivation

1. Initial program 57.0%

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

2. Taylor expanded in t around inf 36.2%

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

3. Final simplification 36.2%

   \[\leadsto a \]

Developer target: 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 2023258 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

  :herbie-target
  (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3.5813117084150564e+153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e+82) (/ 1.0 (/ (+ (+ 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)))