Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.3% → 88.0%

Time: 15.0s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+60} \lor \neg \left(z \leq 26000000000000\right):\\ \;\;\;\;\frac{x \cdot \frac{y}{b - y} + y \cdot \frac{a - t}{{\left(b - y\right)}^{2}}}{z} + \frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{t\_1} + \frac{z \cdot \left(t - a\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y)))))
   (if (or (<= z -1.35e+60) (not (<= z 26000000000000.0)))
     (+
      (/ (+ (* x (/ y (- b y))) (* y (/ (- a t) (pow (- b y) 2.0)))) z)
      (/ (- t a) (- b y)))
     (+ (/ (* x y) t_1) (/ (* z (- t a)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double tmp;
	if ((z <= -1.35e+60) || !(z <= 26000000000000.0)) {
		tmp = (((x * (y / (b - y))) + (y * ((a - t) / pow((b - y), 2.0)))) / z) + ((t - a) / (b - y));
	} else {
		tmp = ((x * y) / t_1) + ((z * (t - a)) / 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 + (z * (b - y))
    if ((z <= (-1.35d+60)) .or. (.not. (z <= 26000000000000.0d0))) then
        tmp = (((x * (y / (b - y))) + (y * ((a - t) / ((b - y) ** 2.0d0)))) / z) + ((t - a) / (b - y))
    else
        tmp = ((x * y) / t_1) + ((z * (t - a)) / 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 + (z * (b - y));
	double tmp;
	if ((z <= -1.35e+60) || !(z <= 26000000000000.0)) {
		tmp = (((x * (y / (b - y))) + (y * ((a - t) / Math.pow((b - y), 2.0)))) / z) + ((t - a) / (b - y));
	} else {
		tmp = ((x * y) / t_1) + ((z * (t - a)) / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (z * (b - y))
	tmp = 0
	if (z <= -1.35e+60) or not (z <= 26000000000000.0):
		tmp = (((x * (y / (b - y))) + (y * ((a - t) / math.pow((b - y), 2.0)))) / z) + ((t - a) / (b - y))
	else:
		tmp = ((x * y) / t_1) + ((z * (t - a)) / t_1)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	tmp = 0.0
	if ((z <= -1.35e+60) || !(z <= 26000000000000.0))
		tmp = Float64(Float64(Float64(Float64(x * Float64(y / Float64(b - y))) + Float64(y * Float64(Float64(a - t) / (Float64(b - y) ^ 2.0)))) / z) + Float64(Float64(t - a) / Float64(b - y)));
	else
		tmp = Float64(Float64(Float64(x * y) / t_1) + Float64(Float64(z * Float64(t - a)) / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (z * (b - y));
	tmp = 0.0;
	if ((z <= -1.35e+60) || ~((z <= 26000000000000.0)))
		tmp = (((x * (y / (b - y))) + (y * ((a - t) / ((b - y) ^ 2.0)))) / z) + ((t - a) / (b - y));
	else
		tmp = ((x * y) / t_1) + ((z * (t - a)) / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[z, -1.35e+60], N[Not[LessEqual[z, 26000000000000.0]], $MachinePrecision]], N[(N[(N[(N[(x * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(N[(a - t), $MachinePrecision] / N[Power[N[(b - y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35e60 or 2.6e13 < z

   1. Initial program 38.1%

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

   3. Taylor expanded in z around -inf 67.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{x \cdot y}{b - y} - -1 \cdot \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z} + \frac{t}{b - y}\right) - \frac{a}{b - y}} \]
   4. Step-by-step derivation

      1. associate--l+ 67.8%

         \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{x \cdot y}{b - y} - -1 \cdot \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z} + \left(\frac{t}{b - y} - \frac{a}{b - y}\right)} \]

      2. mul-1-neg 67.8%

         \[\leadsto \color{blue}{\left(-\frac{-1 \cdot \frac{x \cdot y}{b - y} - -1 \cdot \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z}\right)} + \left(\frac{t}{b - y} - \frac{a}{b - y}\right) \]

      3. distribute-lft-out-- 67.8%

         \[\leadsto \left(-\frac{\color{blue}{-1 \cdot \left(\frac{x \cdot y}{b - y} - \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}\right)}}{z}\right) + \left(\frac{t}{b - y} - \frac{a}{b - y}\right) \]

      4. associate-/l* 71.3%

         \[\leadsto \left(-\frac{-1 \cdot \left(\color{blue}{x \cdot \frac{y}{b - y}} - \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}\right)}{z}\right) + \left(\frac{t}{b - y} - \frac{a}{b - y}\right) \]

      5. associate-/l* 94.7%

         \[\leadsto \left(-\frac{-1 \cdot \left(x \cdot \frac{y}{b - y} - \color{blue}{y \cdot \frac{t - a}{{\left(b - y\right)}^{2}}}\right)}{z}\right) + \left(\frac{t}{b - y} - \frac{a}{b - y}\right) \]

      6. div-sub 95.6%

         \[\leadsto \left(-\frac{-1 \cdot \left(x \cdot \frac{y}{b - y} - y \cdot \frac{t - a}{{\left(b - y\right)}^{2}}\right)}{z}\right) + \color{blue}{\frac{t - a}{b - y}} \]

   5. Simplified 95.6%

      \[\leadsto \color{blue}{\left(-\frac{-1 \cdot \left(x \cdot \frac{y}{b - y} - y \cdot \frac{t - a}{{\left(b - y\right)}^{2}}\right)}{z}\right) + \frac{t - a}{b - y}} \]

   if -1.35e60 < z < 2.6e13

   1. Initial program 87.4%

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

   3. Taylor expanded in x around 0 87.4%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+60} \lor \neg \left(z \leq 26000000000000\right):\\ \;\;\;\;\frac{x \cdot \frac{y}{b - y} + y \cdot \frac{a - t}{{\left(b - y\right)}^{2}}}{z} + \frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\frac{t}{b - y} + \frac{a}{y - b}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{x \cdot y}{t\_1} + \frac{z \cdot \left(t - a\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y)))))
   (if (<= z -2e+15)
     (+ (/ t (- b y)) (/ a (- y b)))
     (if (<= z 8.5e+29)
       (+ (/ (* x y) t_1) (/ (* z (- t a)) t_1))
       (/ (- t a) (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double tmp;
	if (z <= -2e+15) {
		tmp = (t / (b - y)) + (a / (y - b));
	} else if (z <= 8.5e+29) {
		tmp = ((x * y) / t_1) + ((z * (t - a)) / t_1);
	} else {
		tmp = (t - a) / (b - y);
	}
	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 + (z * (b - y))
    if (z <= (-2d+15)) then
        tmp = (t / (b - y)) + (a / (y - b))
    else if (z <= 8.5d+29) then
        tmp = ((x * y) / t_1) + ((z * (t - a)) / t_1)
    else
        tmp = (t - a) / (b - y)
    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 + (z * (b - y));
	double tmp;
	if (z <= -2e+15) {
		tmp = (t / (b - y)) + (a / (y - b));
	} else if (z <= 8.5e+29) {
		tmp = ((x * y) / t_1) + ((z * (t - a)) / t_1);
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (z * (b - y))
	tmp = 0
	if z <= -2e+15:
		tmp = (t / (b - y)) + (a / (y - b))
	elif z <= 8.5e+29:
		tmp = ((x * y) / t_1) + ((z * (t - a)) / t_1)
	else:
		tmp = (t - a) / (b - y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	tmp = 0.0
	if (z <= -2e+15)
		tmp = Float64(Float64(t / Float64(b - y)) + Float64(a / Float64(y - b)));
	elseif (z <= 8.5e+29)
		tmp = Float64(Float64(Float64(x * y) / t_1) + Float64(Float64(z * Float64(t - a)) / t_1));
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (z * (b - y));
	tmp = 0.0;
	if (z <= -2e+15)
		tmp = (t / (b - y)) + (a / (y - b));
	elseif (z <= 8.5e+29)
		tmp = ((x * y) / t_1) + ((z * (t - a)) / t_1);
	else
		tmp = (t - a) / (b - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2e+15], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e+29], N[(N[(N[(x * y), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e15

   1. Initial program 37.1%

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

   3. Taylor expanded in z around inf 80.6%

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

   4. Taylor expanded in t around 0 80.6%

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

      1. +-commutative 80.6%

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

      2. mul-1-neg 80.6%

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

      3. sub-neg 80.6%

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

   6. Simplified 80.6%

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

   if -2e15 < z < 8.5000000000000006e29

   1. Initial program 88.1%

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

   3. Taylor expanded in x around 0 88.1%

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

   if 8.5000000000000006e29 < z

   1. Initial program 37.4%

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

   3. Taylor expanded in z around inf 87.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\frac{t}{b - y} + \frac{a}{y - b}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{x \cdot y}{y + z \cdot \left(b - y\right)} + \frac{z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(a - t\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -2e+15)
   (- (/ t (- b y)) (/ a (- b y)))
   (if (<= z 9e+29)
     (/ (- (* x y) (* z (- a t))) (+ y (* z (- b y))))
     (/ (- t a) (- b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2e+15) {
		tmp = (t / (b - y)) - (a / (b - y));
	} else if (z <= 9e+29) {
		tmp = ((x * y) - (z * (a - t))) / (y + (z * (b - y)));
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-2d+15)) then
        tmp = (t / (b - y)) - (a / (b - y))
    else if (z <= 9d+29) then
        tmp = ((x * y) - (z * (a - t))) / (y + (z * (b - y)))
    else
        tmp = (t - a) / (b - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -2e+15) {
		tmp = (t / (b - y)) - (a / (b - y));
	} else if (z <= 9e+29) {
		tmp = ((x * y) - (z * (a - t))) / (y + (z * (b - y)));
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -2e+15:
		tmp = (t / (b - y)) - (a / (b - y))
	elif z <= 9e+29:
		tmp = ((x * y) - (z * (a - t))) / (y + (z * (b - y)))
	else:
		tmp = (t - a) / (b - y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -2e+15)
		tmp = Float64(Float64(t / Float64(b - y)) - Float64(a / Float64(b - y)));
	elseif (z <= 9e+29)
		tmp = Float64(Float64(Float64(x * y) - Float64(z * Float64(a - t))) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -2e+15)
		tmp = (t / (b - y)) - (a / (b - y));
	elseif (z <= 9e+29)
		tmp = ((x * y) - (z * (a - t))) / (y + (z * (b - y)));
	else
		tmp = (t - a) / (b - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -2e+15], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] - N[(a / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 9e+29], N[(N[(N[(x * y), $MachinePrecision] - N[(z * N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2e15

   1. Initial program 37.1%

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

   3. Taylor expanded in z around inf 80.6%

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

   4. Taylor expanded in t around 0 80.6%

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

      1. +-commutative 80.6%

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

      2. mul-1-neg 80.6%

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

      3. sub-neg 80.6%

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

   6. Simplified 80.6%

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

   if -2e15 < z < 9.0000000000000005e29

   1. Initial program 88.1%

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

   if 9.0000000000000005e29 < z

   1. Initial program 37.4%

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

   3. Taylor expanded in z around inf 87.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\frac{t}{b - y} - \frac{a}{b - y}\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\frac{x \cdot y - z \cdot \left(a - t\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{-14}:\\ \;\;\;\;\frac{t}{b - y} + \frac{a}{y - b}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{z \cdot \left(t - \left(a + x \cdot b\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.18e-14)
   (+ (/ t (- b y)) (/ a (- y b)))
   (if (<= z 2e-17)
     (+ x (/ (* z (- t (+ a (* x b)))) y))
     (/ (- t a) (- b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.18e-14) {
		tmp = (t / (b - y)) + (a / (y - b));
	} else if (z <= 2e-17) {
		tmp = x + ((z * (t - (a + (x * b)))) / y);
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-1.18d-14)) then
        tmp = (t / (b - y)) + (a / (y - b))
    else if (z <= 2d-17) then
        tmp = x + ((z * (t - (a + (x * b)))) / y)
    else
        tmp = (t - a) / (b - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.18e-14) {
		tmp = (t / (b - y)) + (a / (y - b));
	} else if (z <= 2e-17) {
		tmp = x + ((z * (t - (a + (x * b)))) / y);
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.18e-14:
		tmp = (t / (b - y)) + (a / (y - b))
	elif z <= 2e-17:
		tmp = x + ((z * (t - (a + (x * b)))) / y)
	else:
		tmp = (t - a) / (b - y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.18e-14)
		tmp = Float64(Float64(t / Float64(b - y)) + Float64(a / Float64(y - b)));
	elseif (z <= 2e-17)
		tmp = Float64(x + Float64(Float64(z * Float64(t - Float64(a + Float64(x * b)))) / y));
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.18e-14)
		tmp = (t / (b - y)) + (a / (y - b));
	elseif (z <= 2e-17)
		tmp = x + ((z * (t - (a + (x * b)))) / y);
	else
		tmp = (t - a) / (b - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.18e-14], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e-17], N[(x + N[(N[(z * N[(t - N[(a + N[(x * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.17999999999999993e-14

   1. Initial program 45.4%

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

   3. Taylor expanded in z around inf 77.9%

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

   4. Taylor expanded in t around 0 77.9%

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

      1. +-commutative 77.9%

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

      2. mul-1-neg 77.9%

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

      3. sub-neg 77.9%

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

   6. Simplified 77.9%

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

   if -1.17999999999999993e-14 < z < 2.00000000000000014e-17

   1. Initial program 87.2%

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

   3. Taylor expanded in z around 0 53.5%

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

   4. Taylor expanded in y around 0 71.6%

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

   if 2.00000000000000014e-17 < z

   1. Initial program 45.1%

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

   3. Taylor expanded in z around inf 86.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{-14}:\\ \;\;\;\;\frac{t}{b - y} + \frac{a}{y - b}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{z \cdot \left(t - \left(a + x \cdot b\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{t}{b - y} + \frac{a}{y - b}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-139}:\\ \;\;\;\;x + z \cdot \frac{t}{y}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{z \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.9e-11)
   (+ (/ t (- b y)) (/ a (- y b)))
   (if (<= z -1.1e-139)
     (+ x (* z (/ t y)))
     (if (<= z 3.7e-19) (- x (/ (* z a) y)) (/ (- t a) (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.9e-11) {
		tmp = (t / (b - y)) + (a / (y - b));
	} else if (z <= -1.1e-139) {
		tmp = x + (z * (t / y));
	} else if (z <= 3.7e-19) {
		tmp = x - ((z * a) / y);
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-1.9d-11)) then
        tmp = (t / (b - y)) + (a / (y - b))
    else if (z <= (-1.1d-139)) then
        tmp = x + (z * (t / y))
    else if (z <= 3.7d-19) then
        tmp = x - ((z * a) / y)
    else
        tmp = (t - a) / (b - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.9e-11) {
		tmp = (t / (b - y)) + (a / (y - b));
	} else if (z <= -1.1e-139) {
		tmp = x + (z * (t / y));
	} else if (z <= 3.7e-19) {
		tmp = x - ((z * a) / y);
	} else {
		tmp = (t - a) / (b - y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.9e-11:
		tmp = (t / (b - y)) + (a / (y - b))
	elif z <= -1.1e-139:
		tmp = x + (z * (t / y))
	elif z <= 3.7e-19:
		tmp = x - ((z * a) / y)
	else:
		tmp = (t - a) / (b - y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.9e-11)
		tmp = Float64(Float64(t / Float64(b - y)) + Float64(a / Float64(y - b)));
	elseif (z <= -1.1e-139)
		tmp = Float64(x + Float64(z * Float64(t / y)));
	elseif (z <= 3.7e-19)
		tmp = Float64(x - Float64(Float64(z * a) / y));
	else
		tmp = Float64(Float64(t - a) / Float64(b - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.9e-11)
		tmp = (t / (b - y)) + (a / (y - b));
	elseif (z <= -1.1e-139)
		tmp = x + (z * (t / y));
	elseif (z <= 3.7e-19)
		tmp = x - ((z * a) / y);
	else
		tmp = (t - a) / (b - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.9e-11], N[(N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision] + N[(a / N[(y - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.1e-139], N[(x + N[(z * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.7e-19], N[(x - N[(N[(z * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.8999999999999999e-11

   1. Initial program 44.6%

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

   3. Taylor expanded in z around inf 79.0%

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

   4. Taylor expanded in t around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. mul-1-neg 79.0%

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

      3. sub-neg 79.0%

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

   6. Simplified 79.0%

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

   if -1.8999999999999999e-11 < z < -1.10000000000000005e-139

   1. Initial program 81.7%

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

   3. Taylor expanded in z around 0 48.4%

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

   4. Taylor expanded in t around inf 65.4%

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

   if -1.10000000000000005e-139 < z < 3.70000000000000005e-19

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0 54.6%

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

   4. Taylor expanded in a around inf 62.6%

      \[\leadsto x + \color{blue}{-1 \cdot \frac{a \cdot z}{y}} \]
   5. Step-by-step derivation

      1. associate-*r/ 62.6%

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

      2. associate-*r* 62.6%

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

      3. mul-1-neg 62.6%

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

   6. Simplified 62.6%

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

   if 3.70000000000000005e-19 < z

   1. Initial program 45.1%

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

   3. Taylor expanded in z around inf 86.1%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-11}:\\ \;\;\;\;\frac{t}{b - y} + \frac{a}{y - b}\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-139}:\\ \;\;\;\;x + z \cdot \frac{t}{y}\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{-19}:\\ \;\;\;\;x - \frac{z \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.75 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-140}:\\ \;\;\;\;x + z \cdot \frac{t}{y}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;x - \frac{z \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.75e-10)
     t_1
     (if (<= z -4.1e-140)
       (+ x (* z (/ t y)))
       (if (<= z 1.4e-18) (- x (/ (* z a) y)) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.75e-10) {
		tmp = t_1;
	} else if (z <= -4.1e-140) {
		tmp = x + (z * (t / y));
	} else if (z <= 1.4e-18) {
		tmp = x - ((z * a) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-1.75d-10)) then
        tmp = t_1
    else if (z <= (-4.1d-140)) then
        tmp = x + (z * (t / y))
    else if (z <= 1.4d-18) then
        tmp = x - ((z * a) / y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.75e-10) {
		tmp = t_1;
	} else if (z <= -4.1e-140) {
		tmp = x + (z * (t / y));
	} else if (z <= 1.4e-18) {
		tmp = x - ((z * a) / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -1.75e-10:
		tmp = t_1
	elif z <= -4.1e-140:
		tmp = x + (z * (t / y))
	elif z <= 1.4e-18:
		tmp = x - ((z * a) / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.75e-10)
		tmp = t_1;
	elseif (z <= -4.1e-140)
		tmp = Float64(x + Float64(z * Float64(t / y)));
	elseif (z <= 1.4e-18)
		tmp = Float64(x - Float64(Float64(z * a) / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -1.75e-10)
		tmp = t_1;
	elseif (z <= -4.1e-140)
		tmp = x + (z * (t / y));
	elseif (z <= 1.4e-18)
		tmp = x - ((z * a) / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.75e-10], t$95$1, If[LessEqual[z, -4.1e-140], N[(x + N[(z * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.4e-18], N[(x - N[(N[(z * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7499999999999999e-10 or 1.40000000000000006e-18 < z

   1. Initial program 44.8%

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

   3. Taylor expanded in z around inf 82.4%

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

   if -1.7499999999999999e-10 < z < -4.1000000000000001e-140

   1. Initial program 81.7%

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

   3. Taylor expanded in z around 0 48.4%

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

   4. Taylor expanded in t around inf 65.4%

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

   if -4.1000000000000001e-140 < z < 1.40000000000000006e-18

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0 54.6%

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

   4. Taylor expanded in a around inf 62.6%

      \[\leadsto x + \color{blue}{-1 \cdot \frac{a \cdot z}{y}} \]
   5. Step-by-step derivation

      1. associate-*r/ 62.6%

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

      2. associate-*r* 62.6%

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

      3. mul-1-neg 62.6%

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

   6. Simplified 62.6%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-10}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -4.1 \cdot 10^{-140}:\\ \;\;\;\;x + z \cdot \frac{t}{y}\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-18}:\\ \;\;\;\;x - \frac{z \cdot a}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b}\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{-21}:\\ \;\;\;\;x + z \cdot \frac{t}{y}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+267}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) b)))
   (if (<= z -1.9e-12)
     t_1
     (if (<= z 6.1e-21)
       (+ x (* z (/ t y)))
       (if (<= z 1.5e+267) t_1 (/ (- a t) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / b;
	double tmp;
	if (z <= -1.9e-12) {
		tmp = t_1;
	} else if (z <= 6.1e-21) {
		tmp = x + (z * (t / y));
	} else if (z <= 1.5e+267) {
		tmp = t_1;
	} else {
		tmp = (a - t) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / b
    if (z <= (-1.9d-12)) then
        tmp = t_1
    else if (z <= 6.1d-21) then
        tmp = x + (z * (t / y))
    else if (z <= 1.5d+267) then
        tmp = t_1
    else
        tmp = (a - t) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / b;
	double tmp;
	if (z <= -1.9e-12) {
		tmp = t_1;
	} else if (z <= 6.1e-21) {
		tmp = x + (z * (t / y));
	} else if (z <= 1.5e+267) {
		tmp = t_1;
	} else {
		tmp = (a - t) / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / b
	tmp = 0
	if z <= -1.9e-12:
		tmp = t_1
	elif z <= 6.1e-21:
		tmp = x + (z * (t / y))
	elif z <= 1.5e+267:
		tmp = t_1
	else:
		tmp = (a - t) / y
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / b)
	tmp = 0.0
	if (z <= -1.9e-12)
		tmp = t_1;
	elseif (z <= 6.1e-21)
		tmp = Float64(x + Float64(z * Float64(t / y)));
	elseif (z <= 1.5e+267)
		tmp = t_1;
	else
		tmp = Float64(Float64(a - t) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / b;
	tmp = 0.0;
	if (z <= -1.9e-12)
		tmp = t_1;
	elseif (z <= 6.1e-21)
		tmp = x + (z * (t / y));
	elseif (z <= 1.5e+267)
		tmp = t_1;
	else
		tmp = (a - t) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[z, -1.9e-12], t$95$1, If[LessEqual[z, 6.1e-21], N[(x + N[(z * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.5e+267], t$95$1, N[(N[(a - t), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.89999999999999998e-12 or 6.10000000000000013e-21 < z < 1.5e267

   1. Initial program 49.3%

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

   3. Taylor expanded in y around 0 55.5%

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

   if -1.89999999999999998e-12 < z < 6.10000000000000013e-21

   1. Initial program 87.2%

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

   3. Taylor expanded in z around 0 52.7%

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

   4. Taylor expanded in t around inf 59.6%

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

   if 1.5e267 < z

   1. Initial program 9.5%

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

   3. Taylor expanded in z around inf 93.2%

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

   4. Taylor expanded in b around 0 63.8%

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

      1. mul-1-neg 63.8%

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

      2. distribute-neg-frac2 63.8%

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

   6. Simplified 63.8%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{-12}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;z \leq 6.1 \cdot 10^{-21}:\\ \;\;\;\;x + z \cdot \frac{t}{y}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+267}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-9} \lor \neg \left(z \leq 1.52 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{t}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -5.5e-9) (not (<= z 1.52e-18)))
   (/ (- t a) (- b y))
   (+ x (* z (/ t y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -5.5e-9) || !(z <= 1.52e-18)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x + (z * (t / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-5.5d-9)) .or. (.not. (z <= 1.52d-18))) then
        tmp = (t - a) / (b - y)
    else
        tmp = x + (z * (t / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -5.5e-9) || !(z <= 1.52e-18)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = x + (z * (t / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -5.5e-9) or not (z <= 1.52e-18):
		tmp = (t - a) / (b - y)
	else:
		tmp = x + (z * (t / y))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -5.5e-9) || !(z <= 1.52e-18))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(x + Float64(z * Float64(t / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -5.5e-9) || ~((z <= 1.52e-18)))
		tmp = (t - a) / (b - y);
	else
		tmp = x + (z * (t / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -5.5e-9], N[Not[LessEqual[z, 1.52e-18]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x + N[(z * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999996e-9 or 1.52e-18 < z

   1. Initial program 44.8%

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

   3. Taylor expanded in z around inf 82.4%

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

   if -5.4999999999999996e-9 < z < 1.52e-18

   1. Initial program 87.3%

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

   3. Taylor expanded in z around 0 53.1%

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

   4. Taylor expanded in t around inf 59.1%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-9} \lor \neg \left(z \leq 1.52 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{t}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+63} \lor \neg \left(y \leq 2.7 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1e+63) (not (<= y 2.7e+47))) (/ x (- 1.0 z)) (/ (- t a) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1e+63) || !(y <= 2.7e+47)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1d+63)) .or. (.not. (y <= 2.7d+47))) then
        tmp = x / (1.0d0 - z)
    else
        tmp = (t - a) / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1e+63) || !(y <= 2.7e+47)) {
		tmp = x / (1.0 - z);
	} else {
		tmp = (t - a) / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1e+63) or not (y <= 2.7e+47):
		tmp = x / (1.0 - z)
	else:
		tmp = (t - a) / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1e+63) || !(y <= 2.7e+47))
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(Float64(t - a) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1e+63) || ~((y <= 2.7e+47)))
		tmp = x / (1.0 - z);
	else
		tmp = (t - a) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1e+63], N[Not[LessEqual[y, 2.7e+47]], $MachinePrecision]], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.00000000000000006e63 or 2.69999999999999996e47 < y

   1. Initial program 46.3%

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

   3. Taylor expanded in y around inf 56.3%

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg 56.3%

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

      2. unsub-neg 56.3%

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   5. Simplified 56.3%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

   if -1.00000000000000006e63 < y < 2.69999999999999996e47

   1. Initial program 79.1%

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

   3. Taylor expanded in y around 0 54.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+63} \lor \neg \left(y \leq 2.7 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{elif}\;z \leq 3.4:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -7.5e-24)
   (/ (- a) b)
   (if (<= z 3.4) (/ x (- 1.0 z)) (/ t (- b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.5e-24) {
		tmp = -a / b;
	} else if (z <= 3.4) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t / (b - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-7.5d-24)) then
        tmp = -a / b
    else if (z <= 3.4d0) then
        tmp = x / (1.0d0 - z)
    else
        tmp = t / (b - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.5e-24) {
		tmp = -a / b;
	} else if (z <= 3.4) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t / (b - y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -7.5e-24:
		tmp = -a / b
	elif z <= 3.4:
		tmp = x / (1.0 - z)
	else:
		tmp = t / (b - y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -7.5e-24)
		tmp = Float64(Float64(-a) / b);
	elseif (z <= 3.4)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = Float64(t / Float64(b - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -7.5e-24)
		tmp = -a / b;
	elseif (z <= 3.4)
		tmp = x / (1.0 - z);
	else
		tmp = t / (b - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -7.5e-24], N[((-a) / b), $MachinePrecision], If[LessEqual[z, 3.4], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.50000000000000007e-24

   1. Initial program 47.0%

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

   3. Taylor expanded in y around 0 52.5%

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

   4. Taylor expanded in t around 0 36.3%

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

      1. neg-mul-1 36.3%

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

   6. Simplified 36.3%

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

   if -7.50000000000000007e-24 < z < 3.39999999999999991

   1. Initial program 86.5%

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

   3. Taylor expanded in y around inf 47.6%

      \[\leadsto \color{blue}{\frac{x}{1 + -1 \cdot z}} \]
   4. Step-by-step derivation

      1. mul-1-neg 47.6%

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

      2. unsub-neg 47.6%

         \[\leadsto \frac{x}{\color{blue}{1 - z}} \]

   5. Simplified 47.6%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]

   if 3.39999999999999991 < z

   1. Initial program 43.9%

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

   3. Taylor expanded in t around inf 27.1%

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

   4. Taylor expanded in z around inf 50.8%

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

Alternative 11: 41.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;x + z \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -1.8e-26) (/ (- a) b) (if (<= z 0.5) (+ x (* z x)) (/ t (- b y)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.8e-26) {
		tmp = -a / b;
	} else if (z <= 0.5) {
		tmp = x + (z * x);
	} else {
		tmp = t / (b - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-1.8d-26)) then
        tmp = -a / b
    else if (z <= 0.5d0) then
        tmp = x + (z * x)
    else
        tmp = t / (b - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -1.8e-26) {
		tmp = -a / b;
	} else if (z <= 0.5) {
		tmp = x + (z * x);
	} else {
		tmp = t / (b - y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -1.8e-26:
		tmp = -a / b
	elif z <= 0.5:
		tmp = x + (z * x)
	else:
		tmp = t / (b - y)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -1.8e-26)
		tmp = Float64(Float64(-a) / b);
	elseif (z <= 0.5)
		tmp = Float64(x + Float64(z * x));
	else
		tmp = Float64(t / Float64(b - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -1.8e-26)
		tmp = -a / b;
	elseif (z <= 0.5)
		tmp = x + (z * x);
	else
		tmp = t / (b - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -1.8e-26], N[((-a) / b), $MachinePrecision], If[LessEqual[z, 0.5], N[(x + N[(z * x), $MachinePrecision]), $MachinePrecision], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8000000000000001e-26

   1. Initial program 47.0%

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

   3. Taylor expanded in y around 0 52.5%

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

   4. Taylor expanded in t around 0 36.3%

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

      1. neg-mul-1 36.3%

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

   6. Simplified 36.3%

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

   if -1.8000000000000001e-26 < z < 0.5

   1. Initial program 86.5%

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

   3. Taylor expanded in z around 0 51.6%

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

   4. Taylor expanded in y around inf 47.6%

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

      1. *-commutative 47.6%

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

   6. Simplified 47.6%

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

   if 0.5 < z

   1. Initial program 43.9%

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

   3. Taylor expanded in t around inf 27.1%

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

   4. Taylor expanded in z around inf 50.8%

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

Alternative 12: 37.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-24}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;x + z \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -7e-24) (/ (- a) b) (if (<= z 0.5) (+ x (* z x)) (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7e-24) {
		tmp = -a / b;
	} else if (z <= 0.5) {
		tmp = x + (z * x);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-7d-24)) then
        tmp = -a / b
    else if (z <= 0.5d0) then
        tmp = x + (z * x)
    else
        tmp = t / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7e-24) {
		tmp = -a / b;
	} else if (z <= 0.5) {
		tmp = x + (z * x);
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -7e-24:
		tmp = -a / b
	elif z <= 0.5:
		tmp = x + (z * x)
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -7e-24)
		tmp = Float64(Float64(-a) / b);
	elseif (z <= 0.5)
		tmp = Float64(x + Float64(z * x));
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -7e-24)
		tmp = -a / b;
	elseif (z <= 0.5)
		tmp = x + (z * x);
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -7e-24], N[((-a) / b), $MachinePrecision], If[LessEqual[z, 0.5], N[(x + N[(z * x), $MachinePrecision]), $MachinePrecision], N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.9999999999999993e-24

   1. Initial program 47.0%

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

   3. Taylor expanded in y around 0 52.5%

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

   4. Taylor expanded in t around 0 36.3%

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

      1. neg-mul-1 36.3%

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

   6. Simplified 36.3%

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

   if -6.9999999999999993e-24 < z < 0.5

   1. Initial program 86.5%

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

   3. Taylor expanded in z around 0 51.6%

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

   4. Taylor expanded in y around inf 47.6%

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

      1. *-commutative 47.6%

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

   6. Simplified 47.6%

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

   if 0.5 < z

   1. Initial program 43.9%

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

   3. Taylor expanded in t around inf 27.1%

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

   4. Taylor expanded in y around 0 38.0%

      \[\leadsto \color{blue}{\frac{t}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 37.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-7} \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -6e-7) (not (<= z 0.5))) (/ t b) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -6e-7) || !(z <= 0.5)) {
		tmp = t / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-6d-7)) .or. (.not. (z <= 0.5d0))) then
        tmp = t / b
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -6e-7) || !(z <= 0.5)) {
		tmp = t / b;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -6e-7) or not (z <= 0.5):
		tmp = t / b
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -6e-7) || !(z <= 0.5))
		tmp = Float64(t / b);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -6e-7) || ~((z <= 0.5)))
		tmp = t / b;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -6e-7], N[Not[LessEqual[z, 0.5]], $MachinePrecision]], N[(t / b), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -5.9999999999999997e-7 or 0.5 < z

   1. Initial program 44.3%

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

   3. Taylor expanded in t around inf 21.5%

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

   4. Taylor expanded in y around 0 30.2%

      \[\leadsto \color{blue}{\frac{t}{b}} \]

   if -5.9999999999999997e-7 < z < 0.5

   1. Initial program 86.9%

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

   3. Taylor expanded in z around 0 46.6%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-7} \lor \neg \left(z \leq 0.5\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-24}:\\ \;\;\;\;\frac{-a}{b}\\ \mathbf{elif}\;z \leq 0.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -4e-24) (/ (- a) b) (if (<= z 0.5) x (/ t b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4e-24) {
		tmp = -a / b;
	} else if (z <= 0.5) {
		tmp = x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-4d-24)) then
        tmp = -a / b
    else if (z <= 0.5d0) then
        tmp = x
    else
        tmp = t / b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -4e-24) {
		tmp = -a / b;
	} else if (z <= 0.5) {
		tmp = x;
	} else {
		tmp = t / b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -4e-24:
		tmp = -a / b
	elif z <= 0.5:
		tmp = x
	else:
		tmp = t / b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -4e-24)
		tmp = Float64(Float64(-a) / b);
	elseif (z <= 0.5)
		tmp = x;
	else
		tmp = Float64(t / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -4e-24)
		tmp = -a / b;
	elseif (z <= 0.5)
		tmp = x;
	else
		tmp = t / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -4e-24], N[((-a) / b), $MachinePrecision], If[LessEqual[z, 0.5], x, N[(t / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.99999999999999969e-24

   1. Initial program 47.0%

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

   3. Taylor expanded in y around 0 52.5%

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

   4. Taylor expanded in t around 0 36.3%

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

      1. neg-mul-1 36.3%

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

   6. Simplified 36.3%

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

   if -3.99999999999999969e-24 < z < 0.5

   1. Initial program 86.5%

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

   3. Taylor expanded in z around 0 47.5%

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

   if 0.5 < z

   1. Initial program 43.9%

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

   3. Taylor expanded in t around inf 27.1%

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

   4. Taylor expanded in y around 0 38.0%

      \[\leadsto \color{blue}{\frac{t}{b}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 25.5% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.1%

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

3. Taylor expanded in z around 0 25.4%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Developer Target 1: 73.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ (+ (* z t) (* y x)) (+ y (* z (- b y)))) (/ a (+ (- b y) (/ y z)))))

  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))