Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.7% → 93.7%

Time: 12.7s

Alternatives: 15

Speedup: 1.3×

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: 66.7% 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: 93.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-1}{\frac{z}{\frac{y}{y - b} \cdot x}} - \frac{a - t}{b - y}\\ \mathbf{if}\;z \leq -2000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 25000000000:\\ \;\;\;\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (/ -1.0 (/ z (* (/ y (- y b)) x))) (/ (- a t) (- b y)))))
   (if (<= z -2000000000.0)
     t_1
     (if (<= z 25000000000.0)
       (/ (- (* y x) (* (- a t) z)) (+ (* (- b y) z) y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (-1.0 / (z / ((y / (y - b)) * x))) - ((a - t) / (b - y));
	double tmp;
	if (z <= -2000000000.0) {
		tmp = t_1;
	} else if (z <= 25000000000.0) {
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + 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 = ((-1.0d0) / (z / ((y / (y - b)) * x))) - ((a - t) / (b - y))
    if (z <= (-2000000000.0d0)) then
        tmp = t_1
    else if (z <= 25000000000.0d0) then
        tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + 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 = (-1.0 / (z / ((y / (y - b)) * x))) - ((a - t) / (b - y));
	double tmp;
	if (z <= -2000000000.0) {
		tmp = t_1;
	} else if (z <= 25000000000.0) {
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (-1.0 / (z / ((y / (y - b)) * x))) - ((a - t) / (b - y))
	tmp = 0
	if z <= -2000000000.0:
		tmp = t_1
	elif z <= 25000000000.0:
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(-1.0 / Float64(z / Float64(Float64(y / Float64(y - b)) * x))) - Float64(Float64(a - t) / Float64(b - y)))
	tmp = 0.0
	if (z <= -2000000000.0)
		tmp = t_1;
	elseif (z <= 25000000000.0)
		tmp = Float64(Float64(Float64(y * x) - Float64(Float64(a - t) * z)) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (-1.0 / (z / ((y / (y - b)) * x))) - ((a - t) / (b - y));
	tmp = 0.0;
	if (z <= -2000000000.0)
		tmp = t_1;
	elseif (z <= 25000000000.0)
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(-1.0 / N[(z / N[(N[(y / N[(y - b), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(a - t), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2000000000.0], t$95$1, If[LessEqual[z, 25000000000.0], N[(N[(N[(y * x), $MachinePrecision] - N[(N[(a - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2e9 or 2.5e10 < z

   1. Initial program 48.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 92.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{t - a}{b - y} - \frac{-1 \cdot \frac{x \cdot y}{b - y}}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.2%

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

   10. Applied rewrites 99.7%

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

   if -2e9 < z < 2.5e10

   1. Initial program 86.0%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2000000000:\\ \;\;\;\;\frac{-1}{\frac{z}{\frac{y}{y - b} \cdot x}} - \frac{a - t}{b - y}\\ \mathbf{elif}\;z \leq 25000000000:\\ \;\;\;\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{z}{\frac{y}{y - b} \cdot x}} - \frac{a - t}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b} - \frac{\frac{y \cdot x}{y - b}}{z}\\ \mathbf{if}\;z \leq -10200000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (- (/ (- a t) (- y b)) (/ (/ (* y x) (- y b)) z))))
   (if (<= z -10200000000000.0)
     t_1
     (if (<= z 2.7e+15)
       (/ (- (* y x) (* (- a t) z)) (+ (* (- b y) z) y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = ((a - t) / (y - b)) - (((y * x) / (y - b)) / z);
	double tmp;
	if (z <= -10200000000000.0) {
		tmp = t_1;
	} else if (z <= 2.7e+15) {
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + 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 = ((a - t) / (y - b)) - (((y * x) / (y - b)) / z)
    if (z <= (-10200000000000.0d0)) then
        tmp = t_1
    else if (z <= 2.7d+15) then
        tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + 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 = ((a - t) / (y - b)) - (((y * x) / (y - b)) / z);
	double tmp;
	if (z <= -10200000000000.0) {
		tmp = t_1;
	} else if (z <= 2.7e+15) {
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = ((a - t) / (y - b)) - (((y * x) / (y - b)) / z)
	tmp = 0
	if z <= -10200000000000.0:
		tmp = t_1
	elif z <= 2.7e+15:
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(a - t) / Float64(y - b)) - Float64(Float64(Float64(y * x) / Float64(y - b)) / z))
	tmp = 0.0
	if (z <= -10200000000000.0)
		tmp = t_1;
	elseif (z <= 2.7e+15)
		tmp = Float64(Float64(Float64(y * x) - Float64(Float64(a - t) * z)) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((a - t) / (y - b)) - (((y * x) / (y - b)) / z);
	tmp = 0.0;
	if (z <= -10200000000000.0)
		tmp = t_1;
	elseif (z <= 2.7e+15)
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(y * x), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -10200000000000.0], t$95$1, If[LessEqual[z, 2.7e+15], N[(N[(N[(y * x), $MachinePrecision] - N[(N[(a - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.02e13 or 2.7e15 < z

   1. Initial program 48.2%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 92.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{t - a}{b - y} - \frac{-1 \cdot \frac{x \cdot y}{b - y}}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 87.6%

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

   if -1.02e13 < z < 2.7e15

   1. Initial program 85.6%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -10200000000000:\\ \;\;\;\;\frac{a - t}{y - b} - \frac{\frac{y \cdot x}{y - b}}{z}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b} - \frac{\frac{y \cdot x}{y - b}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+23}:\\ \;\;\;\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -3.4e+76)
     t_1
     (if (<= z 2.55e+23)
       (/ (- (* y x) (* (- a t) z)) (+ (* (- b y) z) y))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -3.4e+76) {
		tmp = t_1;
	} else if (z <= 2.55e+23) {
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + 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 = (a - t) / (y - b)
    if (z <= (-3.4d+76)) then
        tmp = t_1
    else if (z <= 2.55d+23) then
        tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + 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 = (a - t) / (y - b);
	double tmp;
	if (z <= -3.4e+76) {
		tmp = t_1;
	} else if (z <= 2.55e+23) {
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (a - t) / (y - b)
	tmp = 0
	if z <= -3.4e+76:
		tmp = t_1
	elif z <= 2.55e+23:
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -3.4e+76)
		tmp = t_1;
	elseif (z <= 2.55e+23)
		tmp = Float64(Float64(Float64(y * x) - Float64(Float64(a - t) * z)) / Float64(Float64(Float64(b - y) * z) + y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (a - t) / (y - b);
	tmp = 0.0;
	if (z <= -3.4e+76)
		tmp = t_1;
	elseif (z <= 2.55e+23)
		tmp = ((y * x) - ((a - t) * z)) / (((b - y) * z) + y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.4e+76], t$95$1, If[LessEqual[z, 2.55e+23], N[(N[(N[(y * x), $MachinePrecision] - N[(N[(a - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b - y), $MachinePrecision] * z), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3999999999999997e76 or 2.5500000000000001e23 < z

   1. Initial program 43.7%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 81.4

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

   5. Applied rewrites 81.4%

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

   if -3.3999999999999997e76 < z < 2.5500000000000001e23

   1. Initial program 85.6%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+76}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{+23}:\\ \;\;\;\;\frac{y \cdot x - \left(a - t\right) \cdot z}{\left(b - y\right) \cdot z + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -2.75 \cdot 10^{-61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y} - \left(-x\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -2.75e-61)
     t_1
     (if (<= z 4.8e-42) (fma (- (/ (- t a) y) (- x)) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -2.75e-61) {
		tmp = t_1;
	} else if (z <= 4.8e-42) {
		tmp = fma((((t - a) / y) - -x), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -2.75e-61)
		tmp = t_1;
	elseif (z <= 4.8e-42)
		tmp = fma(Float64(Float64(Float64(t - a) / y) - Float64(-x)), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.75e-61], t$95$1, If[LessEqual[z, 4.8e-42], N[(N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] - (-x)), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7499999999999998e-61 or 4.80000000000000005e-42 < z

   1. Initial program 53.3%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 76.7

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

   5. Applied rewrites 76.7%

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

   if -2.7499999999999998e-61 < z < 4.80000000000000005e-42

   1. Initial program 84.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate--r+ N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 75.1%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{-61}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y} - \left(-x\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -6 \cdot 10^{-111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -6e-111) t_1 (if (<= z 4.4e-42) (fma (/ (- a) y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -6e-111) {
		tmp = t_1;
	} else if (z <= 4.4e-42) {
		tmp = fma((-a / y), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -6e-111)
		tmp = t_1;
	elseif (z <= 4.4e-42)
		tmp = fma(Float64(Float64(-a) / y), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6e-111], t$95$1, If[LessEqual[z, 4.4e-42], N[(N[((-a) / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.00000000000000016e-111 or 4.4000000000000001e-42 < z

   1. Initial program 55.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 73.9

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

   5. Applied rewrites 73.9%

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

   if -6.00000000000000016e-111 < z < 4.4000000000000001e-42

   1. Initial program 84.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate--r+ N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 69.8%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-111}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-a}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{-112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -1.2e-112) t_1 (if (<= z 4.4e-42) (fma (/ t y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -1.2e-112) {
		tmp = t_1;
	} else if (z <= 4.4e-42) {
		tmp = fma((t / y), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -1.2e-112)
		tmp = t_1;
	elseif (z <= 4.4e-42)
		tmp = fma(Float64(t / y), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.2e-112], t$95$1, If[LessEqual[z, 4.4e-42], N[(N[(t / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e-112 or 4.4000000000000001e-42 < z

   1. Initial program 56.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -1.2e-112 < z < 4.4000000000000001e-42

   1. Initial program 84.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate--r+ N/A

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

      5. div-sub N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 61.4

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

   5. Applied rewrites 61.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 69.2%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-112}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a - t}{y - b}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{-113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a t) (- y b))))
   (if (<= z -7.2e-113) t_1 (if (<= z 7.4e-62) (fma z x x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (a - t) / (y - b);
	double tmp;
	if (z <= -7.2e-113) {
		tmp = t_1;
	} else if (z <= 7.4e-62) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(a - t) / Float64(y - b))
	tmp = 0.0
	if (z <= -7.2e-113)
		tmp = t_1;
	elseif (z <= 7.4e-62)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(a - t), $MachinePrecision] / N[(y - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e-113], t$95$1, If[LessEqual[z, 7.4e-62], N[(z * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.1999999999999995e-113 or 7.3999999999999996e-62 < z

   1. Initial program 56.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 73.5

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

   5. Applied rewrites 73.5%

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

   if -7.1999999999999995e-113 < z < 7.3999999999999996e-62

   1. Initial program 84.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.6%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-113}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-62}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a - t}{y - b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{1 - z}\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{-39}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ x (- 1.0 z))))
   (if (<= y -5.6e+15) t_1 (if (<= y 3.3e-39) (/ (- t a) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -5.6e+15) {
		tmp = t_1;
	} else if (y <= 3.3e-39) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (1.0d0 - z)
    if (y <= (-5.6d+15)) then
        tmp = t_1
    else if (y <= 3.3d-39) then
        tmp = (t - a) / b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = x / (1.0 - z);
	double tmp;
	if (y <= -5.6e+15) {
		tmp = t_1;
	} else if (y <= 3.3e-39) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = x / (1.0 - z)
	tmp = 0
	if y <= -5.6e+15:
		tmp = t_1
	elif y <= 3.3e-39:
		tmp = (t - a) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(x / Float64(1.0 - z))
	tmp = 0.0
	if (y <= -5.6e+15)
		tmp = t_1;
	elseif (y <= 3.3e-39)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = x / (1.0 - z);
	tmp = 0.0;
	if (y <= -5.6e+15)
		tmp = t_1;
	elseif (y <= 3.3e-39)
		tmp = (t - a) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.6e+15], t$95$1, If[LessEqual[y, 3.3e-39], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.6e15 or 3.29999999999999985e-39 < y

   1. Initial program 51.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 56.8

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

   5. Applied rewrites 56.8%

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

   if -5.6e15 < y < 3.29999999999999985e-39

   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

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 55.2

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

   5. Applied rewrites 55.2%

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

Alternative 9: 43.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -2.4e+111) t_1 (if (<= z 6.6e-52) (/ x (- 1.0 z)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -2.4e+111) {
		tmp = t_1;
	} else if (z <= 6.6e-52) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (b - y)
    if (z <= (-2.4d+111)) then
        tmp = t_1
    else if (z <= 6.6d-52) then
        tmp = x / (1.0d0 - z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -2.4e+111) {
		tmp = t_1;
	} else if (z <= 6.6e-52) {
		tmp = x / (1.0 - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if z <= -2.4e+111:
		tmp = t_1
	elif z <= 6.6e-52:
		tmp = x / (1.0 - z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -2.4e+111)
		tmp = t_1;
	elseif (z <= 6.6e-52)
		tmp = Float64(x / Float64(1.0 - z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	tmp = 0.0;
	if (z <= -2.4e+111)
		tmp = t_1;
	elseif (z <= 6.6e-52)
		tmp = x / (1.0 - z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.4e+111], t$95$1, If[LessEqual[z, 6.6e-52], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.40000000000000006e111 or 6.5999999999999999e-52 < z

   1. Initial program 48.8%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 80.4

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 49.1%

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

   if -2.40000000000000006e111 < z < 6.5999999999999999e-52

   1. Initial program 83.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 51.8

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

   5. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{x}{1 - z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 43.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{-112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -1.2e-112) t_1 (if (<= z 6.6e-52) (fma z x x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -1.2e-112) {
		tmp = t_1;
	} else if (z <= 6.6e-52) {
		tmp = fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -1.2e-112)
		tmp = t_1;
	elseif (z <= 6.6e-52)
		tmp = fma(z, x, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.2e-112], t$95$1, If[LessEqual[z, 6.6e-52], N[(z * x + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e-112 or 6.5999999999999999e-52 < z

   1. Initial program 56.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower--.f64 73.5

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 43.7%

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

   if -1.2e-112 < z < 6.5999999999999999e-52

   1. Initial program 84.6%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.6%

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

Alternative 11: 34.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x}{-z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-a}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -5.5e+19) (/ x (- z)) (if (<= z 2e-42) (fma z x x) (/ (- a) b))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -5.5e+19) {
		tmp = x / -z;
	} else if (z <= 2e-42) {
		tmp = fma(z, x, x);
	} else {
		tmp = -a / b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -5.5e+19)
		tmp = Float64(x / Float64(-z));
	elseif (z <= 2e-42)
		tmp = fma(z, x, x);
	else
		tmp = Float64(Float64(-a) / b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -5.5e+19], N[(x / (-z)), $MachinePrecision], If[LessEqual[z, 2e-42], N[(z * x + x), $MachinePrecision], N[((-a) / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5e19

   1. Initial program 40.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 28.5

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

   5. Applied rewrites 28.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 28.5%

      \[\leadsto \frac{x}{-z} \]

   if -5.5e19 < z < 2.00000000000000008e-42

   1. Initial program 84.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 53.8

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 53.8%

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

   if 2.00000000000000008e-42 < z

   1. Initial program 60.8%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 28.6

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 18.4%

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

Alternative 12: 36.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-a}{b}\\ \mathbf{if}\;z \leq -1.16 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- a) b)))
   (if (<= z -1.16e-15) t_1 (if (<= z 2e-42) (/ x 1.0) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -a / b;
	double tmp;
	if (z <= -1.16e-15) {
		tmp = t_1;
	} else if (z <= 2e-42) {
		tmp = x / 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -a / b
    if (z <= (-1.16d-15)) then
        tmp = t_1
    else if (z <= 2d-42) then
        tmp = x / 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -a / b;
	double tmp;
	if (z <= -1.16e-15) {
		tmp = t_1;
	} else if (z <= 2e-42) {
		tmp = x / 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -a / b
	tmp = 0
	if z <= -1.16e-15:
		tmp = t_1
	elif z <= 2e-42:
		tmp = x / 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(-a) / b)
	tmp = 0.0
	if (z <= -1.16e-15)
		tmp = t_1;
	elseif (z <= 2e-42)
		tmp = Float64(x / 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -a / b;
	tmp = 0.0;
	if (z <= -1.16e-15)
		tmp = t_1;
	elseif (z <= 2e-42)
		tmp = x / 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[((-a) / b), $MachinePrecision]}, If[LessEqual[z, -1.16e-15], t$95$1, If[LessEqual[z, 2e-42], N[(x / 1.0), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1599999999999999e-15 or 2.00000000000000008e-42 < z

   1. Initial program 52.7%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. neg-mul-1 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower--.f64 26.8

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

   5. Applied rewrites 26.8%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 21.4%

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

   if -1.1599999999999999e-15 < z < 2.00000000000000008e-42

   1. Initial program 84.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in z around 0

      \[\leadsto \frac{x}{1} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.3%

      \[\leadsto \frac{x}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 25.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(z, x, x\right), z, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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 y around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 35.5

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

5. Applied rewrites 35.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 26.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z, x, x\right), \color{blue}{z}, x\right) \]
9. Add Preprocessing

Alternative 14: 24.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, x, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

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 y around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 35.5

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

5. Applied rewrites 35.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 25.6%

   \[\leadsto \mathsf{fma}\left(z, \color{blue}{x}, x\right) \]
9. Add Preprocessing

Alternative 15: 3.8% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 y around inf

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 35.5

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

5. Applied rewrites 35.5%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 25.6%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 3.4%

    \[\leadsto z \cdot x \]

12. Final simplification 3.4%

    \[\leadsto x \cdot z \]
13. Add Preprocessing

Developer Target 1: 74.2% accurate, 0.6× 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 2024270 
(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)))))