Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 67.3% → 90.7%

Time: 10.5s

Alternatives: 16

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: 67.3% 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: 90.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+192}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+21} \lor \neg \left(z \leq 16800000000\right):\\ \;\;\;\;\frac{\frac{y}{b - y} \cdot \left(x - t\_1\right)}{z} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.3e+192)
     t_1
     (if (or (<= z -4.8e+21) (not (<= z 16800000000.0)))
       (+ (/ (* (/ y (- b y)) (- x t_1)) z) t_1)
       (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))))))

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 <= -1.3e+192) {
		tmp = t_1;
	} else if ((z <= -4.8e+21) || !(z <= 16800000000.0)) {
		tmp = (((y / (b - y)) * (x - t_1)) / z) + t_1;
	} else {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (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 = (t - a) / (b - y)
    if (z <= (-1.3d+192)) then
        tmp = t_1
    else if ((z <= (-4.8d+21)) .or. (.not. (z <= 16800000000.0d0))) then
        tmp = (((y / (b - y)) * (x - t_1)) / z) + t_1
    else
        tmp = ((x * y) + (z * (t - a))) / (y + (z * (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 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.3e+192) {
		tmp = t_1;
	} else if ((z <= -4.8e+21) || !(z <= 16800000000.0)) {
		tmp = (((y / (b - y)) * (x - t_1)) / z) + t_1;
	} else {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -1.3e+192:
		tmp = t_1
	elif (z <= -4.8e+21) or not (z <= 16800000000.0):
		tmp = (((y / (b - y)) * (x - t_1)) / z) + t_1
	else:
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
	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 <= -1.3e+192)
		tmp = t_1;
	elseif ((z <= -4.8e+21) || !(z <= 16800000000.0))
		tmp = Float64(Float64(Float64(Float64(y / Float64(b - y)) * Float64(x - t_1)) / z) + t_1);
	else
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))));
	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 <= -1.3e+192)
		tmp = t_1;
	elseif ((z <= -4.8e+21) || ~((z <= 16800000000.0)))
		tmp = (((y / (b - y)) * (x - t_1)) / z) + t_1;
	else
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	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, -1.3e+192], t$95$1, If[Or[LessEqual[z, -4.8e+21], N[Not[LessEqual[z, 16800000000.0]], $MachinePrecision]], N[(N[(N[(N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision] * N[(x - t$95$1), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + t$95$1), $MachinePrecision], 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]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.30000000000000002e192

   1. Initial program 37.0%

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

   3. Taylor expanded in 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 96.0

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

   5. Applied rewrites 96.0%

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

   if -1.30000000000000002e192 < z < -4.8e21 or 1.68e10 < z

   1. Initial program 47.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 97.9%

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

   if -4.8e21 < z < 1.68e10

   1. Initial program 86.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+192}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+21} \lor \neg \left(z \leq 16800000000\right):\\ \;\;\;\;\frac{\frac{y}{b - y} \cdot \left(x - \frac{t - a}{b - y}\right)}{z} + \frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -9.6 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{+17}:\\ \;\;\;\;\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z} + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -9.6e+22)
     t_1
     (if (<= z 3.3e+17)
       (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y))))
       (+ (/ (- x) z) 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 <= -9.6e+22) {
		tmp = t_1;
	} else if (z <= 3.3e+17) {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	} else {
		tmp = (-x / z) + 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 <= (-9.6d+22)) then
        tmp = t_1
    else if (z <= 3.3d+17) then
        tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
    else
        tmp = (-x / z) + 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 <= -9.6e+22) {
		tmp = t_1;
	} else if (z <= 3.3e+17) {
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	} else {
		tmp = (-x / z) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -9.6e+22:
		tmp = t_1
	elif z <= 3.3e+17:
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
	else:
		tmp = (-x / z) + 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 <= -9.6e+22)
		tmp = t_1;
	elseif (z <= 3.3e+17)
		tmp = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = Float64(Float64(Float64(-x) / z) + 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 <= -9.6e+22)
		tmp = t_1;
	elseif (z <= 3.3e+17)
		tmp = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	else
		tmp = (-x / z) + 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, -9.6e+22], t$95$1, If[LessEqual[z, 3.3e+17], 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], N[(N[((-x) / z), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.6e22

   1. Initial program 51.1%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 89.9

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

   5. Applied rewrites 89.9%

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

   if -9.6e22 < z < 3.3e17

   1. Initial program 86.4%

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

   if 3.3e17 < z

   1. Initial program 38.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 98.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 92.2%

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

Alternative 3: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 55000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{y + z \cdot \left(b - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z} + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.8e+20)
     t_1
     (if (<= z 55000000.0)
       (/ (fma t z (* y x)) (+ y (* z (- b y))))
       (+ (/ (- x) z) 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 <= -1.8e+20) {
		tmp = t_1;
	} else if (z <= 55000000.0) {
		tmp = fma(t, z, (y * x)) / (y + (z * (b - y)));
	} else {
		tmp = (-x / z) + 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 <= -1.8e+20)
		tmp = t_1;
	elseif (z <= 55000000.0)
		tmp = Float64(fma(t, z, Float64(y * x)) / Float64(y + Float64(z * Float64(b - y))));
	else
		tmp = Float64(Float64(Float64(-x) / z) + t_1);
	end
	return 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, -1.8e+20], t$95$1, If[LessEqual[z, 55000000.0], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8e20

   1. Initial program 51.1%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 89.9

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

   5. Applied rewrites 89.9%

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

   if -1.8e20 < z < 5.5e7

   1. Initial program 86.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 63.3

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

   5. Applied rewrites 63.3%

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

   if 5.5e7 < z

   1. Initial program 38.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 98.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 92.2%

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

Alternative 4: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 55000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z} + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.8e+20)
     t_1
     (if (<= z 55000000.0)
       (/ (fma t z (* y x)) (fma (- b y) z y))
       (+ (/ (- x) z) 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 <= -1.8e+20) {
		tmp = t_1;
	} else if (z <= 55000000.0) {
		tmp = fma(t, z, (y * x)) / fma((b - y), z, y);
	} else {
		tmp = (-x / z) + 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 <= -1.8e+20)
		tmp = t_1;
	elseif (z <= 55000000.0)
		tmp = Float64(fma(t, z, Float64(y * x)) / fma(Float64(b - y), z, y));
	else
		tmp = Float64(Float64(Float64(-x) / z) + t_1);
	end
	return 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, -1.8e+20], t$95$1, If[LessEqual[z, 55000000.0], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8e20

   1. Initial program 51.1%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 89.9

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

   5. Applied rewrites 89.9%

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

   if -1.8e20 < z < 5.5e7

   1. Initial program 86.4%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 63.3

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

   5. Applied rewrites 63.3%

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

   if 5.5e7 < z

   1. Initial program 38.6%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 98.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 92.2%

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

Alternative 5: 72.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+20} \lor \neg \left(z \leq 2.4 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.8e+20) (not (<= z 2.4e-14)))
   (/ (- t a) (- b y))
   (/ (fma t z (* y x)) (fma (- b y) z y))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.8e+20) || !(z <= 2.4e-14)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(t, z, (y * x)) / fma((b - y), z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.8e+20) || !(z <= 2.4e-14))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(fma(t, z, Float64(y * x)) / fma(Float64(b - y), z, y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.8e+20], N[Not[LessEqual[z, 2.4e-14]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e20 or 2.4e-14 < z

   1. Initial program 47.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 89.7

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

   5. Applied rewrites 89.7%

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

   if -1.8e20 < z < 2.4e-14

   1. Initial program 85.9%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 63.4

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

   5. Applied rewrites 63.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+20} \lor \neg \left(z \leq 2.4 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{\mathsf{fma}\left(b - y, z, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+20}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-88}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{t\_1}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{t\_1} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -1.8e+20)
     t_2
     (if (<= z -7e-88)
       (/ (* (- t a) z) t_1)
       (if (<= z 3e-14) (* (/ y t_1) x) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.8e+20) {
		tmp = t_2;
	} else if (z <= -7e-88) {
		tmp = ((t - a) * z) / t_1;
	} else if (z <= 3e-14) {
		tmp = (y / t_1) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.8e+20)
		tmp = t_2;
	elseif (z <= -7e-88)
		tmp = Float64(Float64(Float64(t - a) * z) / t_1);
	elseif (z <= 3e-14)
		tmp = Float64(Float64(y / t_1) * x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.8e+20], t$95$2, If[LessEqual[z, -7e-88], N[(N[(N[(t - a), $MachinePrecision] * z), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 3e-14], N[(N[(y / t$95$1), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.8e20 or 2.9999999999999998e-14 < z

   1. Initial program 47.7%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 89.7

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

   5. Applied rewrites 89.7%

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

   if -1.8e20 < z < -7.0000000000000002e-88

   1. Initial program 84.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 84.0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower--.f64 64.8

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

   7. Applied rewrites 64.8%

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

   if -7.0000000000000002e-88 < z < 2.9999999999999998e-14

   1. Initial program 86.3%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 64.8

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

   5. Applied rewrites 64.8%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-88}:\\ \;\;\;\;\frac{\left(t - a\right) \cdot z}{\mathsf{fma}\left(b - y, z, y\right)}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t - a}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-40} \lor \neg \left(z \leq 3 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -2.9e-40) (not (<= z 3e-14)))
   (/ (- t a) (- b y))
   (* (/ y (fma (- b y) z y)) x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -2.9e-40) || !(z <= 3e-14)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = (y / fma((b - y), z, y)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -2.9e-40) || !(z <= 3e-14))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = Float64(Float64(y / fma(Float64(b - y), z, y)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -2.9e-40], N[Not[LessEqual[z, 3e-14]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.8999999999999999e-40 or 2.9999999999999998e-14 < z

   1. Initial program 51.8%

      \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
   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 85.9

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

   5. Applied rewrites 85.9%

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

   if -2.8999999999999999e-40 < z < 2.9999999999999998e-14

   1. Initial program 85.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 62.5

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

   5. Applied rewrites 62.5%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-40} \lor \neg \left(z \leq 3 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(b - y, z, y\right)} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 46.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-41}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+241}:\\ \;\;\;\;\frac{-a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -3.9e-41)
   (/ (- t a) b)
   (if (<= z 3.3e-25)
     (fma x z x)
     (if (<= z 2.35e+241) (/ (- a) (- b y)) (/ t (- b y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -3.9e-41) {
		tmp = (t - a) / b;
	} else if (z <= 3.3e-25) {
		tmp = fma(x, z, x);
	} else if (z <= 2.35e+241) {
		tmp = -a / (b - y);
	} else {
		tmp = t / (b - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -3.9e-41)
		tmp = Float64(Float64(t - a) / b);
	elseif (z <= 3.3e-25)
		tmp = fma(x, z, x);
	elseif (z <= 2.35e+241)
		tmp = Float64(Float64(-a) / Float64(b - y));
	else
		tmp = Float64(t / Float64(b - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -3.9e-41], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[z, 3.3e-25], N[(x * z + x), $MachinePrecision], If[LessEqual[z, 2.35e+241], N[((-a) / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.89999999999999991e-41

   1. Initial program 57.7%

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

   3. Taylor expanded in 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 59.6

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

   5. Applied rewrites 59.6%

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

   if -3.89999999999999991e-41 < z < 3.2999999999999998e-25

   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 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 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 54.9%

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

   if 3.2999999999999998e-25 < z < 2.34999999999999991e241

   1. Initial program 50.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 50.3

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

   4. Applied rewrites 50.3%

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

   5. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 84.6

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

   7. Applied rewrites 84.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 61.0%

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

   if 2.34999999999999991e241 < z

   1. Initial program 34.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 35.7

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

   5. Applied rewrites 35.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 68.1%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-41}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{+241}:\\ \;\;\;\;\frac{-a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{b - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-91} \lor \neg \left(z \leq 3.2 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -6.4e-91) (not (<= z 3.2e-25)))
   (/ (- t a) (- b y))
   (fma x z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -6.4e-91) || !(z <= 3.2e-25)) {
		tmp = (t - a) / (b - y);
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -6.4e-91) || !(z <= 3.2e-25))
		tmp = Float64(Float64(t - a) / Float64(b - y));
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -6.4e-91], N[Not[LessEqual[z, 3.2e-25]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.39999999999999992e-91 or 3.2000000000000001e-25 < z

   1. Initial program 54.9%

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

   3. Taylor expanded in z around 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 80.0

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

   5. Applied rewrites 80.0%

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

   if -6.39999999999999992e-91 < z < 3.2000000000000001e-25

   1. Initial program 85.7%

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

   3. Taylor expanded in 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 57.7

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

   5. Applied rewrites 57.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 57.7%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-91} \lor \neg \left(z \leq 3.2 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{t - a}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-41} \lor \neg \left(z \leq 2.9 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -3.9e-41) (not (<= z 2.9e-33))) (/ (- t a) b) (* x 1.0)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3.9e-41) || !(z <= 2.9e-33)) {
		tmp = (t - a) / b;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-3.9d-41)) .or. (.not. (z <= 2.9d-33))) then
        tmp = (t - a) / b
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -3.9e-41) || !(z <= 2.9e-33)) {
		tmp = (t - a) / b;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -3.9e-41) or not (z <= 2.9e-33):
		tmp = (t - a) / b
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -3.9e-41) || !(z <= 2.9e-33))
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -3.9e-41) || ~((z <= 2.9e-33)))
		tmp = (t - a) / b;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -3.9e-41], N[Not[LessEqual[z, 2.9e-33]], $MachinePrecision]], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.89999999999999991e-41 or 2.90000000000000003e-33 < z

   1. Initial program 53.2%

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

   3. Taylor expanded in y around 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 53.6

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

   5. Applied rewrites 53.6%

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

   if -3.89999999999999991e-41 < z < 2.90000000000000003e-33

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 84.6

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 62.9

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

   7. Applied rewrites 62.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 55.4%

       \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{-41} \lor \neg \left(z \leq 2.9 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-16} \lor \neg \left(z \leq 1.6 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.55e-16) (not (<= z 1.6e+68))) (/ t (- b y)) (/ x (- 1.0 z))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.55e-16) || !(z <= 1.6e+68)) {
		tmp = t / (b - y);
	} else {
		tmp = x / (1.0 - z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-1.55d-16)) .or. (.not. (z <= 1.6d+68))) then
        tmp = t / (b - y)
    else
        tmp = x / (1.0d0 - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.55e-16) || !(z <= 1.6e+68)) {
		tmp = t / (b - y);
	} else {
		tmp = x / (1.0 - z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -1.55e-16) or not (z <= 1.6e+68):
		tmp = t / (b - y)
	else:
		tmp = x / (1.0 - z)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.55e-16) || !(z <= 1.6e+68))
		tmp = Float64(t / Float64(b - y));
	else
		tmp = Float64(x / Float64(1.0 - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -1.55e-16) || ~((z <= 1.6e+68)))
		tmp = t / (b - y);
	else
		tmp = x / (1.0 - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.55e-16], N[Not[LessEqual[z, 1.6e+68]], $MachinePrecision]], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55e-16 or 1.59999999999999997e68 < z

   1. Initial program 48.4%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 33.7

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

   5. Applied rewrites 33.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 47.1%

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

   if -1.55e-16 < z < 1.59999999999999997e68

   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 50.1

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

   5. Applied rewrites 50.1%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-16} \lor \neg \left(z \leq 1.6 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-16} \lor \neg \left(z \leq 6.2 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.55e-16) (not (<= z 6.2e-30))) (/ t (- b y)) (* x 1.0)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.55e-16) || !(z <= 6.2e-30)) {
		tmp = t / (b - y);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((z <= (-1.55d-16)) .or. (.not. (z <= 6.2d-30))) then
        tmp = t / (b - y)
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.55e-16) || !(z <= 6.2e-30)) {
		tmp = t / (b - y);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (z <= -1.55e-16) or not (z <= 6.2e-30):
		tmp = t / (b - y)
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.55e-16) || !(z <= 6.2e-30))
		tmp = Float64(t / Float64(b - y));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((z <= -1.55e-16) || ~((z <= 6.2e-30)))
		tmp = t / (b - y);
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.55e-16], N[Not[LessEqual[z, 6.2e-30]], $MachinePrecision]], N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55e-16 or 6.19999999999999982e-30 < z

   1. Initial program 52.2%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 32.9

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 44.4%

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

   if -1.55e-16 < z < 6.19999999999999982e-30

   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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in x around inf

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

      1. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 61.3

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

   7. Applied rewrites 61.3%

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

   8. Taylor expanded in z around 0

      \[\leadsto x \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 53.2%

       \[\leadsto x \cdot 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-16} \lor \neg \left(z \leq 6.2 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{t}{b - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-16} \lor \neg \left(z \leq 6.5 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= z -1.55e-16) (not (<= z 6.5e-25))) (/ t b) (fma x z x)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((z <= -1.55e-16) || !(z <= 6.5e-25)) {
		tmp = t / b;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((z <= -1.55e-16) || !(z <= 6.5e-25))
		tmp = Float64(t / b);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[z, -1.55e-16], N[Not[LessEqual[z, 6.5e-25]], $MachinePrecision]], N[(t / b), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.55e-16 or 6.5e-25 < z

   1. Initial program 51.8%

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

   3. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower--.f64 33.2

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

   5. Applied rewrites 33.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 29.7%

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

   if -1.55e-16 < z < 6.5e-25

   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 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 52.8%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{-16} \lor \neg \left(z \leq 6.5 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{t}{b}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.4% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 66.9%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 30.3

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

5. Applied rewrites 30.3%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 26.6%

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

Alternative 15: 26.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* x 1.0))

C

double code(double x, double y, double z, double t, double a, double b) {
	return x * 1.0;
}

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 * 1.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x * 1.0;
}

Python

def code(x, y, z, t, a, b):
	return x * 1.0

Julia

function code(x, y, z, t, a, b)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 66.9%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. sub-neg N/A

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

   6. distribute-lft-in N/A

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

   7. associate-+l+ N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 66.9

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

4. Applied rewrites 66.9%

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

5. Taylor expanded in x around inf

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

   1. associate-/l* N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-in N/A

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

   5. mul-1-neg N/A

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

   6. sub-neg N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower--.f64 36.1

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

7. Applied rewrites 36.1%

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

8. Taylor expanded in z around 0

   \[\leadsto x \cdot 1 \]
9. Step-by-step derivation

10. Applied rewrites 26.3%

    \[\leadsto x \cdot 1 \]

11. Final simplification 26.3%

    \[\leadsto x \cdot 1 \]
12. Add Preprocessing

Alternative 16: 3.7% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.9%

   \[\frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)} \]
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 30.3

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

5. Applied rewrites 30.3%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 26.6%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 4.0%

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

Developer Target 1: 73.6% 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 2024322 
(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)))))