Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A

Percentage Accurate: 84.5% → 90.9%

Time: 14.9s

Alternatives: 14

Speedup: 0.7×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y \cdot z}{t - a \cdot z} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = (x - (y * z)) / (t - (a * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y \cdot z}{t - a \cdot z} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = (x - (y * z)) / (t - (a * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ \mathbf{if}\;z \leq -1.56 \cdot 10^{+125}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{t_1} - \frac{z \cdot y}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))))
   (if (<= z -1.56e+125)
     (/ (- y (/ x z)) a)
     (if (<= z 3.1e+126)
       (- (/ x t_1) (/ (* z y) t_1))
       (/ (- y) (- (/ t z) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double tmp;
	if (z <= -1.56e+125) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 3.1e+126) {
		tmp = (x / t_1) - ((z * y) / t_1);
	} else {
		tmp = -y / ((t / z) - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = t - (z * a)
    if (z <= (-1.56d+125)) then
        tmp = (y - (x / z)) / a
    else if (z <= 3.1d+126) then
        tmp = (x / t_1) - ((z * y) / t_1)
    else
        tmp = -y / ((t / z) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double tmp;
	if (z <= -1.56e+125) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 3.1e+126) {
		tmp = (x / t_1) - ((z * y) / t_1);
	} else {
		tmp = -y / ((t / z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	tmp = 0
	if z <= -1.56e+125:
		tmp = (y - (x / z)) / a
	elif z <= 3.1e+126:
		tmp = (x / t_1) - ((z * y) / t_1)
	else:
		tmp = -y / ((t / z) - a)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	tmp = 0.0
	if (z <= -1.56e+125)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 3.1e+126)
		tmp = Float64(Float64(x / t_1) - Float64(Float64(z * y) / t_1));
	else
		tmp = Float64(Float64(-y) / Float64(Float64(t / z) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	tmp = 0.0;
	if (z <= -1.56e+125)
		tmp = (y - (x / z)) / a;
	elseif (z <= 3.1e+126)
		tmp = (x / t_1) - ((z * y) / t_1);
	else
		tmp = -y / ((t / z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.56e+125], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 3.1e+126], N[(N[(x / t$95$1), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.56e125

   1. Initial program 55.3%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in x around 0 55.3%

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

      1. fma-def 55.3%

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

      2. associate-/l* 68.3%

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

      3. *-commutative 68.3%

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

      4. *-commutative 68.3%

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

   6. Simplified 68.3%

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

   7. Taylor expanded in a around inf 83.3%

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

      1. mul-1-neg 83.3%

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

      2. sub-neg 83.3%

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

   9. Simplified 83.3%

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

   if -1.56e125 < z < 3.1e126

   1. Initial program 98.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in x around 0 98.2%

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

   if 3.1e126 < z

   1. Initial program 52.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 52.8%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in x around 0 52.8%

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

      1. fma-def 52.8%

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

      2. associate-/l* 75.2%

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

      3. *-commutative 75.2%

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

      4. *-commutative 75.2%

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

   6. Simplified 75.2%

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

   7. Taylor expanded in t around inf 53.7%

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

   8. Taylor expanded in y around inf 45.7%

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

      1. mul-1-neg 45.7%

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

      2. associate-/l* 68.1%

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

      3. distribute-neg-frac 68.1%

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

      4. *-commutative 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in t around 0 90.4%

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

       1. neg-mul-1 90.4%

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

       2. +-commutative 90.4%

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

       3. unsub-neg 90.4%

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

   13. Simplified 90.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.56 \cdot 10^{+125}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{t - z \cdot a} - \frac{z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \]

Alternative 2: 70.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ t_2 := \frac{-y}{\frac{t}{z} - a}\\ \mathbf{if}\;z \leq -1300000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-108}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))) (t_2 (/ (- y) (- (/ t z) a))))
   (if (<= z -1300000000.0)
     t_2
     (if (<= z 1.32e-158)
       t_1
       (if (<= z 6.8e-108)
         (* z (/ y (- (* z a) t)))
         (if (<= z 3.4e-75)
           (/ (- x (* z y)) t)
           (if (<= z 3.9e-19)
             (/ (- y (/ x z)) a)
             (if (<= z 1.02e+51) t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double t_2 = -y / ((t / z) - a);
	double tmp;
	if (z <= -1300000000.0) {
		tmp = t_2;
	} else if (z <= 1.32e-158) {
		tmp = t_1;
	} else if (z <= 6.8e-108) {
		tmp = z * (y / ((z * a) - t));
	} else if (z <= 3.4e-75) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3.9e-19) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1.02e+51) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / (t - (z * a))
    t_2 = -y / ((t / z) - a)
    if (z <= (-1300000000.0d0)) then
        tmp = t_2
    else if (z <= 1.32d-158) then
        tmp = t_1
    else if (z <= 6.8d-108) then
        tmp = z * (y / ((z * a) - t))
    else if (z <= 3.4d-75) then
        tmp = (x - (z * y)) / t
    else if (z <= 3.9d-19) then
        tmp = (y - (x / z)) / a
    else if (z <= 1.02d+51) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double t_2 = -y / ((t / z) - a);
	double tmp;
	if (z <= -1300000000.0) {
		tmp = t_2;
	} else if (z <= 1.32e-158) {
		tmp = t_1;
	} else if (z <= 6.8e-108) {
		tmp = z * (y / ((z * a) - t));
	} else if (z <= 3.4e-75) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3.9e-19) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1.02e+51) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (z * a))
	t_2 = -y / ((t / z) - a)
	tmp = 0
	if z <= -1300000000.0:
		tmp = t_2
	elif z <= 1.32e-158:
		tmp = t_1
	elif z <= 6.8e-108:
		tmp = z * (y / ((z * a) - t))
	elif z <= 3.4e-75:
		tmp = (x - (z * y)) / t
	elif z <= 3.9e-19:
		tmp = (y - (x / z)) / a
	elif z <= 1.02e+51:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	t_2 = Float64(Float64(-y) / Float64(Float64(t / z) - a))
	tmp = 0.0
	if (z <= -1300000000.0)
		tmp = t_2;
	elseif (z <= 1.32e-158)
		tmp = t_1;
	elseif (z <= 6.8e-108)
		tmp = Float64(z * Float64(y / Float64(Float64(z * a) - t)));
	elseif (z <= 3.4e-75)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 3.9e-19)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 1.02e+51)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (t - (z * a));
	t_2 = -y / ((t / z) - a);
	tmp = 0.0;
	if (z <= -1300000000.0)
		tmp = t_2;
	elseif (z <= 1.32e-158)
		tmp = t_1;
	elseif (z <= 6.8e-108)
		tmp = z * (y / ((z * a) - t));
	elseif (z <= 3.4e-75)
		tmp = (x - (z * y)) / t;
	elseif (z <= 3.9e-19)
		tmp = (y - (x / z)) / a;
	elseif (z <= 1.02e+51)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1300000000.0], t$95$2, If[LessEqual[z, 1.32e-158], t$95$1, If[LessEqual[z, 6.8e-108], N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.4e-75], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3.9e-19], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 1.02e+51], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.3e9 or 1.02e51 < z

   1. Initial program 65.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in x around 0 65.1%

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

      1. fma-def 65.1%

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

      2. associate-/l* 78.1%

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

      3. *-commutative 78.1%

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

      4. *-commutative 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in t around inf 56.6%

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

   8. Taylor expanded in y around inf 54.3%

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

      1. mul-1-neg 54.3%

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

      2. associate-/l* 67.3%

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

      3. distribute-neg-frac 67.3%

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

      4. *-commutative 67.3%

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

   10. Simplified 67.3%

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

   11. Taylor expanded in t around 0 84.2%

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

       1. neg-mul-1 84.2%

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

       2. +-commutative 84.2%

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

       3. unsub-neg 84.2%

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

   13. Simplified 84.2%

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

   if -1.3e9 < z < 1.3200000000000001e-158 or 3.89999999999999995e-19 < z < 1.02e51

   1. Initial program 99.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 83.4%

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

      1. *-commutative 83.4%

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

   6. Simplified 83.4%

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

   if 1.3200000000000001e-158 < z < 6.80000000000000004e-108

   1. Initial program 99.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 92.0%

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

      1. mul-1-neg 92.0%

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

      2. sub-neg 92.0%

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

      3. +-commutative 92.0%

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

      4. distribute-rgt-neg-in 92.0%

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

      5. fma-udef 92.0%

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

      6. associate-*l/ 91.7%

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

      7. distribute-rgt-neg-in 91.7%

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

      8. fma-udef 91.7%

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

      9. distribute-rgt-neg-in 91.7%

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

      10. +-commutative 91.7%

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

      11. sub-neg 91.7%

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

      12. *-commutative 91.7%

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

   6. Simplified 91.7%

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

      1. associate-*l/ 92.0%

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

      2. frac-2neg 92.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 17.2%

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

      5. sqr-neg 17.2%

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

      6. sqrt-unprod 17.2%

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

      7. add-sqr-sqrt 17.2%

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

      8. distribute-rgt-neg-out 17.2%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 87.1%

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

      11. sqr-neg 87.1%

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

      12. sqrt-unprod 91.6%

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

      13. add-sqr-sqrt 92.0%

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

      14. sub-neg 92.0%

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

      15. distribute-neg-in 92.0%

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

      16. distribute-lft-neg-in 92.0%

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

      17. add-sqr-sqrt 0.0%

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

   8. Applied egg-rr 92.0%

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

      1. associate-/l* 77.4%

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

      2. associate-/r/ 91.7%

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

      3. +-commutative 91.7%

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

      4. unsub-neg 91.7%

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

   10. Simplified 91.7%

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

   if 6.80000000000000004e-108 < z < 3.40000000000000015e-75

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around inf 99.7%

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

   if 3.40000000000000015e-75 < z < 3.89999999999999995e-19

   1. Initial program 99.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.7%

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

      2. associate-/l* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in a around inf 82.0%

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

      1. mul-1-neg 82.0%

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

      2. sub-neg 82.0%

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

   9. Simplified 82.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1300000000:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \leq 1.32 \cdot 10^{-158}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-108}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \]

Alternative 3: 70.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ t_2 := \frac{-y}{\frac{t}{z} - a}\\ \mathbf{if}\;z \leq -330000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-108}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+51}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))) (t_2 (/ (- y) (- (/ t z) a))))
   (if (<= z -330000000.0)
     t_2
     (if (<= z 1.8e-158)
       t_1
       (if (<= z 6.8e-108)
         (* z (/ y (- (* z a) t)))
         (if (<= z 3.4e-75)
           (/ (- x (* z y)) t)
           (if (<= z 3.9e-19)
             (- (/ y a) (/ (/ x a) z))
             (if (<= z 1.02e+51) t_1 t_2))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double t_2 = -y / ((t / z) - a);
	double tmp;
	if (z <= -330000000.0) {
		tmp = t_2;
	} else if (z <= 1.8e-158) {
		tmp = t_1;
	} else if (z <= 6.8e-108) {
		tmp = z * (y / ((z * a) - t));
	} else if (z <= 3.4e-75) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3.9e-19) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 1.02e+51) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x / (t - (z * a))
    t_2 = -y / ((t / z) - a)
    if (z <= (-330000000.0d0)) then
        tmp = t_2
    else if (z <= 1.8d-158) then
        tmp = t_1
    else if (z <= 6.8d-108) then
        tmp = z * (y / ((z * a) - t))
    else if (z <= 3.4d-75) then
        tmp = (x - (z * y)) / t
    else if (z <= 3.9d-19) then
        tmp = (y / a) - ((x / a) / z)
    else if (z <= 1.02d+51) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double t_2 = -y / ((t / z) - a);
	double tmp;
	if (z <= -330000000.0) {
		tmp = t_2;
	} else if (z <= 1.8e-158) {
		tmp = t_1;
	} else if (z <= 6.8e-108) {
		tmp = z * (y / ((z * a) - t));
	} else if (z <= 3.4e-75) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3.9e-19) {
		tmp = (y / a) - ((x / a) / z);
	} else if (z <= 1.02e+51) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (z * a))
	t_2 = -y / ((t / z) - a)
	tmp = 0
	if z <= -330000000.0:
		tmp = t_2
	elif z <= 1.8e-158:
		tmp = t_1
	elif z <= 6.8e-108:
		tmp = z * (y / ((z * a) - t))
	elif z <= 3.4e-75:
		tmp = (x - (z * y)) / t
	elif z <= 3.9e-19:
		tmp = (y / a) - ((x / a) / z)
	elif z <= 1.02e+51:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	t_2 = Float64(Float64(-y) / Float64(Float64(t / z) - a))
	tmp = 0.0
	if (z <= -330000000.0)
		tmp = t_2;
	elseif (z <= 1.8e-158)
		tmp = t_1;
	elseif (z <= 6.8e-108)
		tmp = Float64(z * Float64(y / Float64(Float64(z * a) - t)));
	elseif (z <= 3.4e-75)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 3.9e-19)
		tmp = Float64(Float64(y / a) - Float64(Float64(x / a) / z));
	elseif (z <= 1.02e+51)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (t - (z * a));
	t_2 = -y / ((t / z) - a);
	tmp = 0.0;
	if (z <= -330000000.0)
		tmp = t_2;
	elseif (z <= 1.8e-158)
		tmp = t_1;
	elseif (z <= 6.8e-108)
		tmp = z * (y / ((z * a) - t));
	elseif (z <= 3.4e-75)
		tmp = (x - (z * y)) / t;
	elseif (z <= 3.9e-19)
		tmp = (y / a) - ((x / a) / z);
	elseif (z <= 1.02e+51)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -330000000.0], t$95$2, If[LessEqual[z, 1.8e-158], t$95$1, If[LessEqual[z, 6.8e-108], N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.4e-75], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3.9e-19], N[(N[(y / a), $MachinePrecision] - N[(N[(x / a), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.02e+51], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -3.3e8 or 1.02e51 < z

   1. Initial program 65.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in x around 0 65.1%

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

      1. fma-def 65.1%

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

      2. associate-/l* 78.1%

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

      3. *-commutative 78.1%

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

      4. *-commutative 78.1%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in t around inf 56.6%

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

   8. Taylor expanded in y around inf 54.3%

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

      1. mul-1-neg 54.3%

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

      2. associate-/l* 67.3%

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

      3. distribute-neg-frac 67.3%

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

      4. *-commutative 67.3%

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

   10. Simplified 67.3%

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

   11. Taylor expanded in t around 0 84.2%

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

       1. neg-mul-1 84.2%

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

       2. +-commutative 84.2%

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

       3. unsub-neg 84.2%

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

   13. Simplified 84.2%

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

   if -3.3e8 < z < 1.79999999999999995e-158 or 3.89999999999999995e-19 < z < 1.02e51

   1. Initial program 99.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 83.4%

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

      1. *-commutative 83.4%

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

   6. Simplified 83.4%

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

   if 1.79999999999999995e-158 < z < 6.80000000000000004e-108

   1. Initial program 99.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 92.0%

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

      1. mul-1-neg 92.0%

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

      2. sub-neg 92.0%

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

      3. +-commutative 92.0%

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

      4. distribute-rgt-neg-in 92.0%

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

      5. fma-udef 92.0%

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

      6. associate-*l/ 91.7%

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

      7. distribute-rgt-neg-in 91.7%

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

      8. fma-udef 91.7%

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

      9. distribute-rgt-neg-in 91.7%

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

      10. +-commutative 91.7%

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

      11. sub-neg 91.7%

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

      12. *-commutative 91.7%

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

   6. Simplified 91.7%

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

      1. associate-*l/ 92.0%

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

      2. frac-2neg 92.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 17.2%

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

      5. sqr-neg 17.2%

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

      6. sqrt-unprod 17.2%

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

      7. add-sqr-sqrt 17.2%

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

      8. distribute-rgt-neg-out 17.2%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 87.1%

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

      11. sqr-neg 87.1%

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

      12. sqrt-unprod 91.6%

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

      13. add-sqr-sqrt 92.0%

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

      14. sub-neg 92.0%

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

      15. distribute-neg-in 92.0%

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

      16. distribute-lft-neg-in 92.0%

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

      17. add-sqr-sqrt 0.0%

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

   8. Applied egg-rr 92.0%

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

      1. associate-/l* 77.4%

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

      2. associate-/r/ 91.7%

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

      3. +-commutative 91.7%

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

      4. unsub-neg 91.7%

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

   10. Simplified 91.7%

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

   if 6.80000000000000004e-108 < z < 3.40000000000000015e-75

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around inf 99.7%

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

   if 3.40000000000000015e-75 < z < 3.89999999999999995e-19

   1. Initial program 99.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.7%

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

      2. associate-/l* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in t around 0 82.3%

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

      1. +-commutative 82.3%

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

      2. mul-1-neg 82.3%

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

      3. sub-neg 82.3%

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

      4. associate-/r* 82.3%

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

   9. Simplified 82.3%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -330000000:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-158}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-108}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+51}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \]

Alternative 4: 52.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{-y}{t}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 5.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ (- y) t))))
   (if (<= z -9.2e-45)
     (/ y a)
     (if (<= z 7e-157)
       (/ x t)
       (if (<= z 3.1e-110)
         t_1
         (if (<= z 1.2e-77)
           (/ x t)
           (if (<= z 5.35e-14)
             (/ (- (/ x a)) z)
             (if (<= z 4.3e+137) t_1 (/ y a)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (-y / t);
	double tmp;
	if (z <= -9.2e-45) {
		tmp = y / a;
	} else if (z <= 7e-157) {
		tmp = x / t;
	} else if (z <= 3.1e-110) {
		tmp = t_1;
	} else if (z <= 1.2e-77) {
		tmp = x / t;
	} else if (z <= 5.35e-14) {
		tmp = -(x / a) / z;
	} else if (z <= 4.3e+137) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = z * (-y / t)
    if (z <= (-9.2d-45)) then
        tmp = y / a
    else if (z <= 7d-157) then
        tmp = x / t
    else if (z <= 3.1d-110) then
        tmp = t_1
    else if (z <= 1.2d-77) then
        tmp = x / t
    else if (z <= 5.35d-14) then
        tmp = -(x / a) / z
    else if (z <= 4.3d+137) then
        tmp = t_1
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (-y / t);
	double tmp;
	if (z <= -9.2e-45) {
		tmp = y / a;
	} else if (z <= 7e-157) {
		tmp = x / t;
	} else if (z <= 3.1e-110) {
		tmp = t_1;
	} else if (z <= 1.2e-77) {
		tmp = x / t;
	} else if (z <= 5.35e-14) {
		tmp = -(x / a) / z;
	} else if (z <= 4.3e+137) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = z * (-y / t)
	tmp = 0
	if z <= -9.2e-45:
		tmp = y / a
	elif z <= 7e-157:
		tmp = x / t
	elif z <= 3.1e-110:
		tmp = t_1
	elif z <= 1.2e-77:
		tmp = x / t
	elif z <= 5.35e-14:
		tmp = -(x / a) / z
	elif z <= 4.3e+137:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(Float64(-y) / t))
	tmp = 0.0
	if (z <= -9.2e-45)
		tmp = Float64(y / a);
	elseif (z <= 7e-157)
		tmp = Float64(x / t);
	elseif (z <= 3.1e-110)
		tmp = t_1;
	elseif (z <= 1.2e-77)
		tmp = Float64(x / t);
	elseif (z <= 5.35e-14)
		tmp = Float64(Float64(-Float64(x / a)) / z);
	elseif (z <= 4.3e+137)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = z * (-y / t);
	tmp = 0.0;
	if (z <= -9.2e-45)
		tmp = y / a;
	elseif (z <= 7e-157)
		tmp = x / t;
	elseif (z <= 3.1e-110)
		tmp = t_1;
	elseif (z <= 1.2e-77)
		tmp = x / t;
	elseif (z <= 5.35e-14)
		tmp = -(x / a) / z;
	elseif (z <= 4.3e+137)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[((-y) / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e-45], N[(y / a), $MachinePrecision], If[LessEqual[z, 7e-157], N[(x / t), $MachinePrecision], If[LessEqual[z, 3.1e-110], t$95$1, If[LessEqual[z, 1.2e-77], N[(x / t), $MachinePrecision], If[LessEqual[z, 5.35e-14], N[((-N[(x / a), $MachinePrecision]) / z), $MachinePrecision], If[LessEqual[z, 4.3e+137], t$95$1, N[(y / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.19999999999999967e-45 or 4.29999999999999965e137 < z

   1. Initial program 62.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in z around inf 66.4%

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

   if -9.19999999999999967e-45 < z < 7.0000000000000004e-157 or 3.10000000000000007e-110 < z < 1.19999999999999995e-77

   1. Initial program 99.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 72.6%

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

   if 7.0000000000000004e-157 < z < 3.10000000000000007e-110 or 5.3499999999999999e-14 < z < 4.29999999999999965e137

   1. Initial program 93.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around 0 61.3%

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

      1. mul-1-neg 61.3%

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

      2. sub-neg 61.3%

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

      3. +-commutative 61.3%

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

      4. distribute-rgt-neg-in 61.3%

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

      5. fma-udef 61.3%

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

      6. associate-*l/ 65.3%

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

      7. distribute-rgt-neg-in 65.3%

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

      8. fma-udef 65.3%

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

      9. distribute-rgt-neg-in 65.3%

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

      10. +-commutative 65.3%

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

      11. sub-neg 65.3%

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

      12. *-commutative 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in t around inf 51.9%

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

   if 1.19999999999999995e-77 < z < 5.3499999999999999e-14

   1. Initial program 99.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 55.9%

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

      1. *-commutative 55.9%

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

   6. Simplified 55.9%

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

   7. Taylor expanded in t around 0 37.9%

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

      1. mul-1-neg 37.9%

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

      2. associate-/r* 37.8%

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

   9. Simplified 37.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-110}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 5.35 \cdot 10^{-14}:\\ \;\;\;\;\frac{-\frac{x}{a}}{z}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+137}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 5: 52.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{-y}{t}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ (- y) t))))
   (if (<= z -1.1e-44)
     (/ y a)
     (if (<= z 7e-157)
       (/ x t)
       (if (<= z 9.2e-110)
         t_1
         (if (<= z 5.6e-77)
           (/ x t)
           (if (<= z 4.4e-14)
             (/ (- x) (* z a))
             (if (<= z 4.3e+137) t_1 (/ y a)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (-y / t);
	double tmp;
	if (z <= -1.1e-44) {
		tmp = y / a;
	} else if (z <= 7e-157) {
		tmp = x / t;
	} else if (z <= 9.2e-110) {
		tmp = t_1;
	} else if (z <= 5.6e-77) {
		tmp = x / t;
	} else if (z <= 4.4e-14) {
		tmp = -x / (z * a);
	} else if (z <= 4.3e+137) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = z * (-y / t)
    if (z <= (-1.1d-44)) then
        tmp = y / a
    else if (z <= 7d-157) then
        tmp = x / t
    else if (z <= 9.2d-110) then
        tmp = t_1
    else if (z <= 5.6d-77) then
        tmp = x / t
    else if (z <= 4.4d-14) then
        tmp = -x / (z * a)
    else if (z <= 4.3d+137) then
        tmp = t_1
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (-y / t);
	double tmp;
	if (z <= -1.1e-44) {
		tmp = y / a;
	} else if (z <= 7e-157) {
		tmp = x / t;
	} else if (z <= 9.2e-110) {
		tmp = t_1;
	} else if (z <= 5.6e-77) {
		tmp = x / t;
	} else if (z <= 4.4e-14) {
		tmp = -x / (z * a);
	} else if (z <= 4.3e+137) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = z * (-y / t)
	tmp = 0
	if z <= -1.1e-44:
		tmp = y / a
	elif z <= 7e-157:
		tmp = x / t
	elif z <= 9.2e-110:
		tmp = t_1
	elif z <= 5.6e-77:
		tmp = x / t
	elif z <= 4.4e-14:
		tmp = -x / (z * a)
	elif z <= 4.3e+137:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(Float64(-y) / t))
	tmp = 0.0
	if (z <= -1.1e-44)
		tmp = Float64(y / a);
	elseif (z <= 7e-157)
		tmp = Float64(x / t);
	elseif (z <= 9.2e-110)
		tmp = t_1;
	elseif (z <= 5.6e-77)
		tmp = Float64(x / t);
	elseif (z <= 4.4e-14)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 4.3e+137)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = z * (-y / t);
	tmp = 0.0;
	if (z <= -1.1e-44)
		tmp = y / a;
	elseif (z <= 7e-157)
		tmp = x / t;
	elseif (z <= 9.2e-110)
		tmp = t_1;
	elseif (z <= 5.6e-77)
		tmp = x / t;
	elseif (z <= 4.4e-14)
		tmp = -x / (z * a);
	elseif (z <= 4.3e+137)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[((-y) / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e-44], N[(y / a), $MachinePrecision], If[LessEqual[z, 7e-157], N[(x / t), $MachinePrecision], If[LessEqual[z, 9.2e-110], t$95$1, If[LessEqual[z, 5.6e-77], N[(x / t), $MachinePrecision], If[LessEqual[z, 4.4e-14], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.3e+137], t$95$1, N[(y / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.10000000000000006e-44 or 4.29999999999999965e137 < z

   1. Initial program 62.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in z around inf 66.4%

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

   if -1.10000000000000006e-44 < z < 7.0000000000000004e-157 or 9.2000000000000006e-110 < z < 5.5999999999999999e-77

   1. Initial program 99.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 72.6%

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

   if 7.0000000000000004e-157 < z < 9.2000000000000006e-110 or 4.4000000000000002e-14 < z < 4.29999999999999965e137

   1. Initial program 93.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around 0 61.3%

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

      1. mul-1-neg 61.3%

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

      2. sub-neg 61.3%

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

      3. +-commutative 61.3%

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

      4. distribute-rgt-neg-in 61.3%

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

      5. fma-udef 61.3%

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

      6. associate-*l/ 65.3%

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

      7. distribute-rgt-neg-in 65.3%

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

      8. fma-udef 65.3%

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

      9. distribute-rgt-neg-in 65.3%

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

      10. +-commutative 65.3%

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

      11. sub-neg 65.3%

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

      12. *-commutative 65.3%

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

   6. Simplified 65.3%

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

   7. Taylor expanded in t around inf 51.9%

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

   if 5.5999999999999999e-77 < z < 4.4000000000000002e-14

   1. Initial program 99.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.7%

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

      2. associate-/l* 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in t around 0 63.0%

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

      1. +-commutative 63.0%

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

      2. mul-1-neg 63.0%

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

      3. sub-neg 63.0%

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

      4. associate-/r* 62.9%

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

   9. Simplified 62.9%

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

   10. Taylor expanded in y around 0 37.9%

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

       1. associate-*r/ 37.9%

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

       2. neg-mul-1 37.9%

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

       3. *-commutative 37.9%

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

   12. Simplified 37.9%

       \[\leadsto \color{blue}{\frac{-x}{z \cdot a}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-110}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+137}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 6: 52.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{-z}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+137}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.2e-45)
   (/ y a)
   (if (<= z 7e-157)
     (/ x t)
     (if (<= z 2.2e-110)
       (/ (- z) (/ t y))
       (if (<= z 4.4e-77)
         (/ x t)
         (if (<= z 2.1e-14)
           (/ (- x) (* z a))
           (if (<= z 4e+137) (* z (/ (- y) t)) (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.2e-45) {
		tmp = y / a;
	} else if (z <= 7e-157) {
		tmp = x / t;
	} else if (z <= 2.2e-110) {
		tmp = -z / (t / y);
	} else if (z <= 4.4e-77) {
		tmp = x / t;
	} else if (z <= 2.1e-14) {
		tmp = -x / (z * a);
	} else if (z <= 4e+137) {
		tmp = z * (-y / t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-9.2d-45)) then
        tmp = y / a
    else if (z <= 7d-157) then
        tmp = x / t
    else if (z <= 2.2d-110) then
        tmp = -z / (t / y)
    else if (z <= 4.4d-77) then
        tmp = x / t
    else if (z <= 2.1d-14) then
        tmp = -x / (z * a)
    else if (z <= 4d+137) then
        tmp = z * (-y / t)
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.2e-45) {
		tmp = y / a;
	} else if (z <= 7e-157) {
		tmp = x / t;
	} else if (z <= 2.2e-110) {
		tmp = -z / (t / y);
	} else if (z <= 4.4e-77) {
		tmp = x / t;
	} else if (z <= 2.1e-14) {
		tmp = -x / (z * a);
	} else if (z <= 4e+137) {
		tmp = z * (-y / t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -9.2e-45:
		tmp = y / a
	elif z <= 7e-157:
		tmp = x / t
	elif z <= 2.2e-110:
		tmp = -z / (t / y)
	elif z <= 4.4e-77:
		tmp = x / t
	elif z <= 2.1e-14:
		tmp = -x / (z * a)
	elif z <= 4e+137:
		tmp = z * (-y / t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.2e-45)
		tmp = Float64(y / a);
	elseif (z <= 7e-157)
		tmp = Float64(x / t);
	elseif (z <= 2.2e-110)
		tmp = Float64(Float64(-z) / Float64(t / y));
	elseif (z <= 4.4e-77)
		tmp = Float64(x / t);
	elseif (z <= 2.1e-14)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 4e+137)
		tmp = Float64(z * Float64(Float64(-y) / t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -9.2e-45)
		tmp = y / a;
	elseif (z <= 7e-157)
		tmp = x / t;
	elseif (z <= 2.2e-110)
		tmp = -z / (t / y);
	elseif (z <= 4.4e-77)
		tmp = x / t;
	elseif (z <= 2.1e-14)
		tmp = -x / (z * a);
	elseif (z <= 4e+137)
		tmp = z * (-y / t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.2e-45], N[(y / a), $MachinePrecision], If[LessEqual[z, 7e-157], N[(x / t), $MachinePrecision], If[LessEqual[z, 2.2e-110], N[((-z) / N[(t / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.4e-77], N[(x / t), $MachinePrecision], If[LessEqual[z, 2.1e-14], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+137], N[(z * N[((-y) / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -9.19999999999999967e-45 or 4.0000000000000001e137 < z

   1. Initial program 62.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in z around inf 66.4%

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

   if -9.19999999999999967e-45 < z < 7.0000000000000004e-157 or 2.1999999999999999e-110 < z < 4.40000000000000014e-77

   1. Initial program 99.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 72.6%

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

   if 7.0000000000000004e-157 < z < 2.1999999999999999e-110

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around inf 82.1%

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

   5. Taylor expanded in x around 0 82.4%

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

      1. mul-1-neg 82.4%

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

      2. *-commutative 82.4%

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

      3. associate-/l* 82.6%

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

      4. distribute-neg-frac 82.6%

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

   7. Simplified 82.6%

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

   if 4.40000000000000014e-77 < z < 2.0999999999999999e-14

   1. Initial program 99.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.7%

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

      2. associate-/l* 99.6%

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

      3. *-commutative 99.6%

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

      4. *-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in t around 0 63.0%

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

      1. +-commutative 63.0%

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

      2. mul-1-neg 63.0%

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

      3. sub-neg 63.0%

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

      4. associate-/r* 62.9%

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

   9. Simplified 62.9%

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

   10. Taylor expanded in y around 0 37.9%

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

       1. associate-*r/ 37.9%

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

       2. neg-mul-1 37.9%

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

       3. *-commutative 37.9%

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

   12. Simplified 37.9%

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

   if 2.0999999999999999e-14 < z < 4.0000000000000001e137

   1. Initial program 91.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in x around 0 52.0%

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

      1. mul-1-neg 52.0%

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

      2. sub-neg 52.0%

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

      3. +-commutative 52.0%

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

      4. distribute-rgt-neg-in 52.0%

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

      5. fma-udef 52.0%

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

      6. associate-*l/ 57.5%

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

      7. distribute-rgt-neg-in 57.5%

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

      8. fma-udef 57.5%

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

      9. distribute-rgt-neg-in 57.5%

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

      10. +-commutative 57.5%

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

      11. sub-neg 57.5%

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

      12. *-commutative 57.5%

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

   6. Simplified 57.5%

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

   7. Taylor expanded in t around inf 42.3%

      \[\leadsto \color{blue}{\frac{y}{t}} \cdot \left(-z\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{-z}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+137}:\\ \;\;\;\;z \cdot \frac{-y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 7: 71.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-y}{\frac{t}{z} - a}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y) (- (/ t z) a))))
   (if (<= z -1.1e-44)
     t_1
     (if (<= z 3.4e-75)
       (/ (- x (* z y)) t)
       (if (<= z 3.9e-19)
         (/ (- y (/ x z)) a)
         (if (<= z 1.6e+52) (/ x (- t (* z a))) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -y / ((t / z) - a);
	double tmp;
	if (z <= -1.1e-44) {
		tmp = t_1;
	} else if (z <= 3.4e-75) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3.9e-19) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1.6e+52) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = -y / ((t / z) - a)
    if (z <= (-1.1d-44)) then
        tmp = t_1
    else if (z <= 3.4d-75) then
        tmp = (x - (z * y)) / t
    else if (z <= 3.9d-19) then
        tmp = (y - (x / z)) / a
    else if (z <= 1.6d+52) then
        tmp = x / (t - (z * a))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = -y / ((t / z) - a);
	double tmp;
	if (z <= -1.1e-44) {
		tmp = t_1;
	} else if (z <= 3.4e-75) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 3.9e-19) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1.6e+52) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = -y / ((t / z) - a)
	tmp = 0
	if z <= -1.1e-44:
		tmp = t_1
	elif z <= 3.4e-75:
		tmp = (x - (z * y)) / t
	elif z <= 3.9e-19:
		tmp = (y - (x / z)) / a
	elif z <= 1.6e+52:
		tmp = x / (t - (z * a))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(-y) / Float64(Float64(t / z) - a))
	tmp = 0.0
	if (z <= -1.1e-44)
		tmp = t_1;
	elseif (z <= 3.4e-75)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 3.9e-19)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 1.6e+52)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = -y / ((t / z) - a);
	tmp = 0.0;
	if (z <= -1.1e-44)
		tmp = t_1;
	elseif (z <= 3.4e-75)
		tmp = (x - (z * y)) / t;
	elseif (z <= 3.9e-19)
		tmp = (y - (x / z)) / a;
	elseif (z <= 1.6e+52)
		tmp = x / (t - (z * a));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e-44], t$95$1, If[LessEqual[z, 3.4e-75], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3.9e-19], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 1.6e+52], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.10000000000000006e-44 or 1.6e52 < z

   1. Initial program 66.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in x around 0 66.7%

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

      1. fma-def 66.7%

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

      2. associate-/l* 79.2%

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

      3. *-commutative 79.2%

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

      4. *-commutative 79.2%

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

   6. Simplified 79.2%

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

   7. Taylor expanded in t around inf 57.1%

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

   8. Taylor expanded in y around inf 54.1%

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

      1. mul-1-neg 54.1%

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

      2. associate-/l* 66.6%

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

      3. distribute-neg-frac 66.6%

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

      4. *-commutative 66.6%

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

   10. Simplified 66.6%

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

   11. Taylor expanded in t around 0 82.6%

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

       1. neg-mul-1 82.6%

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

       2. +-commutative 82.6%

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

       3. unsub-neg 82.6%

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

   13. Simplified 82.6%

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

   if -1.10000000000000006e-44 < z < 3.40000000000000015e-75

   1. Initial program 99.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 83.9%

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

   if 3.40000000000000015e-75 < z < 3.89999999999999995e-19

   1. Initial program 99.6%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. fma-def 99.7%

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

      2. associate-/l* 99.7%

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

      3. *-commutative 99.7%

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

      4. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in a around inf 82.0%

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

      1. mul-1-neg 82.0%

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

      2. sub-neg 82.0%

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

   9. Simplified 82.0%

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

   if 3.89999999999999995e-19 < z < 1.6e52

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 70.4%

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

      1. *-commutative 70.4%

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

   6. Simplified 70.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \]

Alternative 8: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-44} \lor \neg \left(z \leq 3.4 \cdot 10^{-75}\right) \land \left(z \leq 3.9 \cdot 10^{-19} \lor \neg \left(z \leq 1.1 \cdot 10^{+92}\right)\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.1e-44)
         (and (not (<= z 3.4e-75)) (or (<= z 3.9e-19) (not (<= z 1.1e+92)))))
   (/ (- y (/ x z)) a)
   (/ (- x (* z y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.1e-44) || (!(z <= 3.4e-75) && ((z <= 3.9e-19) || !(z <= 1.1e+92)))) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((z <= (-1.1d-44)) .or. (.not. (z <= 3.4d-75)) .and. (z <= 3.9d-19) .or. (.not. (z <= 1.1d+92))) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (z * y)) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.1e-44) || (!(z <= 3.4e-75) && ((z <= 3.9e-19) || !(z <= 1.1e+92)))) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (z * y)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.1e-44) or (not (z <= 3.4e-75) and ((z <= 3.9e-19) or not (z <= 1.1e+92))):
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (z * y)) / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.1e-44) || (!(z <= 3.4e-75) && ((z <= 3.9e-19) || !(z <= 1.1e+92))))
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.1e-44) || (~((z <= 3.4e-75)) && ((z <= 3.9e-19) || ~((z <= 1.1e+92)))))
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (z * y)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.1e-44], And[N[Not[LessEqual[z, 3.4e-75]], $MachinePrecision], Or[LessEqual[z, 3.9e-19], N[Not[LessEqual[z, 1.1e+92]], $MachinePrecision]]]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000006e-44 or 3.40000000000000015e-75 < z < 3.89999999999999995e-19 or 1.09999999999999996e92 < z

   1. Initial program 67.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in x around 0 67.2%

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

      1. fma-def 67.2%

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

      2. associate-/l* 79.8%

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

      3. *-commutative 79.8%

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

      4. *-commutative 79.8%

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

   6. Simplified 79.8%

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

   7. Taylor expanded in a around inf 75.8%

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

      1. mul-1-neg 75.8%

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

      2. sub-neg 75.8%

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

   9. Simplified 75.8%

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

   if -1.10000000000000006e-44 < z < 3.40000000000000015e-75 or 3.89999999999999995e-19 < z < 1.09999999999999996e92

   1. Initial program 99.1%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in t around inf 80.7%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-44} \lor \neg \left(z \leq 3.4 \cdot 10^{-75}\right) \land \left(z \leq 3.9 \cdot 10^{-19} \lor \neg \left(z \leq 1.1 \cdot 10^{+92}\right)\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \end{array} \]

Alternative 9: 62.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;z \leq -7200000000:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{-z}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))))
   (if (<= z -7200000000.0)
     (/ y a)
     (if (<= z 7e-157)
       t_1
       (if (<= z 2.4e-110)
         (/ (- z) (/ t y))
         (if (<= z 4e+137) t_1 (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (z <= -7200000000.0) {
		tmp = y / a;
	} else if (z <= 7e-157) {
		tmp = t_1;
	} else if (z <= 2.4e-110) {
		tmp = -z / (t / y);
	} else if (z <= 4e+137) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: tmp
    t_1 = x / (t - (z * a))
    if (z <= (-7200000000.0d0)) then
        tmp = y / a
    else if (z <= 7d-157) then
        tmp = t_1
    else if (z <= 2.4d-110) then
        tmp = -z / (t / y)
    else if (z <= 4d+137) then
        tmp = t_1
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (z <= -7200000000.0) {
		tmp = y / a;
	} else if (z <= 7e-157) {
		tmp = t_1;
	} else if (z <= 2.4e-110) {
		tmp = -z / (t / y);
	} else if (z <= 4e+137) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (z * a))
	tmp = 0
	if z <= -7200000000.0:
		tmp = y / a
	elif z <= 7e-157:
		tmp = t_1
	elif z <= 2.4e-110:
		tmp = -z / (t / y)
	elif z <= 4e+137:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (z <= -7200000000.0)
		tmp = Float64(y / a);
	elseif (z <= 7e-157)
		tmp = t_1;
	elseif (z <= 2.4e-110)
		tmp = Float64(Float64(-z) / Float64(t / y));
	elseif (z <= 4e+137)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (t - (z * a));
	tmp = 0.0;
	if (z <= -7200000000.0)
		tmp = y / a;
	elseif (z <= 7e-157)
		tmp = t_1;
	elseif (z <= 2.4e-110)
		tmp = -z / (t / y);
	elseif (z <= 4e+137)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7200000000.0], N[(y / a), $MachinePrecision], If[LessEqual[z, 7e-157], t$95$1, If[LessEqual[z, 2.4e-110], N[((-z) / N[(t / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4e+137], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.2e9 or 4.0000000000000001e137 < z

   1. Initial program 60.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 60.2%

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

   3. Simplified 60.2%

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

   4. Taylor expanded in z around inf 68.3%

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

   if -7.2e9 < z < 7.0000000000000004e-157 or 2.40000000000000006e-110 < z < 4.0000000000000001e137

   1. Initial program 97.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 97.9%

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

   3. Simplified 97.9%

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

   4. Taylor expanded in x around inf 73.3%

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

      1. *-commutative 73.3%

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

   6. Simplified 73.3%

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

   if 7.0000000000000004e-157 < z < 2.40000000000000006e-110

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around inf 82.1%

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

   5. Taylor expanded in x around 0 82.4%

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

      1. mul-1-neg 82.4%

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

      2. *-commutative 82.4%

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

      3. associate-/l* 82.6%

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

      4. distribute-neg-frac 82.6%

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

   7. Simplified 82.6%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7200000000:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{-z}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+137}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 10: 91.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+127}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.4e+125)
   (/ (- y (/ x z)) a)
   (if (<= z 1.35e+127)
     (/ (- x (* z y)) (- t (* z a)))
     (/ (- y) (- (/ t z) a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e+125) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1.35e+127) {
		tmp = (x - (z * y)) / (t - (z * a));
	} else {
		tmp = -y / ((t / z) - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-1.4d+125)) then
        tmp = (y - (x / z)) / a
    else if (z <= 1.35d+127) then
        tmp = (x - (z * y)) / (t - (z * a))
    else
        tmp = -y / ((t / z) - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e+125) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1.35e+127) {
		tmp = (x - (z * y)) / (t - (z * a));
	} else {
		tmp = -y / ((t / z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.4e+125:
		tmp = (y - (x / z)) / a
	elif z <= 1.35e+127:
		tmp = (x - (z * y)) / (t - (z * a))
	else:
		tmp = -y / ((t / z) - a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.4e+125)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 1.35e+127)
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(z * a)));
	else
		tmp = Float64(Float64(-y) / Float64(Float64(t / z) - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.4e+125)
		tmp = (y - (x / z)) / a;
	elseif (z <= 1.35e+127)
		tmp = (x - (z * y)) / (t - (z * a));
	else
		tmp = -y / ((t / z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.4e+125], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 1.35e+127], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4e125

   1. Initial program 55.3%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 55.3%

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

   3. Simplified 55.3%

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

   4. Taylor expanded in x around 0 55.3%

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

      1. fma-def 55.3%

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

      2. associate-/l* 68.3%

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

      3. *-commutative 68.3%

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

      4. *-commutative 68.3%

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

   6. Simplified 68.3%

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

   7. Taylor expanded in a around inf 83.3%

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

      1. mul-1-neg 83.3%

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

      2. sub-neg 83.3%

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

   9. Simplified 83.3%

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

   if -1.4e125 < z < 1.3500000000000001e127

   1. Initial program 98.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]

   if 1.3500000000000001e127 < z

   1. Initial program 52.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 52.8%

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

   3. Simplified 52.8%

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

   4. Taylor expanded in x around 0 52.8%

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

      1. fma-def 52.8%

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

      2. associate-/l* 75.2%

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

      3. *-commutative 75.2%

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

      4. *-commutative 75.2%

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

   6. Simplified 75.2%

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

   7. Taylor expanded in t around inf 53.7%

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

   8. Taylor expanded in y around inf 45.7%

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

      1. mul-1-neg 45.7%

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

      2. associate-/l* 68.1%

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

      3. distribute-neg-frac 68.1%

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

      4. *-commutative 68.1%

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

   10. Simplified 68.1%

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

   11. Taylor expanded in t around 0 90.4%

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

       1. neg-mul-1 90.4%

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

       2. +-commutative 90.4%

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

       3. unsub-neg 90.4%

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

   13. Simplified 90.4%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+127}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \]

Alternative 11: 51.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+137}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.08e-44)
   (/ y a)
   (if (<= z 6.8e-157) (/ x t) (if (<= z 4e+137) (* (- y) (/ z t)) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.08e-44) {
		tmp = y / a;
	} else if (z <= 6.8e-157) {
		tmp = x / t;
	} else if (z <= 4e+137) {
		tmp = -y * (z / t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if (z <= (-1.08d-44)) then
        tmp = y / a
    else if (z <= 6.8d-157) then
        tmp = x / t
    else if (z <= 4d+137) then
        tmp = -y * (z / t)
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.08e-44) {
		tmp = y / a;
	} else if (z <= 6.8e-157) {
		tmp = x / t;
	} else if (z <= 4e+137) {
		tmp = -y * (z / t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.08e-44:
		tmp = y / a
	elif z <= 6.8e-157:
		tmp = x / t
	elif z <= 4e+137:
		tmp = -y * (z / t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.08e-44)
		tmp = Float64(y / a);
	elseif (z <= 6.8e-157)
		tmp = Float64(x / t);
	elseif (z <= 4e+137)
		tmp = Float64(Float64(-y) * Float64(z / t));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.08e-44)
		tmp = y / a;
	elseif (z <= 6.8e-157)
		tmp = x / t;
	elseif (z <= 4e+137)
		tmp = -y * (z / t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.08e-44], N[(y / a), $MachinePrecision], If[LessEqual[z, 6.8e-157], N[(x / t), $MachinePrecision], If[LessEqual[z, 4e+137], N[((-y) * N[(z / t), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.07999999999999994e-44 or 4.0000000000000001e137 < z

   1. Initial program 62.5%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 62.5%

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

   3. Simplified 62.5%

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

   4. Taylor expanded in z around inf 66.4%

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

   if -1.07999999999999994e-44 < z < 6.79999999999999955e-157

   1. Initial program 99.9%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 72.4%

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

   if 6.79999999999999955e-157 < z < 4.0000000000000001e137

   1. Initial program 95.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in t around inf 63.0%

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

      1. div-inv 62.9%

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

      2. *-commutative 62.9%

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

   6. Applied egg-rr 62.9%

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

   7. Taylor expanded in x around 0 43.1%

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

      1. mul-1-neg 43.1%

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

      2. associate-*r/ 40.2%

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

      3. distribute-rgt-neg-in 40.2%

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

   9. Simplified 40.2%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.08 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{+137}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 12: 63.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -108000000 \lor \neg \left(z \leq 1.3 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -108000000.0) (not (<= z 1.3e+146)))
   (/ y a)
   (/ (- x (* z y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -108000000.0) || !(z <= 1.3e+146)) {
		tmp = y / a;
	} else {
		tmp = (x - (z * y)) / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((z <= (-108000000.0d0)) .or. (.not. (z <= 1.3d+146))) then
        tmp = y / a
    else
        tmp = (x - (z * y)) / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -108000000.0) || !(z <= 1.3e+146)) {
		tmp = y / a;
	} else {
		tmp = (x - (z * y)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -108000000.0) or not (z <= 1.3e+146):
		tmp = y / a
	else:
		tmp = (x - (z * y)) / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -108000000.0) || !(z <= 1.3e+146))
		tmp = Float64(y / a);
	else
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -108000000.0) || ~((z <= 1.3e+146)))
		tmp = y / a;
	else
		tmp = (x - (z * y)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -108000000.0], N[Not[LessEqual[z, 1.3e+146]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.08e8 or 1.30000000000000007e146 < z

   1. Initial program 60.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in z around inf 68.6%

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

   if -1.08e8 < z < 1.30000000000000007e146

   1. Initial program 97.4%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in t around inf 73.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -108000000 \lor \neg \left(z \leq 1.3 \cdot 10^{+146}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \end{array} \]

Alternative 13: 54.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-44} \lor \neg \left(z \leq 1.9 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.1e-44) (not (<= z 1.9e+85))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.1e-44) || !(z <= 1.9e+85)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: tmp
    if ((z <= (-1.1d-44)) .or. (.not. (z <= 1.9d+85))) then
        tmp = y / a
    else
        tmp = x / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.1e-44) || !(z <= 1.9e+85)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.1e-44) or not (z <= 1.9e+85):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.1e-44) || !(z <= 1.9e+85))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.1e-44) || ~((z <= 1.9e+85)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.1e-44], N[Not[LessEqual[z, 1.9e+85]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.10000000000000006e-44 or 1.89999999999999996e85 < z

   1. Initial program 64.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 64.7%

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

   3. Simplified 64.7%

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

   4. Taylor expanded in z around inf 63.1%

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

   if -1.10000000000000006e-44 < z < 1.89999999999999996e85

   1. Initial program 99.2%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Step-by-step derivation

      1. *-commutative 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in z around 0 57.5%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{-44} \lor \neg \left(z \leq 1.9 \cdot 10^{+85}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 14: 34.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    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
    code = x / t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x / t), $MachinePrecision]

Derivation

1. Initial program 83.6%

   \[\frac{x - y \cdot z}{t - a \cdot z} \]
2. Step-by-step derivation

   1. *-commutative 83.6%

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

3. Simplified 83.6%

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

4. Taylor expanded in z around 0 37.1%

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

5. Final simplification 37.1%

   \[\leadsto \frac{x}{t} \]

Developer target: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - a \cdot z\\ t_2 := \frac{x}{t_1} - \frac{y}{\frac{t}{z} - a}\\ \mathbf{if}\;z < -32113435955957344:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z < 3.5139522372978296 \cdot 10^{-86}:\\ \;\;\;\;\left(x - y \cdot z\right) \cdot \frac{1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* a z))) (t_2 (- (/ x t_1) (/ y (- (/ t z) a)))))
   (if (< z -32113435955957344.0)
     t_2
     (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 t_1)) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = t - (a * z)
    t_2 = (x / t_1) - (y / ((t / z) - a))
    if (z < (-32113435955957344.0d0)) then
        tmp = t_2
    else if (z < 3.5139522372978296d-86) then
        tmp = (x - (y * z)) * (1.0d0 / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (a * z);
	double t_2 = (x / t_1) - (y / ((t / z) - a));
	double tmp;
	if (z < -32113435955957344.0) {
		tmp = t_2;
	} else if (z < 3.5139522372978296e-86) {
		tmp = (x - (y * z)) * (1.0 / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (a * z)
	t_2 = (x / t_1) - (y / ((t / z) - a))
	tmp = 0
	if z < -32113435955957344.0:
		tmp = t_2
	elif z < 3.5139522372978296e-86:
		tmp = (x - (y * z)) * (1.0 / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(a * z))
	t_2 = Float64(Float64(x / t_1) - Float64(y / Float64(Float64(t / z) - a)))
	tmp = 0.0
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = Float64(Float64(x - Float64(y * z)) * Float64(1.0 / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (a * z);
	t_2 = (x / t_1) - (y / ((t / z) - a));
	tmp = 0.0;
	if (z < -32113435955957344.0)
		tmp = t_2;
	elseif (z < 3.5139522372978296e-86)
		tmp = (x - (y * z)) * (1.0 / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(a * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / t$95$1), $MachinePrecision] - N[(y / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -32113435955957344.0], t$95$2, If[Less[z, 3.5139522372978296e-86], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2023322 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :herbie-target
  (if (< z -32113435955957344.0) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 3.5139522372978296e-86) (* (- x (* y z)) (/ 1.0 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a)))))

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