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

Percentage Accurate: 85.2% → 90.8%

Time: 12.8s

Alternatives: 12

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: 85.2% 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: 90.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+177} \lor \neg \left(z \leq 4.5 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.05e+177) (not (<= z 4.5e+105)))
   (/ (- y) (- (/ t z) a))
   (/ (- x (* z y)) (fma (- z) a t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.05e+177) || !(z <= 4.5e+105)) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = (x - (z * y)) / fma(-z, a, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.05e+177) || !(z <= 4.5e+105))
		tmp = Float64(Float64(-y) / Float64(Float64(t / z) - a));
	else
		tmp = Float64(Float64(x - Float64(z * y)) / fma(Float64(-z), a, t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.05e+177], N[Not[LessEqual[z, 4.5e+105]], $MachinePrecision]], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[((-z) * a + t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05000000000000006e177 or 4.5000000000000001e105 < z

   1. Initial program 45.8%

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

      1. *-commutative 45.8%

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

   3. Simplified 45.8%

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

      1. clear-num 45.9%

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

      2. associate-/r/ 45.9%

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

      3. sub-neg 45.9%

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

      4. +-commutative 45.9%

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

      5. *-commutative 45.9%

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

      6. distribute-rgt-neg-in 45.9%

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

      7. fma-def 45.9%

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

   5. Applied egg-rr 45.9%

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

   6. Taylor expanded in x around 0 39.8%

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

      1. mul-1-neg 39.8%

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

      2. associate-/l* 58.1%

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

      3. distribute-neg-frac 58.1%

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

      4. mul-1-neg 58.1%

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

      5. sub-neg 58.1%

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

      6. div-sub 58.1%

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

      7. associate-/l* 87.7%

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

      8. *-inverses 87.7%

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

      9. /-rgt-identity 87.7%

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

   8. Simplified 87.7%

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

   if -1.05000000000000006e177 < z < 4.5000000000000001e105

   1. Initial program 94.3%

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

      1. *-commutative 94.3%

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

   3. Simplified 94.3%

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

      1. sub-neg 94.3%

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

      2. +-commutative 94.3%

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

      3. distribute-lft-neg-in 94.3%

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

      4. fma-def 94.3%

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

   5. Applied egg-rr 94.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+177} \lor \neg \left(z \leq 4.5 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{\mathsf{fma}\left(-z, a, t\right)}\\ \end{array} \]

Alternative 2: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+172} \lor \neg \left(z \leq 5 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -9e+172) (not (<= z 5e+105)))
   (/ (- y) (- (/ t z) a))
   (/ (- x (* z y)) (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -9e+172) || !(z <= 5e+105)) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = (x - (z * 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 <= (-9d+172)) .or. (.not. (z <= 5d+105))) then
        tmp = -y / ((t / z) - a)
    else
        tmp = (x - (z * 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 <= -9e+172) || !(z <= 5e+105)) {
		tmp = -y / ((t / z) - a);
	} else {
		tmp = (x - (z * y)) / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -9e+172) or not (z <= 5e+105):
		tmp = -y / ((t / z) - a)
	else:
		tmp = (x - (z * y)) / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -9e+172) || !(z <= 5e+105))
		tmp = Float64(Float64(-y) / Float64(Float64(t / z) - a));
	else
		tmp = Float64(Float64(x - Float64(z * y)) / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -9e+172) || ~((z <= 5e+105)))
		tmp = -y / ((t / z) - a);
	else
		tmp = (x - (z * y)) / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -9e+172], N[Not[LessEqual[z, 5e+105]], $MachinePrecision]], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.0000000000000004e172 or 5.00000000000000046e105 < z

   1. Initial program 45.8%

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

      1. *-commutative 45.8%

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

   3. Simplified 45.8%

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

      1. clear-num 45.9%

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

      2. associate-/r/ 45.9%

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

      3. sub-neg 45.9%

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

      4. +-commutative 45.9%

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

      5. *-commutative 45.9%

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

      6. distribute-rgt-neg-in 45.9%

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

      7. fma-def 45.9%

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

   5. Applied egg-rr 45.9%

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

   6. Taylor expanded in x around 0 39.8%

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

      1. mul-1-neg 39.8%

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

      2. associate-/l* 58.1%

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

      3. distribute-neg-frac 58.1%

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

      4. mul-1-neg 58.1%

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

      5. sub-neg 58.1%

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

      6. div-sub 58.1%

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

      7. associate-/l* 87.7%

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

      8. *-inverses 87.7%

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

      9. /-rgt-identity 87.7%

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

   8. Simplified 87.7%

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

   if -9.0000000000000004e172 < z < 5.00000000000000046e105

   1. Initial program 94.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+172} \lor \neg \left(z \leq 5 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \]

Alternative 3: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-76}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-251}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.15e-76)
   (/ (- y (/ x z)) a)
   (if (<= z 2.75e-251)
     (/ (- x (* z y)) t)
     (if (<= z 2e+25) (/ x (- t (* z a))) (/ (- y) (- (/ t z) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.15e-76) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 2.75e-251) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 2e+25) {
		tmp = x / (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 <= (-1.15d-76)) then
        tmp = (y - (x / z)) / a
    else if (z <= 2.75d-251) then
        tmp = (x - (z * y)) / t
    else if (z <= 2d+25) then
        tmp = x / (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 <= -1.15e-76) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 2.75e-251) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 2e+25) {
		tmp = x / (t - (z * a));
	} else {
		tmp = -y / ((t / z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.15e-76:
		tmp = (y - (x / z)) / a
	elif z <= 2.75e-251:
		tmp = (x - (z * y)) / t
	elif z <= 2e+25:
		tmp = x / (t - (z * a))
	else:
		tmp = -y / ((t / z) - a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.15e-76)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 2.75e-251)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 2e+25)
		tmp = Float64(x / 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 <= -1.15e-76)
		tmp = (y - (x / z)) / a;
	elseif (z <= 2.75e-251)
		tmp = (x - (z * y)) / t;
	elseif (z <= 2e+25)
		tmp = x / (t - (z * a));
	else
		tmp = -y / ((t / z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.15e-76], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 2.75e-251], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2e+25], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.15000000000000003e-76

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in t around 0 48.1%

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

      1. associate-*r/ 48.1%

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

      2. neg-mul-1 48.1%

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

      3. neg-sub0 48.1%

         \[\leadsto \frac{\color{blue}{0 - \left(x - y \cdot z\right)}}{a \cdot z} \]

      4. sub-neg 48.1%

         \[\leadsto \frac{0 - \color{blue}{\left(x + \left(-y \cdot z\right)\right)}}{a \cdot z} \]

      5. distribute-rgt-neg-out 48.1%

         \[\leadsto \frac{0 - \left(x + \color{blue}{y \cdot \left(-z\right)}\right)}{a \cdot z} \]

      6. +-commutative 48.1%

         \[\leadsto \frac{0 - \color{blue}{\left(y \cdot \left(-z\right) + x\right)}}{a \cdot z} \]

      7. associate--r+ 48.1%

         \[\leadsto \frac{\color{blue}{\left(0 - y \cdot \left(-z\right)\right) - x}}{a \cdot z} \]

      8. neg-sub0 48.1%

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

      9. distribute-rgt-neg-out 48.1%

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

      10. remove-double-neg 48.1%

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

      11. *-commutative 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in y around 0 66.3%

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

      1. +-commutative 66.3%

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

      2. mul-1-neg 66.3%

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

      3. sub-neg 66.3%

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

      4. *-commutative 66.3%

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

      5. associate-/r* 71.8%

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

      6. div-sub 74.7%

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

   9. Simplified 74.7%

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

   if -1.15000000000000003e-76 < z < 2.75e-251

   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 t around inf 82.9%

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

   if 2.75e-251 < z < 2.00000000000000018e25

   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 82.2%

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

      1. *-commutative 82.2%

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

   6. Simplified 82.2%

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

   if 2.00000000000000018e25 < z

   1. Initial program 57.7%

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

      1. *-commutative 57.7%

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

   3. Simplified 57.7%

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

      1. clear-num 57.7%

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

      2. associate-/r/ 57.7%

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

      3. sub-neg 57.7%

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

      4. +-commutative 57.7%

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

      5. *-commutative 57.7%

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

      6. distribute-rgt-neg-in 57.7%

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

      7. fma-def 57.7%

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

   5. Applied egg-rr 57.7%

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

   6. Taylor expanded in x around 0 42.9%

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

      1. mul-1-neg 42.9%

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

      2. associate-/l* 59.3%

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

      3. distribute-neg-frac 59.3%

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

      4. mul-1-neg 59.3%

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

      5. sub-neg 59.3%

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

      6. div-sub 59.3%

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

      7. associate-/l* 81.5%

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

      8. *-inverses 81.5%

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

      9. /-rgt-identity 81.5%

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

   8. Simplified 81.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-76}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 2.75 \cdot 10^{-251}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \]

Alternative 4: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-251}:\\ \;\;\;\;\frac{x}{t} - \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.8e-77)
   (/ (- y (/ x z)) a)
   (if (<= z 2.8e-251)
     (- (/ x t) (/ (* z y) t))
     (if (<= z 1.45e+25) (/ x (- t (* z a))) (/ (- y) (- (/ t z) a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.8e-77) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 2.8e-251) {
		tmp = (x / t) - ((z * y) / t);
	} else if (z <= 1.45e+25) {
		tmp = x / (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 <= (-3.8d-77)) then
        tmp = (y - (x / z)) / a
    else if (z <= 2.8d-251) then
        tmp = (x / t) - ((z * y) / t)
    else if (z <= 1.45d+25) then
        tmp = x / (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 <= -3.8e-77) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 2.8e-251) {
		tmp = (x / t) - ((z * y) / t);
	} else if (z <= 1.45e+25) {
		tmp = x / (t - (z * a));
	} else {
		tmp = -y / ((t / z) - a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.8e-77:
		tmp = (y - (x / z)) / a
	elif z <= 2.8e-251:
		tmp = (x / t) - ((z * y) / t)
	elif z <= 1.45e+25:
		tmp = x / (t - (z * a))
	else:
		tmp = -y / ((t / z) - a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.8e-77)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 2.8e-251)
		tmp = Float64(Float64(x / t) - Float64(Float64(z * y) / t));
	elseif (z <= 1.45e+25)
		tmp = Float64(x / 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 <= -3.8e-77)
		tmp = (y - (x / z)) / a;
	elseif (z <= 2.8e-251)
		tmp = (x / t) - ((z * y) / t);
	elseif (z <= 1.45e+25)
		tmp = x / (t - (z * a));
	else
		tmp = -y / ((t / z) - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.8e-77], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 2.8e-251], N[(N[(x / t), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.45e+25], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-y) / N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -3.7999999999999999e-77

   1. Initial program 70.5%

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

      1. *-commutative 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in t around 0 48.1%

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

      1. associate-*r/ 48.1%

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

      2. neg-mul-1 48.1%

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

      3. neg-sub0 48.1%

         \[\leadsto \frac{\color{blue}{0 - \left(x - y \cdot z\right)}}{a \cdot z} \]

      4. sub-neg 48.1%

         \[\leadsto \frac{0 - \color{blue}{\left(x + \left(-y \cdot z\right)\right)}}{a \cdot z} \]

      5. distribute-rgt-neg-out 48.1%

         \[\leadsto \frac{0 - \left(x + \color{blue}{y \cdot \left(-z\right)}\right)}{a \cdot z} \]

      6. +-commutative 48.1%

         \[\leadsto \frac{0 - \color{blue}{\left(y \cdot \left(-z\right) + x\right)}}{a \cdot z} \]

      7. associate--r+ 48.1%

         \[\leadsto \frac{\color{blue}{\left(0 - y \cdot \left(-z\right)\right) - x}}{a \cdot z} \]

      8. neg-sub0 48.1%

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

      9. distribute-rgt-neg-out 48.1%

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

      10. remove-double-neg 48.1%

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

      11. *-commutative 48.1%

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

   6. Simplified 48.1%

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

   7. Taylor expanded in y around 0 66.3%

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

      1. +-commutative 66.3%

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

      2. mul-1-neg 66.3%

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

      3. sub-neg 66.3%

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

      4. *-commutative 66.3%

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

      5. associate-/r* 71.8%

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

      6. div-sub 74.7%

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

   9. Simplified 74.7%

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

   if -3.7999999999999999e-77 < z < 2.79999999999999989e-251

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. add-cube-cbrt 99.3%

         \[\leadsto \frac{\left(-y\right) \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} + x}{t - z \cdot a} \]

      5. associate-*r* 99.3%

         \[\leadsto \frac{\color{blue}{\left(\left(-y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}} + x}{t - z \cdot a} \]

      6. fma-def 99.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right), \sqrt[3]{z}, x\right)}}{t - z \cdot a} \]

      7. pow2 99.3%

         \[\leadsto \frac{\mathsf{fma}\left(\left(-y\right) \cdot \color{blue}{{\left(\sqrt[3]{z}\right)}^{2}}, \sqrt[3]{z}, x\right)}{t - z \cdot a} \]

   5. Applied egg-rr 99.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-y\right) \cdot {\left(\sqrt[3]{z}\right)}^{2}, \sqrt[3]{z}, x\right)}}{t - z \cdot a} \]
   6. Step-by-step derivation

      1. fma-udef 99.3%

         \[\leadsto \frac{\color{blue}{\left(\left(-y\right) \cdot {\left(\sqrt[3]{z}\right)}^{2}\right) \cdot \sqrt[3]{z} + x}}{t - z \cdot a} \]

      2. +-commutative 99.3%

         \[\leadsto \frac{\color{blue}{x + \left(\left(-y\right) \cdot {\left(\sqrt[3]{z}\right)}^{2}\right) \cdot \sqrt[3]{z}}}{t - z \cdot a} \]

      3. associate-*l* 99.3%

         \[\leadsto \frac{x + \color{blue}{\left(-y\right) \cdot \left({\left(\sqrt[3]{z}\right)}^{2} \cdot \sqrt[3]{z}\right)}}{t - z \cdot a} \]

      4. pow-plus 99.3%

         \[\leadsto \frac{x + \left(-y\right) \cdot \color{blue}{{\left(\sqrt[3]{z}\right)}^{\left(2 + 1\right)}}}{t - z \cdot a} \]

      5. metadata-eval 99.3%

         \[\leadsto \frac{x + \left(-y\right) \cdot {\left(\sqrt[3]{z}\right)}^{\color{blue}{3}}}{t - z \cdot a} \]

   7. Simplified 99.3%

      \[\leadsto \frac{\color{blue}{x + \left(-y\right) \cdot {\left(\sqrt[3]{z}\right)}^{3}}}{t - z \cdot a} \]

   8. Taylor expanded in t around inf 82.5%

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

      1. rem-cube-cbrt 82.9%

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

      2. cancel-sign-sub-inv 82.9%

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

      3. div-sub 82.9%

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

      4. *-commutative 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 2.79999999999999989e-251 < z < 1.44999999999999995e25

   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 82.2%

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

      1. *-commutative 82.2%

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

   6. Simplified 82.2%

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

   if 1.44999999999999995e25 < z

   1. Initial program 57.7%

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

      1. *-commutative 57.7%

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

   3. Simplified 57.7%

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

      1. clear-num 57.7%

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

      2. associate-/r/ 57.7%

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

      3. sub-neg 57.7%

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

      4. +-commutative 57.7%

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

      5. *-commutative 57.7%

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

      6. distribute-rgt-neg-in 57.7%

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

      7. fma-def 57.7%

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

   5. Applied egg-rr 57.7%

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

   6. Taylor expanded in x around 0 42.9%

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

      1. mul-1-neg 42.9%

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

      2. associate-/l* 59.3%

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

      3. distribute-neg-frac 59.3%

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

      4. mul-1-neg 59.3%

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

      5. sub-neg 59.3%

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

      6. div-sub 59.3%

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

      7. associate-/l* 81.5%

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

      8. *-inverses 81.5%

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

      9. /-rgt-identity 81.5%

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

   8. Simplified 81.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-77}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-251}:\\ \;\;\;\;\frac{x}{t} - \frac{z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{t}{z} - a}\\ \end{array} \]

Alternative 5: 65.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-251}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.6e+49)
   (/ y a)
   (if (<= z 4e-251)
     (/ (- x (* z y)) t)
     (if (<= z 1.05e+101) (/ x (- t (* z a))) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e+49) {
		tmp = y / a;
	} else if (z <= 4e-251) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1.05e+101) {
		tmp = x / (t - (z * a));
	} 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 <= (-2.6d+49)) then
        tmp = y / a
    else if (z <= 4d-251) then
        tmp = (x - (z * y)) / t
    else if (z <= 1.05d+101) then
        tmp = x / (t - (z * a))
    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 <= -2.6e+49) {
		tmp = y / a;
	} else if (z <= 4e-251) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1.05e+101) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.6e+49:
		tmp = y / a
	elif z <= 4e-251:
		tmp = (x - (z * y)) / t
	elif z <= 1.05e+101:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.6e+49)
		tmp = Float64(y / a);
	elseif (z <= 4e-251)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 1.05e+101)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.6e+49)
		tmp = y / a;
	elseif (z <= 4e-251)
		tmp = (x - (z * y)) / t;
	elseif (z <= 1.05e+101)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.6e+49], N[(y / a), $MachinePrecision], If[LessEqual[z, 4e-251], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.05e+101], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.59999999999999989e49 or 1.05e101 < z

   1. Initial program 55.7%

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

      1. *-commutative 55.7%

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

   3. Simplified 55.7%

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

   4. Taylor expanded in z around inf 67.6%

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

   if -2.59999999999999989e49 < z < 4.00000000000000006e-251

   1. Initial program 96.6%

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

      1. *-commutative 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in t around inf 73.7%

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

   if 4.00000000000000006e-251 < z < 1.05e101

   1. Initial program 96.1%

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

      1. *-commutative 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in x around inf 74.4%

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

      1. *-commutative 74.4%

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

   6. Simplified 74.4%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+49}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-251}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 6: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - \frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -2.65 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-251}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y (/ x z)) a)))
   (if (<= z -2.65e-76)
     t_1
     (if (<= z 3e-251)
       (/ (- x (* z y)) t)
       (if (<= z 1.5e+30) (/ x (- t (* z a))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - (x / z)) / a;
	double tmp;
	if (z <= -2.65e-76) {
		tmp = t_1;
	} else if (z <= 3e-251) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1.5e+30) {
		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 - (x / z)) / a
    if (z <= (-2.65d-76)) then
        tmp = t_1
    else if (z <= 3d-251) then
        tmp = (x - (z * y)) / t
    else if (z <= 1.5d+30) 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 - (x / z)) / a;
	double tmp;
	if (z <= -2.65e-76) {
		tmp = t_1;
	} else if (z <= 3e-251) {
		tmp = (x - (z * y)) / t;
	} else if (z <= 1.5e+30) {
		tmp = x / (t - (z * a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - (x / z)) / a
	tmp = 0
	if z <= -2.65e-76:
		tmp = t_1
	elif z <= 3e-251:
		tmp = (x - (z * y)) / t
	elif z <= 1.5e+30:
		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(x / z)) / a)
	tmp = 0.0
	if (z <= -2.65e-76)
		tmp = t_1;
	elseif (z <= 3e-251)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	elseif (z <= 1.5e+30)
		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 - (x / z)) / a;
	tmp = 0.0;
	if (z <= -2.65e-76)
		tmp = t_1;
	elseif (z <= 3e-251)
		tmp = (x - (z * y)) / t;
	elseif (z <= 1.5e+30)
		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[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[z, -2.65e-76], t$95$1, If[LessEqual[z, 3e-251], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 1.5e+30], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.65e-76 or 1.49999999999999989e30 < 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 t around 0 44.8%

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

      1. associate-*r/ 44.8%

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

      2. neg-mul-1 44.8%

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

      3. neg-sub0 44.8%

         \[\leadsto \frac{\color{blue}{0 - \left(x - y \cdot z\right)}}{a \cdot z} \]

      4. sub-neg 44.8%

         \[\leadsto \frac{0 - \color{blue}{\left(x + \left(-y \cdot z\right)\right)}}{a \cdot z} \]

      5. distribute-rgt-neg-out 44.8%

         \[\leadsto \frac{0 - \left(x + \color{blue}{y \cdot \left(-z\right)}\right)}{a \cdot z} \]

      6. +-commutative 44.8%

         \[\leadsto \frac{0 - \color{blue}{\left(y \cdot \left(-z\right) + x\right)}}{a \cdot z} \]

      7. associate--r+ 44.8%

         \[\leadsto \frac{\color{blue}{\left(0 - y \cdot \left(-z\right)\right) - x}}{a \cdot z} \]

      8. neg-sub0 44.8%

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

      9. distribute-rgt-neg-out 44.8%

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

      10. remove-double-neg 44.8%

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

      11. *-commutative 44.8%

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

   6. Simplified 44.8%

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

   7. Taylor expanded in y around 0 68.8%

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

      1. +-commutative 68.8%

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

      2. mul-1-neg 68.8%

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

      3. sub-neg 68.8%

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

      4. *-commutative 68.8%

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

      5. associate-/r* 73.4%

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

      6. div-sub 75.8%

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

   9. Simplified 75.8%

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

   if -2.65e-76 < z < 2.9999999999999999e-251

   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 t around inf 82.9%

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

   if 2.9999999999999999e-251 < z < 1.49999999999999989e30

   1. Initial program 98.4%

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

      1. *-commutative 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in x around inf 81.0%

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

      1. *-commutative 81.0%

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

   6. Simplified 81.0%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{-76}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-251}:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]

Alternative 7: 54.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+63}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.2e-74)
   (/ y a)
   (if (<= z 3.6e-19)
     (/ x t)
     (if (<= z 1.7e+63)
       (/ (- x) (* z a))
       (if (<= z 8.5e+67) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.2e-74) {
		tmp = y / a;
	} else if (z <= 3.6e-19) {
		tmp = x / t;
	} else if (z <= 1.7e+63) {
		tmp = -x / (z * a);
	} else if (z <= 8.5e+67) {
		tmp = x / 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 <= (-3.2d-74)) then
        tmp = y / a
    else if (z <= 3.6d-19) then
        tmp = x / t
    else if (z <= 1.7d+63) then
        tmp = -x / (z * a)
    else if (z <= 8.5d+67) then
        tmp = x / 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 <= -3.2e-74) {
		tmp = y / a;
	} else if (z <= 3.6e-19) {
		tmp = x / t;
	} else if (z <= 1.7e+63) {
		tmp = -x / (z * a);
	} else if (z <= 8.5e+67) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.2e-74:
		tmp = y / a
	elif z <= 3.6e-19:
		tmp = x / t
	elif z <= 1.7e+63:
		tmp = -x / (z * a)
	elif z <= 8.5e+67:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.2e-74)
		tmp = Float64(y / a);
	elseif (z <= 3.6e-19)
		tmp = Float64(x / t);
	elseif (z <= 1.7e+63)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 8.5e+67)
		tmp = Float64(x / 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 <= -3.2e-74)
		tmp = y / a;
	elseif (z <= 3.6e-19)
		tmp = x / t;
	elseif (z <= 1.7e+63)
		tmp = -x / (z * a);
	elseif (z <= 8.5e+67)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.2e-74], N[(y / a), $MachinePrecision], If[LessEqual[z, 3.6e-19], N[(x / t), $MachinePrecision], If[LessEqual[z, 1.7e+63], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e+67], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1999999999999999e-74 or 8.50000000000000038e67 < z

   1. Initial program 63.0%

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

      1. *-commutative 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in z around inf 61.3%

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

   if -3.1999999999999999e-74 < z < 3.6000000000000001e-19 or 1.6999999999999999e63 < z < 8.50000000000000038e67

   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 z around 0 63.0%

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

   if 3.6000000000000001e-19 < z < 1.6999999999999999e63

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in t around 0 54.1%

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

      1. associate-*r/ 54.1%

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

      2. neg-mul-1 54.1%

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

      3. neg-sub0 54.1%

         \[\leadsto \frac{\color{blue}{0 - \left(x - y \cdot z\right)}}{a \cdot z} \]

      4. sub-neg 54.1%

         \[\leadsto \frac{0 - \color{blue}{\left(x + \left(-y \cdot z\right)\right)}}{a \cdot z} \]

      5. distribute-rgt-neg-out 54.1%

         \[\leadsto \frac{0 - \left(x + \color{blue}{y \cdot \left(-z\right)}\right)}{a \cdot z} \]

      6. +-commutative 54.1%

         \[\leadsto \frac{0 - \color{blue}{\left(y \cdot \left(-z\right) + x\right)}}{a \cdot z} \]

      7. associate--r+ 54.1%

         \[\leadsto \frac{\color{blue}{\left(0 - y \cdot \left(-z\right)\right) - x}}{a \cdot z} \]

      8. neg-sub0 54.1%

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

      9. distribute-rgt-neg-out 54.1%

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

      10. remove-double-neg 54.1%

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

      11. *-commutative 54.1%

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

   6. Simplified 54.1%

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

   7. Taylor expanded in y around 0 48.3%

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

      1. associate-*r/ 48.3%

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

      2. neg-mul-1 48.3%

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

      3. *-commutative 48.3%

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

   9. Simplified 48.3%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+63}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 8: 54.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{-y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.2e-75)
   (/ y a)
   (if (<= z 8.5e-14)
     (/ x t)
     (if (<= z 8.2e+67)
       (/ (- y) (/ t z))
       (if (<= z 3.2e+68) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.2e-75) {
		tmp = y / a;
	} else if (z <= 8.5e-14) {
		tmp = x / t;
	} else if (z <= 8.2e+67) {
		tmp = -y / (t / z);
	} else if (z <= 3.2e+68) {
		tmp = x / 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 <= (-5.2d-75)) then
        tmp = y / a
    else if (z <= 8.5d-14) then
        tmp = x / t
    else if (z <= 8.2d+67) then
        tmp = -y / (t / z)
    else if (z <= 3.2d+68) then
        tmp = x / 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 <= -5.2e-75) {
		tmp = y / a;
	} else if (z <= 8.5e-14) {
		tmp = x / t;
	} else if (z <= 8.2e+67) {
		tmp = -y / (t / z);
	} else if (z <= 3.2e+68) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -5.2e-75:
		tmp = y / a
	elif z <= 8.5e-14:
		tmp = x / t
	elif z <= 8.2e+67:
		tmp = -y / (t / z)
	elif z <= 3.2e+68:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.2e-75)
		tmp = Float64(y / a);
	elseif (z <= 8.5e-14)
		tmp = Float64(x / t);
	elseif (z <= 8.2e+67)
		tmp = Float64(Float64(-y) / Float64(t / z));
	elseif (z <= 3.2e+68)
		tmp = Float64(x / 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 <= -5.2e-75)
		tmp = y / a;
	elseif (z <= 8.5e-14)
		tmp = x / t;
	elseif (z <= 8.2e+67)
		tmp = -y / (t / z);
	elseif (z <= 3.2e+68)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.2e-75], N[(y / a), $MachinePrecision], If[LessEqual[z, 8.5e-14], N[(x / t), $MachinePrecision], If[LessEqual[z, 8.2e+67], N[((-y) / N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.2e+68], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2e-75 or 3.19999999999999994e68 < z

   1. Initial program 63.0%

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

      1. *-commutative 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in z around inf 61.3%

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

   if -5.2e-75 < z < 8.50000000000000038e-14 or 8.19999999999999959e67 < z < 3.19999999999999994e68

   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 z around 0 63.5%

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

   if 8.50000000000000038e-14 < z < 8.19999999999999959e67

   1. Initial program 89.2%

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

      1. *-commutative 89.2%

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

   3. Simplified 89.2%

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

      1. clear-num 89.2%

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

      2. associate-/r/ 89.0%

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

      3. sub-neg 89.0%

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

      4. +-commutative 89.0%

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

      5. *-commutative 89.0%

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

      6. distribute-rgt-neg-in 89.0%

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

      7. fma-def 88.9%

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

   5. Applied egg-rr 88.9%

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

   6. Taylor expanded in x around 0 47.0%

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

      1. mul-1-neg 47.0%

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

      2. associate-/l* 57.2%

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

      3. distribute-neg-frac 57.2%

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

      4. mul-1-neg 57.2%

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

      5. sub-neg 57.2%

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

      6. div-sub 57.2%

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

      7. associate-/l* 57.2%

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

      8. *-inverses 57.2%

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

      9. /-rgt-identity 57.2%

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

   8. Simplified 57.2%

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

   9. Taylor expanded in t around inf 46.3%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+67}:\\ \;\;\;\;\frac{-y}{\frac{t}{z}}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 9: 54.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{-z}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.1e-74)
   (/ y a)
   (if (<= z 2.4e-12)
     (/ x t)
     (if (<= z 4.6e+66)
       (/ (- z) (/ t y))
       (if (<= z 8.5e+67) (/ x t) (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.1e-74) {
		tmp = y / a;
	} else if (z <= 2.4e-12) {
		tmp = x / t;
	} else if (z <= 4.6e+66) {
		tmp = -z / (t / y);
	} else if (z <= 8.5e+67) {
		tmp = x / 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 <= (-2.1d-74)) then
        tmp = y / a
    else if (z <= 2.4d-12) then
        tmp = x / t
    else if (z <= 4.6d+66) then
        tmp = -z / (t / y)
    else if (z <= 8.5d+67) then
        tmp = x / 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 <= -2.1e-74) {
		tmp = y / a;
	} else if (z <= 2.4e-12) {
		tmp = x / t;
	} else if (z <= 4.6e+66) {
		tmp = -z / (t / y);
	} else if (z <= 8.5e+67) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.1e-74:
		tmp = y / a
	elif z <= 2.4e-12:
		tmp = x / t
	elif z <= 4.6e+66:
		tmp = -z / (t / y)
	elif z <= 8.5e+67:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.1e-74)
		tmp = Float64(y / a);
	elseif (z <= 2.4e-12)
		tmp = Float64(x / t);
	elseif (z <= 4.6e+66)
		tmp = Float64(Float64(-z) / Float64(t / y));
	elseif (z <= 8.5e+67)
		tmp = Float64(x / 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 <= -2.1e-74)
		tmp = y / a;
	elseif (z <= 2.4e-12)
		tmp = x / t;
	elseif (z <= 4.6e+66)
		tmp = -z / (t / y);
	elseif (z <= 8.5e+67)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.1e-74], N[(y / a), $MachinePrecision], If[LessEqual[z, 2.4e-12], N[(x / t), $MachinePrecision], If[LessEqual[z, 4.6e+66], N[((-z) / N[(t / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e+67], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.1e-74 or 8.50000000000000038e67 < z

   1. Initial program 63.0%

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

      1. *-commutative 63.0%

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

   3. Simplified 63.0%

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

   4. Taylor expanded in z around inf 61.3%

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

   if -2.1e-74 < z < 2.39999999999999987e-12 or 4.6e66 < z < 8.50000000000000038e67

   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 z around 0 63.5%

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

   if 2.39999999999999987e-12 < z < 4.6e66

   1. Initial program 89.2%

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

      1. *-commutative 89.2%

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

   3. Simplified 89.2%

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

      1. sub-neg 89.2%

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

      2. +-commutative 89.2%

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

      3. distribute-lft-neg-in 89.2%

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

      4. add-cube-cbrt 88.6%

         \[\leadsto \frac{\left(-y\right) \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} + x}{t - z \cdot a} \]

      5. associate-*r* 88.7%

         \[\leadsto \frac{\color{blue}{\left(\left(-y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}} + x}{t - z \cdot a} \]

      6. fma-def 88.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-y\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right), \sqrt[3]{z}, x\right)}}{t - z \cdot a} \]

      7. pow2 88.7%

         \[\leadsto \frac{\mathsf{fma}\left(\left(-y\right) \cdot \color{blue}{{\left(\sqrt[3]{z}\right)}^{2}}, \sqrt[3]{z}, x\right)}{t - z \cdot a} \]

   5. Applied egg-rr 88.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(-y\right) \cdot {\left(\sqrt[3]{z}\right)}^{2}, \sqrt[3]{z}, x\right)}}{t - z \cdot a} \]
   6. Step-by-step derivation

      1. fma-udef 88.7%

         \[\leadsto \frac{\color{blue}{\left(\left(-y\right) \cdot {\left(\sqrt[3]{z}\right)}^{2}\right) \cdot \sqrt[3]{z} + x}}{t - z \cdot a} \]

      2. +-commutative 88.7%

         \[\leadsto \frac{\color{blue}{x + \left(\left(-y\right) \cdot {\left(\sqrt[3]{z}\right)}^{2}\right) \cdot \sqrt[3]{z}}}{t - z \cdot a} \]

      3. associate-*l* 88.6%

         \[\leadsto \frac{x + \color{blue}{\left(-y\right) \cdot \left({\left(\sqrt[3]{z}\right)}^{2} \cdot \sqrt[3]{z}\right)}}{t - z \cdot a} \]

      4. pow-plus 88.8%

         \[\leadsto \frac{x + \left(-y\right) \cdot \color{blue}{{\left(\sqrt[3]{z}\right)}^{\left(2 + 1\right)}}}{t - z \cdot a} \]

      5. metadata-eval 88.8%

         \[\leadsto \frac{x + \left(-y\right) \cdot {\left(\sqrt[3]{z}\right)}^{\color{blue}{3}}}{t - z \cdot a} \]

   7. Simplified 88.8%

      \[\leadsto \frac{\color{blue}{x + \left(-y\right) \cdot {\left(\sqrt[3]{z}\right)}^{3}}}{t - z \cdot a} \]

   8. Taylor expanded in t around inf 46.4%

      \[\leadsto \frac{x + \left(-y\right) \cdot {\left(\sqrt[3]{z}\right)}^{3}}{\color{blue}{t}} \]

   9. Taylor expanded in x around 0 36.0%

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

       1. mul-1-neg 36.0%

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

       2. *-commutative 36.0%

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

       3. associate-/l* 46.4%

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

       4. distribute-neg-frac 46.4%

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

   11. Simplified 46.4%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{-74}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{-z}{\frac{t}{y}}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 10: 65.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+34} \lor \neg \left(z \leq 1.05 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.1e+34) (not (<= z 1.05e+101))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.1e+34) || !(z <= 1.05e+101)) {
		tmp = y / a;
	} else {
		tmp = x / (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 <= (-2.1d+34)) .or. (.not. (z <= 1.05d+101))) then
        tmp = y / a
    else
        tmp = x / (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 <= -2.1e+34) || !(z <= 1.05e+101)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.1e+34) or not (z <= 1.05e+101):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.1e+34) || !(z <= 1.05e+101))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / Float64(t - Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -2.1e+34) || ~((z <= 1.05e+101)))
		tmp = y / a;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.1e+34], N[Not[LessEqual[z, 1.05e+101]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.10000000000000017e34 or 1.05e101 < z

   1. Initial program 57.4%

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

      1. *-commutative 57.4%

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

   3. Simplified 57.4%

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

   4. Taylor expanded in z around inf 66.3%

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

   if -2.10000000000000017e34 < z < 1.05e101

   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 x around inf 72.2%

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

      1. *-commutative 72.2%

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

   6. Simplified 72.2%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+34} \lor \neg \left(z \leq 1.05 \cdot 10^{+101}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 11: 55.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-74} \lor \neg \left(z \leq 1.2 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.3e-74) (not (<= z 1.2e+30))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.3e-74) || !(z <= 1.2e+30)) {
		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 <= (-3.3d-74)) .or. (.not. (z <= 1.2d+30))) 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 <= -3.3e-74) || !(z <= 1.2e+30)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.3e-74) or not (z <= 1.2e+30):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.3e-74) || !(z <= 1.2e+30))
		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 <= -3.3e-74) || ~((z <= 1.2e+30)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.3e-74], N[Not[LessEqual[z, 1.2e+30]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.29999999999999996e-74 or 1.2e30 < z

   1. Initial program 64.8%

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

      1. *-commutative 64.8%

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

   3. Simplified 64.8%

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

   4. Taylor expanded in z around inf 58.7%

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

   if -3.29999999999999996e-74 < z < 1.2e30

   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 z around 0 60.1%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-74} \lor \neg \left(z \leq 1.2 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 12: 35.3% 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 82.4%

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

   1. *-commutative 82.4%

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

3. Simplified 82.4%

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

4. Taylor expanded in z around 0 35.2%

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

5. Final simplification 35.2%

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

Developer target: 97.3% 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 2023305 
(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))))