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

Percentage Accurate: 84.5% → 96.3%

Time: 13.9s

Alternatives: 10

Speedup: 0.5×

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: 96.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot a - t\\ t_2 := t - z \cdot a\\ t_3 := \frac{x - y \cdot z}{t\_2}\\ t_4 := \frac{x}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{t\_1} + t\_4\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-261}:\\ \;\;\;\;\frac{y \cdot z}{t\_1} + t\_4\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{-1}{z \cdot \frac{\frac{t}{z} - a}{y \cdot z - x}}\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+299}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* z a) t))
        (t_2 (- t (* z a)))
        (t_3 (/ (- x (* y z)) t_2))
        (t_4 (/ x t_2)))
   (if (<= t_3 (- INFINITY))
     (+ (* z (/ y t_1)) t_4)
     (if (<= t_3 -2e-261)
       (+ (/ (* y z) t_1) t_4)
       (if (<= t_3 0.0)
         (/ -1.0 (* z (/ (- (/ t z) a) (- (* y z) x))))
         (if (<= t_3 4e+299) t_3 (/ y (- a (/ t z)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * a) - t;
	double t_2 = t - (z * a);
	double t_3 = (x - (y * z)) / t_2;
	double t_4 = x / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (z * (y / t_1)) + t_4;
	} else if (t_3 <= -2e-261) {
		tmp = ((y * z) / t_1) + t_4;
	} else if (t_3 <= 0.0) {
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)));
	} else if (t_3 <= 4e+299) {
		tmp = t_3;
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * a) - t;
	double t_2 = t - (z * a);
	double t_3 = (x - (y * z)) / t_2;
	double t_4 = x / t_2;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (z * (y / t_1)) + t_4;
	} else if (t_3 <= -2e-261) {
		tmp = ((y * z) / t_1) + t_4;
	} else if (t_3 <= 0.0) {
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)));
	} else if (t_3 <= 4e+299) {
		tmp = t_3;
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * a) - t
	t_2 = t - (z * a)
	t_3 = (x - (y * z)) / t_2
	t_4 = x / t_2
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (z * (y / t_1)) + t_4
	elif t_3 <= -2e-261:
		tmp = ((y * z) / t_1) + t_4
	elif t_3 <= 0.0:
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)))
	elif t_3 <= 4e+299:
		tmp = t_3
	else:
		tmp = y / (a - (t / z))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * a) - t)
	t_2 = Float64(t - Float64(z * a))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_2)
	t_4 = Float64(x / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(z * Float64(y / t_1)) + t_4);
	elseif (t_3 <= -2e-261)
		tmp = Float64(Float64(Float64(y * z) / t_1) + t_4);
	elseif (t_3 <= 0.0)
		tmp = Float64(-1.0 / Float64(z * Float64(Float64(Float64(t / z) - a) / Float64(Float64(y * z) - x))));
	elseif (t_3 <= 4e+299)
		tmp = t_3;
	else
		tmp = Float64(y / Float64(a - Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * a) - t;
	t_2 = t - (z * a);
	t_3 = (x - (y * z)) / t_2;
	t_4 = x / t_2;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (z * (y / t_1)) + t_4;
	elseif (t_3 <= -2e-261)
		tmp = ((y * z) / t_1) + t_4;
	elseif (t_3 <= 0.0)
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)));
	elseif (t_3 <= 4e+299)
		tmp = t_3;
	else
		tmp = y / (a - (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(x / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(z * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$3, -2e-261], N[(N[(N[(y * z), $MachinePrecision] / t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(-1.0 / N[(z * N[(N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+299], t$95$3, N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0

   1. Initial program 64.5%

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

      1. *-commutative 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in x around 0 64.5%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*r/ 64.5%

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

      2. *-commutative 64.5%

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

      3. associate-*r/ 99.8%

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

   9. Simplified 99.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.99999999999999997e-261

   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. Add Preprocessing

   5. Taylor expanded in x around 0 99.7%

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

   if -1.99999999999999997e-261 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in z around inf 51.7%

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

      1. clear-num 51.7%

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

      2. inv-pow 51.7%

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

   7. Applied egg-rr 51.7%

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

      1. unpow-1 51.7%

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

      2. associate-/l* 97.0%

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

   9. Simplified 97.0%

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.0000000000000002e299

   1. Initial program 99.8%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   if 4.0000000000000002e299 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 33.0%

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

      1. *-commutative 33.0%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in z around inf 33.0%

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

   6. Taylor expanded in x around 0 89.6%

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

      1. associate-*r/ 89.6%

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

      2. neg-mul-1 89.6%

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

   8. Simplified 89.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t} + \frac{x}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-261}:\\ \;\;\;\;\frac{y \cdot z}{z \cdot a - t} + \frac{x}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{-1}{z \cdot \frac{\frac{t}{z} - a}{y \cdot z - x}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 4 \cdot 10^{+299}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t} + \frac{x}{t\_1}\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-261}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{-1}{z \cdot \frac{\frac{t}{z} - a}{y \cdot z - x}}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+299}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 (- INFINITY))
     (+ (* z (/ y (- (* z a) t))) (/ x t_1))
     (if (<= t_2 -2e-261)
       t_2
       (if (<= t_2 0.0)
         (/ -1.0 (* z (/ (- (/ t z) a) (- (* y z) x))))
         (if (<= t_2 4e+299) t_2 (/ y (- a (/ t z)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = (z * (y / ((z * a) - t))) + (x / t_1);
	} else if (t_2 <= -2e-261) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)));
	} else if (t_2 <= 4e+299) {
		tmp = t_2;
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = (z * (y / ((z * a) - t))) + (x / t_1);
	} else if (t_2 <= -2e-261) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)));
	} else if (t_2 <= 4e+299) {
		tmp = t_2;
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = (x - (y * z)) / t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = (z * (y / ((z * a) - t))) + (x / t_1)
	elif t_2 <= -2e-261:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)))
	elif t_2 <= 4e+299:
		tmp = t_2
	else:
		tmp = y / (a - (t / z))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(z * Float64(y / Float64(Float64(z * a) - t))) + Float64(x / t_1));
	elseif (t_2 <= -2e-261)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(-1.0 / Float64(z * Float64(Float64(Float64(t / z) - a) / Float64(Float64(y * z) - x))));
	elseif (t_2 <= 4e+299)
		tmp = t_2;
	else
		tmp = Float64(y / Float64(a - Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = (z * (y / ((z * a) - t))) + (x / t_1);
	elseif (t_2 <= -2e-261)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)));
	elseif (t_2 <= 4e+299)
		tmp = t_2;
	else
		tmp = y / (a - (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-261], t$95$2, If[LessEqual[t$95$2, 0.0], N[(-1.0 / N[(z * N[(N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+299], t$95$2, N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0

   1. Initial program 64.5%

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

      1. *-commutative 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in x around 0 64.5%

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

      1. associate-/l* 99.7%

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

      2. *-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. associate-*r/ 64.5%

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

      2. *-commutative 64.5%

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

      3. associate-*r/ 99.8%

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

   9. Simplified 99.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.99999999999999997e-261 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.0000000000000002e299

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   if -1.99999999999999997e-261 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in z around inf 51.7%

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

      1. clear-num 51.7%

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

      2. inv-pow 51.7%

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

   7. Applied egg-rr 51.7%

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

      1. unpow-1 51.7%

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

      2. associate-/l* 97.0%

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

   9. Simplified 97.0%

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

   if 4.0000000000000002e299 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 33.0%

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

      1. *-commutative 33.0%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in z around inf 33.0%

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

   6. Taylor expanded in x around 0 89.6%

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

      1. associate-*r/ 89.6%

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

      2. neg-mul-1 89.6%

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

   8. Simplified 89.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t} + \frac{x}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-261}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{-1}{z \cdot \frac{\frac{t}{z} - a}{y \cdot z - x}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 4 \cdot 10^{+299}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;y \cdot \left(\frac{z}{z \cdot a - t} + \frac{x}{y \cdot t\_1}\right)\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{-261}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{-1}{z \cdot \frac{\frac{t}{z} - a}{y \cdot z - x}}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+299}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))) (t_2 (/ (- x (* y z)) t_1)))
   (if (<= t_2 (- INFINITY))
     (* y (+ (/ z (- (* z a) t)) (/ x (* y t_1))))
     (if (<= t_2 -2e-261)
       t_2
       (if (<= t_2 0.0)
         (/ -1.0 (* z (/ (- (/ t z) a) (- (* y z) x))))
         (if (<= t_2 4e+299) t_2 (/ y (- a (/ t z)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = y * ((z / ((z * a) - t)) + (x / (y * t_1)));
	} else if (t_2 <= -2e-261) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)));
	} else if (t_2 <= 4e+299) {
		tmp = t_2;
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = (x - (y * z)) / t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = y * ((z / ((z * a) - t)) + (x / (y * t_1)));
	} else if (t_2 <= -2e-261) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)));
	} else if (t_2 <= 4e+299) {
		tmp = t_2;
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = (x - (y * z)) / t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = y * ((z / ((z * a) - t)) + (x / (y * t_1)))
	elif t_2 <= -2e-261:
		tmp = t_2
	elif t_2 <= 0.0:
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)))
	elif t_2 <= 4e+299:
		tmp = t_2
	else:
		tmp = y / (a - (t / z))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(y * Float64(Float64(z / Float64(Float64(z * a) - t)) + Float64(x / Float64(y * t_1))));
	elseif (t_2 <= -2e-261)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(-1.0 / Float64(z * Float64(Float64(Float64(t / z) - a) / Float64(Float64(y * z) - x))));
	elseif (t_2 <= 4e+299)
		tmp = t_2;
	else
		tmp = Float64(y / Float64(a - Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = y * ((z / ((z * a) - t)) + (x / (y * t_1)));
	elseif (t_2 <= -2e-261)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = -1.0 / (z * (((t / z) - a) / ((y * z) - x)));
	elseif (t_2 <= 4e+299)
		tmp = t_2;
	else
		tmp = y / (a - (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(y * N[(N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2e-261], t$95$2, If[LessEqual[t$95$2, 0.0], N[(-1.0 / N[(z * N[(N[(N[(t / z), $MachinePrecision] - a), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+299], t$95$2, N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0

   1. Initial program 64.5%

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

      1. *-commutative 64.5%

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

   3. Simplified 64.5%

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

   5. Taylor expanded in y around inf 99.7%

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

   7. Simplified 99.7%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.99999999999999997e-261 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 4.0000000000000002e299

   1. Initial program 99.7%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   if -1.99999999999999997e-261 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 51.7%

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

      1. *-commutative 51.7%

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

   3. Simplified 51.7%

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

   5. Taylor expanded in z around inf 51.7%

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

      1. clear-num 51.7%

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

      2. inv-pow 51.7%

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

   7. Applied egg-rr 51.7%

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

      1. unpow-1 51.7%

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

      2. associate-/l* 97.0%

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

   9. Simplified 97.0%

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

   if 4.0000000000000002e299 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 33.0%

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

      1. *-commutative 33.0%

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

   3. Simplified 33.0%

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

   5. Taylor expanded in z around inf 33.0%

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

   6. Taylor expanded in x around 0 89.6%

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

      1. associate-*r/ 89.6%

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

      2. neg-mul-1 89.6%

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

   8. Simplified 89.6%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;y \cdot \left(\frac{z}{z \cdot a - t} + \frac{x}{y \cdot \left(t - z \cdot a\right)}\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-261}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{-1}{z \cdot \frac{\frac{t}{z} - a}{y \cdot z - x}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 4 \cdot 10^{+299}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+158}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+167}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+209}:\\ \;\;\;\;\frac{\frac{-x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.6e+19)
   (/ y a)
   (if (<= z 3.5e+78)
     (/ x (- t (* z a)))
     (if (<= z 4.1e+158)
       (/ y a)
       (if (<= z 1.65e+167)
         (/ (- x (* y z)) t)
         (if (<= z 2.5e+209) (/ (/ (- x) z) a) (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e+19) {
		tmp = y / a;
	} else if (z <= 3.5e+78) {
		tmp = x / (t - (z * a));
	} else if (z <= 4.1e+158) {
		tmp = y / a;
	} else if (z <= 1.65e+167) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 2.5e+209) {
		tmp = (-x / 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 <= (-1.6d+19)) then
        tmp = y / a
    else if (z <= 3.5d+78) then
        tmp = x / (t - (z * a))
    else if (z <= 4.1d+158) then
        tmp = y / a
    else if (z <= 1.65d+167) then
        tmp = (x - (y * z)) / t
    else if (z <= 2.5d+209) then
        tmp = (-x / 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 <= -1.6e+19) {
		tmp = y / a;
	} else if (z <= 3.5e+78) {
		tmp = x / (t - (z * a));
	} else if (z <= 4.1e+158) {
		tmp = y / a;
	} else if (z <= 1.65e+167) {
		tmp = (x - (y * z)) / t;
	} else if (z <= 2.5e+209) {
		tmp = (-x / z) / a;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.6e+19:
		tmp = y / a
	elif z <= 3.5e+78:
		tmp = x / (t - (z * a))
	elif z <= 4.1e+158:
		tmp = y / a
	elif z <= 1.65e+167:
		tmp = (x - (y * z)) / t
	elif z <= 2.5e+209:
		tmp = (-x / z) / a
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.6e+19)
		tmp = Float64(y / a);
	elseif (z <= 3.5e+78)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 4.1e+158)
		tmp = Float64(y / a);
	elseif (z <= 1.65e+167)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (z <= 2.5e+209)
		tmp = Float64(Float64(Float64(-x) / 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 <= -1.6e+19)
		tmp = y / a;
	elseif (z <= 3.5e+78)
		tmp = x / (t - (z * a));
	elseif (z <= 4.1e+158)
		tmp = y / a;
	elseif (z <= 1.65e+167)
		tmp = (x - (y * z)) / t;
	elseif (z <= 2.5e+209)
		tmp = (-x / z) / a;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.6e+19], N[(y / a), $MachinePrecision], If[LessEqual[z, 3.5e+78], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.1e+158], N[(y / a), $MachinePrecision], If[LessEqual[z, 1.65e+167], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 2.5e+209], N[(N[((-x) / z), $MachinePrecision] / a), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.6e19 or 3.5000000000000001e78 < z < 4.10000000000000004e158 or 2.49999999999999982e209 < z

   1. Initial program 58.1%

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

      1. *-commutative 58.1%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in z around inf 69.0%

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

   if -1.6e19 < z < 3.5000000000000001e78

   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. Add Preprocessing

   5. Taylor expanded in x around inf 75.2%

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

      1. *-commutative 75.2%

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

   7. Simplified 75.2%

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

   if 4.10000000000000004e158 < z < 1.65000000000000009e167

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 100.0%

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

   if 1.65000000000000009e167 < z < 2.49999999999999982e209

   1. Initial program 35.8%

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

      1. *-commutative 35.8%

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

   3. Simplified 35.8%

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

   5. Taylor expanded in x around 0 35.8%

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

      1. associate-/l* 57.1%

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

      2. *-commutative 57.1%

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

   7. Applied egg-rr 57.1%

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

      1. associate-*r/ 35.8%

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

      2. *-commutative 35.8%

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

      3. associate-*r/ 57.0%

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

   9. Simplified 57.0%

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

   10. Taylor expanded in a around inf 57.9%

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

       1. mul-1-neg 57.9%

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

       2. unsub-neg 57.9%

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

   12. Simplified 57.9%

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

   13. Taylor expanded in y around 0 48.1%

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

       1. neg-mul-1 48.1%

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

       2. distribute-neg-frac2 48.1%

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

   15. Simplified 48.1%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+19}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+158}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+167}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+209}:\\ \;\;\;\;\frac{\frac{-x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+122}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.25e+122)
   (/ (- y (/ x z)) a)
   (if (<= z 3.5e+105) (/ (- x (* y z)) (- t (* z a))) (/ y (- a (/ t z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.25e+122) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 3.5e+105) {
		tmp = (x - (y * z)) / (t - (z * a));
	} else {
		tmp = y / (a - (t / z));
	}
	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 <= (-1.25d+122)) then
        tmp = (y - (x / z)) / a
    else if (z <= 3.5d+105) then
        tmp = (x - (y * z)) / (t - (z * a))
    else
        tmp = y / (a - (t / z))
    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 <= -1.25e+122) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 3.5e+105) {
		tmp = (x - (y * z)) / (t - (z * a));
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.25e+122:
		tmp = (y - (x / z)) / a
	elif z <= 3.5e+105:
		tmp = (x - (y * z)) / (t - (z * a))
	else:
		tmp = y / (a - (t / z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.25e+122)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 3.5e+105)
		tmp = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / Float64(a - Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.25e+122)
		tmp = (y - (x / z)) / a;
	elseif (z <= 3.5e+105)
		tmp = (x - (y * z)) / (t - (z * a));
	else
		tmp = y / (a - (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.25e+122], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 3.5e+105], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.24999999999999997e122

   1. Initial program 52.5%

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

      1. *-commutative 52.5%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in x around 0 52.5%

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

      1. associate-/l* 70.1%

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

      2. *-commutative 70.1%

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

   7. Applied egg-rr 70.1%

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

      1. associate-*r/ 52.5%

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

      2. *-commutative 52.5%

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

      3. associate-*r/ 66.5%

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

   9. Simplified 66.5%

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

   10. Taylor expanded in a around inf 86.7%

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

       1. mul-1-neg 86.7%

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

       2. unsub-neg 86.7%

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

   12. Simplified 86.7%

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

   if -1.24999999999999997e122 < z < 3.49999999999999991e105

   1. Initial program 97.0%

      \[\frac{x - y \cdot z}{t - a \cdot z} \]
   2. Add Preprocessing

   if 3.49999999999999991e105 < z

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

   3. Simplified 47.5%

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

   5. Taylor expanded in z around inf 47.6%

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

   6. Taylor expanded in x around 0 85.7%

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

      1. associate-*r/ 85.7%

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

      2. neg-mul-1 85.7%

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

   8. Simplified 85.7%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+122}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-30} \lor \neg \left(z \leq 1.32 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.2e-30) (not (<= z 1.32e+73)))
   (/ (- y (/ x z)) a)
   (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.2e-30) || !(z <= 1.32e+73)) {
		tmp = (y - (x / z)) / 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 <= (-1.2d-30)) .or. (.not. (z <= 1.32d+73))) then
        tmp = (y - (x / z)) / 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 <= -1.2e-30) || !(z <= 1.32e+73)) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.2e-30) or not (z <= 1.32e+73):
		tmp = (y - (x / z)) / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.2e-30) || !(z <= 1.32e+73))
		tmp = Float64(Float64(y - Float64(x / z)) / 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 <= -1.2e-30) || ~((z <= 1.32e+73)))
		tmp = (y - (x / z)) / a;
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.2e-30], N[Not[LessEqual[z, 1.32e+73]], $MachinePrecision]], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.19999999999999992e-30 or 1.32e73 < z

   1. Initial program 61.8%

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

      1. *-commutative 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in x around 0 61.8%

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

      1. associate-/l* 76.8%

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

      2. *-commutative 76.8%

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

   7. Applied egg-rr 76.8%

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

      1. associate-*r/ 61.8%

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

      2. *-commutative 61.8%

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

      3. associate-*r/ 74.5%

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

   9. Simplified 74.5%

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

   10. Taylor expanded in a around inf 74.2%

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

       1. mul-1-neg 74.2%

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

       2. unsub-neg 74.2%

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

   12. Simplified 74.2%

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

   if -1.19999999999999992e-30 < z < 1.32e73

   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. Add Preprocessing

   5. Taylor expanded in x around inf 78.0%

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

      1. *-commutative 78.0%

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

   7. Simplified 78.0%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{-30} \lor \neg \left(z \leq 1.32 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+19} \lor \neg \left(z \leq 4.7 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.7e+19) (not (<= z 4.7e+77))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.7e+19) || !(z <= 4.7e+77)) {
		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 <= (-1.7d+19)) .or. (.not. (z <= 4.7d+77))) 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 <= -1.7e+19) || !(z <= 4.7e+77)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.7e+19) or not (z <= 4.7e+77):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.7e+19) || !(z <= 4.7e+77))
		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 <= -1.7e+19) || ~((z <= 4.7e+77)))
		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, -1.7e+19], N[Not[LessEqual[z, 4.7e+77]], $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 < -1.7e19 or 4.7000000000000001e77 < 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. Add Preprocessing

   5. Taylor expanded in z around inf 63.8%

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

   if -1.7e19 < z < 4.7000000000000001e77

   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. Add Preprocessing

   5. Taylor expanded in x around inf 75.2%

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

      1. *-commutative 75.2%

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

   7. Simplified 75.2%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+19} \lor \neg \left(z \leq 4.7 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 10^{+20}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.3e-30)
   (/ (- y (/ x z)) a)
   (if (<= z 1e+20) (/ x (- t (* z a))) (/ y (- a (/ t z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.3e-30) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1e+20) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / (a - (t / z));
	}
	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 <= (-1.3d-30)) then
        tmp = (y - (x / z)) / a
    else if (z <= 1d+20) then
        tmp = x / (t - (z * a))
    else
        tmp = y / (a - (t / z))
    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 <= -1.3e-30) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1e+20) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / (a - (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.3e-30:
		tmp = (y - (x / z)) / a
	elif z <= 1e+20:
		tmp = x / (t - (z * a))
	else:
		tmp = y / (a - (t / z))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.3e-30)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 1e+20)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / Float64(a - Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.3e-30)
		tmp = (y - (x / z)) / a;
	elseif (z <= 1e+20)
		tmp = x / (t - (z * a));
	else
		tmp = y / (a - (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.3e-30], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 1e+20], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.29999999999999993e-30

   1. Initial program 69.5%

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

      1. *-commutative 69.5%

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

   3. Simplified 69.5%

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

   5. Taylor expanded in x around 0 69.5%

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

      1. associate-/l* 83.0%

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

      2. *-commutative 83.0%

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

   7. Applied egg-rr 83.0%

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

      1. associate-*r/ 69.5%

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

      2. *-commutative 69.5%

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

      3. associate-*r/ 81.0%

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

   9. Simplified 81.0%

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

   10. Taylor expanded in a around inf 77.5%

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

       1. mul-1-neg 77.5%

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

       2. unsub-neg 77.5%

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

   12. Simplified 77.5%

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

   if -1.29999999999999993e-30 < z < 1e20

   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. Add Preprocessing

   5. Taylor expanded in x around inf 80.5%

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

      1. *-commutative 80.5%

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

   7. Simplified 80.5%

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

   if 1e20 < z

   1. Initial program 55.5%

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

      1. *-commutative 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in z around inf 55.5%

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

   6. Taylor expanded in x around 0 79.9%

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

      1. associate-*r/ 79.9%

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

      2. neg-mul-1 79.9%

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

   8. Simplified 79.9%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-30}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 10^{+20}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-31} \lor \neg \left(z \leq 5.2 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.65e-31) (not (<= z 5.2e+73))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.65e-31) || !(z <= 5.2e+73)) {
		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 <= (-1.65d-31)) .or. (.not. (z <= 5.2d+73))) 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 <= -1.65e-31) || !(z <= 5.2e+73)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.65e-31) or not (z <= 5.2e+73):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.65e-31) || !(z <= 5.2e+73))
		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 <= -1.65e-31) || ~((z <= 5.2e+73)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.65e-31], N[Not[LessEqual[z, 5.2e+73]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.65e-31 or 5.2000000000000001e73 < z

   1. Initial program 61.8%

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

      1. *-commutative 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in z around inf 61.3%

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

   if -1.65e-31 < z < 5.2000000000000001e73

   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. Add Preprocessing

   5. Taylor expanded in z around 0 60.0%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{-31} \lor \neg \left(z \leq 5.2 \cdot 10^{+73}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.5% 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 81.1%

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

   1. *-commutative 81.1%

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

3. Simplified 81.1%

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

5. Taylor expanded in z around 0 36.0%

   \[\leadsto \color{blue}{\frac{x}{t}} \]
6. Add Preprocessing

Developer target: 97.6% accurate, 0.4× 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 2024091 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (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))))