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

Percentage Accurate: 84.7% → 97.5%

Time: 12.5s

Alternatives: 11

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: 84.7% 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: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := y \cdot \left(\frac{z}{z \cdot a - t} + \frac{x}{y \cdot t\_1}\right)\\ t_3 := \frac{x - z \cdot y}{t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-313}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a)))
        (t_2 (* y (+ (/ z (- (* z a) t)) (/ x (* y t_1)))))
        (t_3 (/ (- x (* z y)) t_1)))
   (if (<= t_3 (- INFINITY))
     t_2
     (if (<= t_3 -2e-313)
       t_3
       (if (<= t_3 0.0)
         (/ y (- a (/ t z)))
         (if (<= t_3 2e+272) t_3 (if (<= t_3 INFINITY) t_2 (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = y * ((z / ((z * a) - t)) + (x / (y * t_1)));
	double t_3 = (x - (z * y)) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_3 <= -2e-313) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = y / (a - (t / z));
	} else if (t_3 <= 2e+272) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	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 = y * ((z / ((z * a) - t)) + (x / (y * t_1)));
	double t_3 = (x - (z * y)) / t_1;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_3 <= -2e-313) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = y / (a - (t / z));
	} else if (t_3 <= 2e+272) {
		tmp = t_3;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = y * ((z / ((z * a) - t)) + (x / (y * t_1)))
	t_3 = (x - (z * y)) / t_1
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_2
	elif t_3 <= -2e-313:
		tmp = t_3
	elif t_3 <= 0.0:
		tmp = y / (a - (t / z))
	elif t_3 <= 2e+272:
		tmp = t_3
	elif t_3 <= math.inf:
		tmp = t_2
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(y * Float64(Float64(z / Float64(Float64(z * a) - t)) + Float64(x / Float64(y * t_1))))
	t_3 = Float64(Float64(x - Float64(z * y)) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_3 <= -2e-313)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = Float64(y / Float64(a - Float64(t / z)));
	elseif (t_3 <= 2e+272)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = y * ((z / ((z * a) - t)) + (x / (y * t_1)));
	t_3 = (x - (z * y)) / t_1;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_2;
	elseif (t_3 <= -2e-313)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = y / (a - (t / z));
	elseif (t_3 <= 2e+272)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = y / a;
	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[(y * N[(N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$2, If[LessEqual[t$95$3, -2e-313], t$95$3, If[LessEqual[t$95$3, 0.0], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+272], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$2, N[(y / a), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or 2.0000000000000001e272 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 67.1%

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

      1. *-commutative 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in y around inf 99.8%

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

   7. Simplified 99.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.99999999998e-313 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.0000000000000001e272

   1. Initial program 99.7%

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

   if -1.99999999998e-313 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in x around inf 70.0%

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

   7. Simplified 69.9%

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

   8. Taylor expanded in z around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. neg-mul-1 69.9%

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

   10. Simplified 69.9%

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

   11. Taylor expanded in y around 0 85.9%

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

       1. associate-/r* 83.4%

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

       2. mul-1-neg 83.4%

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

       3. distribute-frac-neg 83.4%

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

       4. distribute-frac-neg 83.4%

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

       5. unsub-neg 83.4%

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

   13. Simplified 83.4%

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

   14. Taylor expanded in x around 0 85.9%

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

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

   1. Initial program 0.0%

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

      1. *-commutative 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in z around inf 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - z \cdot y}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;y \cdot \left(\frac{z}{z \cdot a - t} + \frac{x}{y \cdot \left(t - z \cdot a\right)}\right)\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - z \cdot a} \leq -2 \cdot 10^{-313}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - z \cdot a} \leq 2 \cdot 10^{+272}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - z \cdot y}{t - z \cdot a} \leq \infty:\\ \;\;\;\;y \cdot \left(\frac{z}{z \cdot a - t} + \frac{x}{y \cdot \left(t - z \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ \mathbf{if}\;z \leq -7.6 \cdot 10^{+100} \lor \neg \left(z \leq 1.15 \cdot 10^{+54}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\frac{y}{x}}{a - \frac{t}{z}}, \frac{x}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))))
   (if (or (<= z -7.6e+100) (not (<= z 1.15e+54)))
     (fma x (/ (/ y x) (- a (/ t z))) (/ x t_1))
     (/ (- x (* z y)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double tmp;
	if ((z <= -7.6e+100) || !(z <= 1.15e+54)) {
		tmp = fma(x, ((y / x) / (a - (t / z))), (x / t_1));
	} else {
		tmp = (x - (z * y)) / t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	tmp = 0.0
	if ((z <= -7.6e+100) || !(z <= 1.15e+54))
		tmp = fma(x, Float64(Float64(y / x) / Float64(a - Float64(t / z))), Float64(x / t_1));
	else
		tmp = Float64(Float64(x - Float64(z * y)) / t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[z, -7.6e+100], N[Not[LessEqual[z, 1.15e+54]], $MachinePrecision]], N[(x * N[(N[(y / x), $MachinePrecision] / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.59999999999999927e100 or 1.14999999999999997e54 < z

   1. Initial program 62.6%

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

      1. *-commutative 62.6%

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

   3. Simplified 62.6%

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

   5. Taylor expanded in x around inf 57.1%

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

   7. Simplified 69.4%

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

   8. Taylor expanded in z around inf 69.4%

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

      1. associate-*r/ 69.4%

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

      2. neg-mul-1 69.4%

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

   10. Simplified 69.4%

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

   11. Taylor expanded in y around 0 85.5%

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

       1. associate-/r* 86.4%

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

       2. mul-1-neg 86.4%

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

       3. distribute-frac-neg 86.4%

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

       4. distribute-frac-neg 86.4%

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

       5. unsub-neg 86.4%

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

   13. Simplified 86.4%

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

   if -7.59999999999999927e100 < z < 1.14999999999999997e54

   1. Initial program 99.2%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+100} \lor \neg \left(z \leq 1.15 \cdot 10^{+54}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\frac{y}{x}}{a - \frac{t}{z}}, \frac{x}{t - z \cdot a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+132} \lor \neg \left(z \leq -9.5 \cdot 10^{+88} \lor \neg \left(z \leq -1.8 \cdot 10^{-13}\right) \land \left(z \leq 2.75 \cdot 10^{-78} \lor \neg \left(z \leq 3.2 \cdot 10^{-32}\right) \land z \leq 4000000000\right)\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.8e+132)
         (not
          (or (<= z -9.5e+88)
              (and (not (<= z -1.8e-13))
                   (or (<= z 2.75e-78)
                       (and (not (<= z 3.2e-32)) (<= z 4000000000.0)))))))
   (/ y (- a (/ t z)))
   (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.8e+132) || !((z <= -9.5e+88) || (!(z <= -1.8e-13) && ((z <= 2.75e-78) || (!(z <= 3.2e-32) && (z <= 4000000000.0)))))) {
		tmp = y / (a - (t / z));
	} 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 <= (-2.8d+132)) .or. (.not. (z <= (-9.5d+88)) .or. (.not. (z <= (-1.8d-13))) .and. (z <= 2.75d-78) .or. (.not. (z <= 3.2d-32)) .and. (z <= 4000000000.0d0))) then
        tmp = y / (a - (t / z))
    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 <= -2.8e+132) || !((z <= -9.5e+88) || (!(z <= -1.8e-13) && ((z <= 2.75e-78) || (!(z <= 3.2e-32) && (z <= 4000000000.0)))))) {
		tmp = y / (a - (t / z));
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.8e+132) or not ((z <= -9.5e+88) or (not (z <= -1.8e-13) and ((z <= 2.75e-78) or (not (z <= 3.2e-32) and (z <= 4000000000.0))))):
		tmp = y / (a - (t / z))
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.8e+132) || !((z <= -9.5e+88) || (!(z <= -1.8e-13) && ((z <= 2.75e-78) || (!(z <= 3.2e-32) && (z <= 4000000000.0))))))
		tmp = Float64(y / Float64(a - Float64(t / z)));
	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 <= -2.8e+132) || ~(((z <= -9.5e+88) || (~((z <= -1.8e-13)) && ((z <= 2.75e-78) || (~((z <= 3.2e-32)) && (z <= 4000000000.0)))))))
		tmp = y / (a - (t / z));
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.8e+132], N[Not[Or[LessEqual[z, -9.5e+88], And[N[Not[LessEqual[z, -1.8e-13]], $MachinePrecision], Or[LessEqual[z, 2.75e-78], And[N[Not[LessEqual[z, 3.2e-32]], $MachinePrecision], LessEqual[z, 4000000000.0]]]]]], $MachinePrecision]], N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7999999999999999e132 or -9.50000000000000059e88 < z < -1.7999999999999999e-13 or 2.75000000000000009e-78 < z < 3.2000000000000002e-32 or 4e9 < z

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in x around inf 61.6%

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

   7. Simplified 71.3%

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

   8. Taylor expanded in z around inf 71.3%

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

      1. associate-*r/ 71.3%

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

      2. neg-mul-1 71.3%

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

   10. Simplified 71.3%

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

   11. Taylor expanded in y around 0 84.9%

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

       1. associate-/r* 84.3%

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

       2. mul-1-neg 84.3%

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

       3. distribute-frac-neg 84.3%

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

       4. distribute-frac-neg 84.3%

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

       5. unsub-neg 84.3%

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

   13. Simplified 84.3%

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

   14. Taylor expanded in x around 0 81.6%

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

   if -2.7999999999999999e132 < z < -9.50000000000000059e88 or -1.7999999999999999e-13 < z < 2.75000000000000009e-78 or 3.2000000000000002e-32 < z < 4e9

   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. Add Preprocessing

   5. Taylor expanded in x around inf 80.6%

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

      1. *-commutative 80.6%

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

   7. Simplified 80.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+132} \lor \neg \left(z \leq -9.5 \cdot 10^{+88} \lor \neg \left(z \leq -1.8 \cdot 10^{-13}\right) \land \left(z \leq 2.75 \cdot 10^{-78} \lor \neg \left(z \leq 3.2 \cdot 10^{-32}\right) \land z \leq 4000000000\right)\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{t - z \cdot a}\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq 950000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+137}:\\ \;\;\;\;z \cdot \frac{y}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ x (- t (* z a)))))
   (if (<= z -7.2e+133)
     (/ y a)
     (if (<= z -1.65e+87)
       t_1
       (if (<= z -8.5e+28)
         (/ (* z y) (- t))
         (if (<= z 950000000000.0)
           t_1
           (if (<= z 7.2e+137) (* z (/ y (- t))) (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (z <= -7.2e+133) {
		tmp = y / a;
	} else if (z <= -1.65e+87) {
		tmp = t_1;
	} else if (z <= -8.5e+28) {
		tmp = (z * y) / -t;
	} else if (z <= 950000000000.0) {
		tmp = t_1;
	} else if (z <= 7.2e+137) {
		tmp = z * (y / -t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x / (t - (z * a))
    if (z <= (-7.2d+133)) then
        tmp = y / a
    else if (z <= (-1.65d+87)) then
        tmp = t_1
    else if (z <= (-8.5d+28)) then
        tmp = (z * y) / -t
    else if (z <= 950000000000.0d0) then
        tmp = t_1
    else if (z <= 7.2d+137) then
        tmp = z * (y / -t)
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x / (t - (z * a));
	double tmp;
	if (z <= -7.2e+133) {
		tmp = y / a;
	} else if (z <= -1.65e+87) {
		tmp = t_1;
	} else if (z <= -8.5e+28) {
		tmp = (z * y) / -t;
	} else if (z <= 950000000000.0) {
		tmp = t_1;
	} else if (z <= 7.2e+137) {
		tmp = z * (y / -t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x / (t - (z * a))
	tmp = 0
	if z <= -7.2e+133:
		tmp = y / a
	elif z <= -1.65e+87:
		tmp = t_1
	elif z <= -8.5e+28:
		tmp = (z * y) / -t
	elif z <= 950000000000.0:
		tmp = t_1
	elif z <= 7.2e+137:
		tmp = z * (y / -t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (z <= -7.2e+133)
		tmp = Float64(y / a);
	elseif (z <= -1.65e+87)
		tmp = t_1;
	elseif (z <= -8.5e+28)
		tmp = Float64(Float64(z * y) / Float64(-t));
	elseif (z <= 950000000000.0)
		tmp = t_1;
	elseif (z <= 7.2e+137)
		tmp = Float64(z * Float64(y / Float64(-t)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x / (t - (z * a));
	tmp = 0.0;
	if (z <= -7.2e+133)
		tmp = y / a;
	elseif (z <= -1.65e+87)
		tmp = t_1;
	elseif (z <= -8.5e+28)
		tmp = (z * y) / -t;
	elseif (z <= 950000000000.0)
		tmp = t_1;
	elseif (z <= 7.2e+137)
		tmp = z * (y / -t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -7.2e+133], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.65e+87], t$95$1, If[LessEqual[z, -8.5e+28], N[(N[(z * y), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[z, 950000000000.0], t$95$1, If[LessEqual[z, 7.2e+137], N[(z * N[(y / (-t)), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -7.19999999999999956e133 or 7.1999999999999999e137 < z

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

   3. Simplified 60.8%

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

   5. Taylor expanded in z around inf 65.4%

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

   if -7.19999999999999956e133 < z < -1.6500000000000001e87 or -8.49999999999999954e28 < z < 9.5e11

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in x around inf 75.0%

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

      1. *-commutative 75.0%

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

   7. Simplified 75.0%

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

   if -1.6500000000000001e87 < z < -8.49999999999999954e28

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in x around 0 82.2%

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

      1. mul-1-neg 82.2%

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

      2. associate-/l* 82.1%

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

      3. distribute-rgt-neg-in 82.1%

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

      4. distribute-neg-frac2 82.1%

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

      5. cancel-sign-sub-inv 82.1%

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

      6. *-commutative 82.1%

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

      7. +-commutative 82.1%

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

      8. distribute-rgt-neg-out 82.1%

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

      9. distribute-lft-neg-in 82.1%

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

      10. *-commutative 82.1%

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

      11. fma-undefine 82.1%

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

      12. neg-sub0 82.1%

          \[\leadsto y \cdot \frac{z}{\color{blue}{0 - \mathsf{fma}\left(a, -z, t\right)}} \]

      13. fma-undefine 82.1%

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

      14. distribute-rgt-neg-in 82.1%

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

      15. mul-1-neg 82.1%

          \[\leadsto y \cdot \frac{z}{0 - \left(\color{blue}{-1 \cdot \left(a \cdot z\right)} + t\right)} \]

      16. associate-*r* 82.1%

          \[\leadsto y \cdot \frac{z}{0 - \left(\color{blue}{\left(-1 \cdot a\right) \cdot z} + t\right)} \]

      17. neg-mul-1 82.1%

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

      18. *-commutative 82.1%

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

      19. associate--r+ 82.1%

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

      20. neg-sub0 82.1%

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

      21. distribute-rgt-neg-out 82.1%

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

      22. remove-double-neg 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in z around 0 73.3%

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

      1. associate-*r/ 73.3%

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

      2. associate-*r* 73.3%

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

      3. neg-mul-1 73.3%

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

   10. Simplified 73.3%

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

   if 9.5e11 < z < 7.1999999999999999e137

   1. Initial program 67.4%

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

      1. *-commutative 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in t around inf 44.4%

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

   6. Taylor expanded in x around 0 35.9%

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

      1. mul-1-neg 35.9%

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

      2. *-commutative 35.9%

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

      3. associate-/l* 49.7%

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

      4. distribute-rgt-neg-in 49.7%

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

      5. distribute-neg-frac2 49.7%

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

   8. Simplified 49.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.65 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq -8.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq 950000000000:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+137}:\\ \;\;\;\;z \cdot \frac{y}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -4.9 \cdot 10^{+29}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq -0.04:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 840000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+137}:\\ \;\;\;\;z \cdot \frac{y}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.4e+92)
   (/ y a)
   (if (<= z -4.9e+29)
     (/ (* z y) (- t))
     (if (<= z -0.04)
       (/ y a)
       (if (<= z 840000000000.0)
         (/ x t)
         (if (<= z 7e+137) (* z (/ y (- t))) (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.4e+92) {
		tmp = y / a;
	} else if (z <= -4.9e+29) {
		tmp = (z * y) / -t;
	} else if (z <= -0.04) {
		tmp = y / a;
	} else if (z <= 840000000000.0) {
		tmp = x / t;
	} else if (z <= 7e+137) {
		tmp = z * (y / -t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-4.4d+92)) then
        tmp = y / a
    else if (z <= (-4.9d+29)) then
        tmp = (z * y) / -t
    else if (z <= (-0.04d0)) then
        tmp = y / a
    else if (z <= 840000000000.0d0) then
        tmp = x / t
    else if (z <= 7d+137) then
        tmp = z * (y / -t)
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.4e+92) {
		tmp = y / a;
	} else if (z <= -4.9e+29) {
		tmp = (z * y) / -t;
	} else if (z <= -0.04) {
		tmp = y / a;
	} else if (z <= 840000000000.0) {
		tmp = x / t;
	} else if (z <= 7e+137) {
		tmp = z * (y / -t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -4.4e+92:
		tmp = y / a
	elif z <= -4.9e+29:
		tmp = (z * y) / -t
	elif z <= -0.04:
		tmp = y / a
	elif z <= 840000000000.0:
		tmp = x / t
	elif z <= 7e+137:
		tmp = z * (y / -t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.4e+92)
		tmp = Float64(y / a);
	elseif (z <= -4.9e+29)
		tmp = Float64(Float64(z * y) / Float64(-t));
	elseif (z <= -0.04)
		tmp = Float64(y / a);
	elseif (z <= 840000000000.0)
		tmp = Float64(x / t);
	elseif (z <= 7e+137)
		tmp = Float64(z * Float64(y / Float64(-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 <= -4.4e+92)
		tmp = y / a;
	elseif (z <= -4.9e+29)
		tmp = (z * y) / -t;
	elseif (z <= -0.04)
		tmp = y / a;
	elseif (z <= 840000000000.0)
		tmp = x / t;
	elseif (z <= 7e+137)
		tmp = z * (y / -t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.4e+92], N[(y / a), $MachinePrecision], If[LessEqual[z, -4.9e+29], N[(N[(z * y), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[z, -0.04], N[(y / a), $MachinePrecision], If[LessEqual[z, 840000000000.0], N[(x / t), $MachinePrecision], If[LessEqual[z, 7e+137], N[(z * N[(y / (-t)), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.39999999999999984e92 or -4.9000000000000001e29 < z < -0.0400000000000000008 or 7.0000000000000002e137 < z

   1. Initial program 68.2%

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

      1. *-commutative 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in z around inf 60.7%

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

   if -4.39999999999999984e92 < z < -4.9000000000000001e29

   1. Initial program 92.4%

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

      1. *-commutative 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in x around 0 77.7%

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

      1. mul-1-neg 77.7%

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

      2. associate-/l* 77.6%

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

      3. distribute-rgt-neg-in 77.6%

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

      4. distribute-neg-frac2 77.6%

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

      5. cancel-sign-sub-inv 77.6%

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

      6. *-commutative 77.6%

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

      7. +-commutative 77.6%

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

      8. distribute-rgt-neg-out 77.6%

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

      9. distribute-lft-neg-in 77.6%

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

      10. *-commutative 77.6%

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

      11. fma-undefine 77.6%

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

      12. neg-sub0 77.6%

          \[\leadsto y \cdot \frac{z}{\color{blue}{0 - \mathsf{fma}\left(a, -z, t\right)}} \]

      13. fma-undefine 77.6%

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

      14. distribute-rgt-neg-in 77.6%

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

      15. mul-1-neg 77.6%

          \[\leadsto y \cdot \frac{z}{0 - \left(\color{blue}{-1 \cdot \left(a \cdot z\right)} + t\right)} \]

      16. associate-*r* 77.6%

          \[\leadsto y \cdot \frac{z}{0 - \left(\color{blue}{\left(-1 \cdot a\right) \cdot z} + t\right)} \]

      17. neg-mul-1 77.6%

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

      18. *-commutative 77.6%

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

      19. associate--r+ 77.6%

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

      20. neg-sub0 77.6%

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

      21. distribute-rgt-neg-out 77.6%

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

      22. remove-double-neg 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in z around 0 70.2%

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

      1. associate-*r/ 70.2%

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

      2. associate-*r* 70.2%

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

      3. neg-mul-1 70.2%

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

   10. Simplified 70.2%

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

   if -0.0400000000000000008 < z < 8.4e11

   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. Add Preprocessing

   5. Taylor expanded in z around 0 63.0%

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

   if 8.4e11 < z < 7.0000000000000002e137

   1. Initial program 67.4%

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

      1. *-commutative 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in t around inf 44.4%

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

   6. Taylor expanded in x around 0 35.9%

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

      1. mul-1-neg 35.9%

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

      2. *-commutative 35.9%

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

      3. associate-/l* 49.7%

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

      4. distribute-rgt-neg-in 49.7%

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

      5. distribute-neg-frac2 49.7%

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

   8. Simplified 49.7%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -4.9 \cdot 10^{+29}:\\ \;\;\;\;\frac{z \cdot y}{-t}\\ \mathbf{elif}\;z \leq -0.04:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 840000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+137}:\\ \;\;\;\;z \cdot \frac{y}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{+29}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq -0.048:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 650000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+138}:\\ \;\;\;\;z \cdot \frac{y}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.9e+92)
   (/ y a)
   (if (<= z -2e+29)
     (* (- y) (/ z t))
     (if (<= z -0.048)
       (/ y a)
       (if (<= z 650000000000.0)
         (/ x t)
         (if (<= z 2.2e+138) (* z (/ y (- t))) (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e+92) {
		tmp = y / a;
	} else if (z <= -2e+29) {
		tmp = -y * (z / t);
	} else if (z <= -0.048) {
		tmp = y / a;
	} else if (z <= 650000000000.0) {
		tmp = x / t;
	} else if (z <= 2.2e+138) {
		tmp = z * (y / -t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.9d+92)) then
        tmp = y / a
    else if (z <= (-2d+29)) then
        tmp = -y * (z / t)
    else if (z <= (-0.048d0)) then
        tmp = y / a
    else if (z <= 650000000000.0d0) then
        tmp = x / t
    else if (z <= 2.2d+138) then
        tmp = z * (y / -t)
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e+92) {
		tmp = y / a;
	} else if (z <= -2e+29) {
		tmp = -y * (z / t);
	} else if (z <= -0.048) {
		tmp = y / a;
	} else if (z <= 650000000000.0) {
		tmp = x / t;
	} else if (z <= 2.2e+138) {
		tmp = z * (y / -t);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.9e+92:
		tmp = y / a
	elif z <= -2e+29:
		tmp = -y * (z / t)
	elif z <= -0.048:
		tmp = y / a
	elif z <= 650000000000.0:
		tmp = x / t
	elif z <= 2.2e+138:
		tmp = z * (y / -t)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.9e+92)
		tmp = Float64(y / a);
	elseif (z <= -2e+29)
		tmp = Float64(Float64(-y) * Float64(z / t));
	elseif (z <= -0.048)
		tmp = Float64(y / a);
	elseif (z <= 650000000000.0)
		tmp = Float64(x / t);
	elseif (z <= 2.2e+138)
		tmp = Float64(z * Float64(y / Float64(-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 <= -2.9e+92)
		tmp = y / a;
	elseif (z <= -2e+29)
		tmp = -y * (z / t);
	elseif (z <= -0.048)
		tmp = y / a;
	elseif (z <= 650000000000.0)
		tmp = x / t;
	elseif (z <= 2.2e+138)
		tmp = z * (y / -t);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.9e+92], N[(y / a), $MachinePrecision], If[LessEqual[z, -2e+29], N[((-y) * N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -0.048], N[(y / a), $MachinePrecision], If[LessEqual[z, 650000000000.0], N[(x / t), $MachinePrecision], If[LessEqual[z, 2.2e+138], N[(z * N[(y / (-t)), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.9000000000000001e92 or -1.99999999999999983e29 < z < -0.048000000000000001 or 2.2000000000000001e138 < z

   1. Initial program 68.2%

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

      1. *-commutative 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in z around inf 60.7%

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

   if -2.9000000000000001e92 < z < -1.99999999999999983e29

   1. Initial program 92.4%

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

      1. *-commutative 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in x around 0 77.7%

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

      1. mul-1-neg 77.7%

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

      2. associate-/l* 77.6%

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

      3. distribute-rgt-neg-in 77.6%

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

      4. distribute-neg-frac2 77.6%

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

      5. cancel-sign-sub-inv 77.6%

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

      6. *-commutative 77.6%

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

      7. +-commutative 77.6%

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

      8. distribute-rgt-neg-out 77.6%

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

      9. distribute-lft-neg-in 77.6%

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

      10. *-commutative 77.6%

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

      11. fma-undefine 77.6%

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

      12. neg-sub0 77.6%

          \[\leadsto y \cdot \frac{z}{\color{blue}{0 - \mathsf{fma}\left(a, -z, t\right)}} \]

      13. fma-undefine 77.6%

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

      14. distribute-rgt-neg-in 77.6%

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

      15. mul-1-neg 77.6%

          \[\leadsto y \cdot \frac{z}{0 - \left(\color{blue}{-1 \cdot \left(a \cdot z\right)} + t\right)} \]

      16. associate-*r* 77.6%

          \[\leadsto y \cdot \frac{z}{0 - \left(\color{blue}{\left(-1 \cdot a\right) \cdot z} + t\right)} \]

      17. neg-mul-1 77.6%

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

      18. *-commutative 77.6%

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

      19. associate--r+ 77.6%

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

      20. neg-sub0 77.6%

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

      21. distribute-rgt-neg-out 77.6%

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

      22. remove-double-neg 77.6%

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

   7. Simplified 77.6%

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

   8. Taylor expanded in z around 0 70.0%

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

      1. associate-*r/ 70.0%

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

      2. mul-1-neg 70.0%

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

   10. Simplified 70.0%

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

   if -0.048000000000000001 < z < 6.5e11

   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. Add Preprocessing

   5. Taylor expanded in z around 0 63.0%

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

   if 6.5e11 < z < 2.2000000000000001e138

   1. Initial program 67.4%

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

      1. *-commutative 67.4%

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

   3. Simplified 67.4%

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

   5. Taylor expanded in t around inf 44.4%

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

   6. Taylor expanded in x around 0 35.9%

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

      1. mul-1-neg 35.9%

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

      2. *-commutative 35.9%

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

      3. associate-/l* 49.7%

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

      4. distribute-rgt-neg-in 49.7%

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

      5. distribute-neg-frac2 49.7%

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

   8. Simplified 49.7%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{+29}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq -0.048:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 650000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+138}:\\ \;\;\;\;z \cdot \frac{y}{-t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-y\right) \cdot \frac{z}{t}\\ \mathbf{if}\;z \leq -4.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -0.054:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 68000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- y) (/ z t))))
   (if (<= z -4.8e+92)
     (/ y a)
     (if (<= z -4.5e+29)
       t_1
       (if (<= z -0.054)
         (/ y a)
         (if (<= z 68000000000.0)
           (/ x t)
           (if (<= z 7.2e+137) t_1 (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -y * (z / t);
	double tmp;
	if (z <= -4.8e+92) {
		tmp = y / a;
	} else if (z <= -4.5e+29) {
		tmp = t_1;
	} else if (z <= -0.054) {
		tmp = y / a;
	} else if (z <= 68000000000.0) {
		tmp = x / t;
	} else if (z <= 7.2e+137) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -y * (z / t)
    if (z <= (-4.8d+92)) then
        tmp = y / a
    else if (z <= (-4.5d+29)) then
        tmp = t_1
    else if (z <= (-0.054d0)) then
        tmp = y / a
    else if (z <= 68000000000.0d0) then
        tmp = x / t
    else if (z <= 7.2d+137) then
        tmp = t_1
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = -y * (z / t);
	double tmp;
	if (z <= -4.8e+92) {
		tmp = y / a;
	} else if (z <= -4.5e+29) {
		tmp = t_1;
	} else if (z <= -0.054) {
		tmp = y / a;
	} else if (z <= 68000000000.0) {
		tmp = x / t;
	} else if (z <= 7.2e+137) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = -y * (z / t)
	tmp = 0
	if z <= -4.8e+92:
		tmp = y / a
	elif z <= -4.5e+29:
		tmp = t_1
	elif z <= -0.054:
		tmp = y / a
	elif z <= 68000000000.0:
		tmp = x / t
	elif z <= 7.2e+137:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(-y) * Float64(z / t))
	tmp = 0.0
	if (z <= -4.8e+92)
		tmp = Float64(y / a);
	elseif (z <= -4.5e+29)
		tmp = t_1;
	elseif (z <= -0.054)
		tmp = Float64(y / a);
	elseif (z <= 68000000000.0)
		tmp = Float64(x / t);
	elseif (z <= 7.2e+137)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = -y * (z / t);
	tmp = 0.0;
	if (z <= -4.8e+92)
		tmp = y / a;
	elseif (z <= -4.5e+29)
		tmp = t_1;
	elseif (z <= -0.054)
		tmp = y / a;
	elseif (z <= 68000000000.0)
		tmp = x / t;
	elseif (z <= 7.2e+137)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-y) * N[(z / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.8e+92], N[(y / a), $MachinePrecision], If[LessEqual[z, -4.5e+29], t$95$1, If[LessEqual[z, -0.054], N[(y / a), $MachinePrecision], If[LessEqual[z, 68000000000.0], N[(x / t), $MachinePrecision], If[LessEqual[z, 7.2e+137], t$95$1, N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.80000000000000009e92 or -4.5000000000000002e29 < z < -0.0539999999999999994 or 7.1999999999999999e137 < z

   1. Initial program 68.2%

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

      1. *-commutative 68.2%

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

   3. Simplified 68.2%

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

   5. Taylor expanded in z around inf 60.7%

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

   if -4.80000000000000009e92 < z < -4.5000000000000002e29 or 6.8e10 < z < 7.1999999999999999e137

   1. Initial program 77.0%

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

      1. *-commutative 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in x around 0 57.3%

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

      1. mul-1-neg 57.3%

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

      2. associate-/l* 65.8%

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

      3. distribute-rgt-neg-in 65.8%

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

      4. distribute-neg-frac2 65.8%

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

      5. cancel-sign-sub-inv 65.8%

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

      6. *-commutative 65.8%

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

      7. +-commutative 65.8%

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

      8. distribute-rgt-neg-out 65.8%

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

      9. distribute-lft-neg-in 65.8%

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

      10. *-commutative 65.8%

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

      11. fma-undefine 65.8%

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

      12. neg-sub0 65.8%

          \[\leadsto y \cdot \frac{z}{\color{blue}{0 - \mathsf{fma}\left(a, -z, t\right)}} \]

      13. fma-undefine 65.8%

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

      14. distribute-rgt-neg-in 65.8%

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

      15. mul-1-neg 65.8%

          \[\leadsto y \cdot \frac{z}{0 - \left(\color{blue}{-1 \cdot \left(a \cdot z\right)} + t\right)} \]

      16. associate-*r* 65.8%

          \[\leadsto y \cdot \frac{z}{0 - \left(\color{blue}{\left(-1 \cdot a\right) \cdot z} + t\right)} \]

      17. neg-mul-1 65.8%

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

      18. *-commutative 65.8%

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

      19. associate--r+ 65.8%

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

      20. neg-sub0 65.8%

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

      21. distribute-rgt-neg-out 65.8%

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

      22. remove-double-neg 65.8%

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

   7. Simplified 65.8%

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

   8. Taylor expanded in z around 0 57.3%

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

      1. associate-*r/ 57.3%

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

      2. mul-1-neg 57.3%

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

   10. Simplified 57.3%

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

   if -0.0539999999999999994 < z < 6.8e10

   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. Add Preprocessing

   5. Taylor expanded in z around 0 63.0%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+92}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -4.5 \cdot 10^{+29}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq -0.054:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq 68000000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+137}:\\ \;\;\;\;\left(-y\right) \cdot \frac{z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - \frac{t}{z}}\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-184}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 2000000000:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ y (- a (/ t z)))))
   (if (<= z -8.2e-14)
     t_1
     (if (<= z -4.3e-184)
       (/ x (- t (* z a)))
       (if (<= z 2000000000.0) (/ (- x (* z y)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y / (a - (t / z));
	double tmp;
	if (z <= -8.2e-14) {
		tmp = t_1;
	} else if (z <= -4.3e-184) {
		tmp = x / (t - (z * a));
	} else if (z <= 2000000000.0) {
		tmp = (x - (z * y)) / t;
	} 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 / (a - (t / z))
    if (z <= (-8.2d-14)) then
        tmp = t_1
    else if (z <= (-4.3d-184)) then
        tmp = x / (t - (z * a))
    else if (z <= 2000000000.0d0) then
        tmp = (x - (z * y)) / t
    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 / (a - (t / z));
	double tmp;
	if (z <= -8.2e-14) {
		tmp = t_1;
	} else if (z <= -4.3e-184) {
		tmp = x / (t - (z * a));
	} else if (z <= 2000000000.0) {
		tmp = (x - (z * y)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y / (a - (t / z))
	tmp = 0
	if z <= -8.2e-14:
		tmp = t_1
	elif z <= -4.3e-184:
		tmp = x / (t - (z * a))
	elif z <= 2000000000.0:
		tmp = (x - (z * y)) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y / Float64(a - Float64(t / z)))
	tmp = 0.0
	if (z <= -8.2e-14)
		tmp = t_1;
	elseif (z <= -4.3e-184)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 2000000000.0)
		tmp = Float64(Float64(x - Float64(z * y)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y / (a - (t / z));
	tmp = 0.0;
	if (z <= -8.2e-14)
		tmp = t_1;
	elseif (z <= -4.3e-184)
		tmp = x / (t - (z * a));
	elseif (z <= 2000000000.0)
		tmp = (x - (z * y)) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y / N[(a - N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.2e-14], t$95$1, If[LessEqual[z, -4.3e-184], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2000000000.0], N[(N[(x - N[(z * y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.2000000000000004e-14 or 2e9 < z

   1. Initial program 71.4%

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

      1. *-commutative 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in x around inf 65.1%

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

   7. Simplified 73.4%

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

   8. Taylor expanded in z around inf 73.4%

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

      1. associate-*r/ 73.4%

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

      2. neg-mul-1 73.4%

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

   10. Simplified 73.4%

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

   11. Taylor expanded in y around 0 87.3%

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

       1. associate-/r* 86.1%

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

       2. mul-1-neg 86.1%

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

       3. distribute-frac-neg 86.1%

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

       4. distribute-frac-neg 86.1%

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

       5. unsub-neg 86.1%

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

   13. Simplified 86.1%

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

   14. Taylor expanded in x around 0 77.4%

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

   if -8.2000000000000004e-14 < z < -4.30000000000000007e-184

   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. Add Preprocessing

   5. Taylor expanded in x around inf 80.9%

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

      1. *-commutative 80.9%

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

   7. Simplified 80.9%

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

   if -4.30000000000000007e-184 < z < 2e9

   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. Add Preprocessing

   5. Taylor expanded in t around inf 85.8%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-184}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 2000000000:\\ \;\;\;\;\frac{x - z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+134} \lor \neg \left(z \leq 1.05 \cdot 10^{+121}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.7e+134) (not (<= z 1.05e+121)))
   (/ y (- a (/ t z)))
   (/ (- x (* z y)) (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.7e+134) || !(z <= 1.05e+121)) {
		tmp = y / (a - (t / z));
	} 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 <= (-3.7d+134)) .or. (.not. (z <= 1.05d+121))) then
        tmp = y / (a - (t / z))
    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 <= -3.7e+134) || !(z <= 1.05e+121)) {
		tmp = y / (a - (t / z));
	} else {
		tmp = (x - (z * y)) / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.7e+134) or not (z <= 1.05e+121):
		tmp = y / (a - (t / z))
	else:
		tmp = (x - (z * y)) / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.7e+134) || !(z <= 1.05e+121))
		tmp = Float64(y / Float64(a - Float64(t / z)));
	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 <= -3.7e+134) || ~((z <= 1.05e+121)))
		tmp = y / (a - (t / z));
	else
		tmp = (x - (z * y)) / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3.7e+134], N[Not[LessEqual[z, 1.05e+121]], $MachinePrecision]], N[(y / N[(a - N[(t / z), $MachinePrecision]), $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 < -3.70000000000000013e134 or 1.0500000000000001e121 < z

   1. Initial program 59.6%

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

      1. *-commutative 59.6%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in x around inf 53.8%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in z around inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. neg-mul-1 67.1%

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

   10. Simplified 67.1%

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

   11. Taylor expanded in y around 0 85.2%

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

       1. associate-/r* 86.4%

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

       2. mul-1-neg 86.4%

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

       3. distribute-frac-neg 86.4%

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

       4. distribute-frac-neg 86.4%

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

       5. unsub-neg 86.4%

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

   13. Simplified 86.4%

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

   14. Taylor expanded in x around 0 85.3%

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

   if -3.70000000000000013e134 < z < 1.0500000000000001e121

   1. Initial program 96.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+134} \lor \neg \left(z \leq 1.05 \cdot 10^{+121}\right):\\ \;\;\;\;\frac{y}{a - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - z \cdot y}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.055 \lor \neg \left(z \leq 37000000000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -0.055) (not (<= z 37000000000.0))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -0.055) || !(z <= 37000000000.0)) {
		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 <= (-0.055d0)) .or. (.not. (z <= 37000000000.0d0))) 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 <= -0.055) || !(z <= 37000000000.0)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -0.055) or not (z <= 37000000000.0):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -0.055) || !(z <= 37000000000.0))
		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 <= -0.055) || ~((z <= 37000000000.0)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -0.055], N[Not[LessEqual[z, 37000000000.0]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0550000000000000003 or 3.7e10 < z

   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. Add Preprocessing

   5. Taylor expanded in z around inf 51.8%

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

   if -0.0550000000000000003 < z < 3.7e10

   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. Add Preprocessing

   5. Taylor expanded in z around 0 63.0%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.055 \lor \neg \left(z \leq 37000000000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 84.9%

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

   1. *-commutative 84.9%

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

3. Simplified 84.9%

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

5. Taylor expanded in z around 0 40.3%

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

Developer target: 97.4% 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 2024085 
(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))))