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

Percentage Accurate: 85.1% → 99.6%

Time: 18.6s

Alternatives: 15

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: 85.1% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} - z\\ t_2 := x - y \cdot z\\ t_3 := t - z \cdot a\\ t_4 := \frac{t\_2}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{y}{t\_3} \cdot t\_1\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{t\_2}{a} \cdot \frac{-1}{z - \frac{t}{a}}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{t\_1}{\frac{t\_3}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (/ x y) z))
        (t_2 (- x (* y z)))
        (t_3 (- t (* z a)))
        (t_4 (/ t_2 t_3)))
   (if (<= t_4 (- INFINITY))
     (* (/ y t_3) t_1)
     (if (<= t_4 -2e-308)
       t_4
       (if (<= t_4 0.0)
         (* (/ t_2 a) (/ -1.0 (- z (/ t a))))
         (if (<= t_4 2e+306)
           t_4
           (if (<= t_4 INFINITY) (/ t_1 (/ t_3 y)) (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x / y) - z;
	double t_2 = x - (y * z);
	double t_3 = t - (z * a);
	double t_4 = t_2 / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (y / t_3) * t_1;
	} else if (t_4 <= -2e-308) {
		tmp = t_4;
	} else if (t_4 <= 0.0) {
		tmp = (t_2 / a) * (-1.0 / (z - (t / a)));
	} else if (t_4 <= 2e+306) {
		tmp = t_4;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1 / (t_3 / y);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x / y) - z;
	double t_2 = x - (y * z);
	double t_3 = t - (z * a);
	double t_4 = t_2 / t_3;
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = (y / t_3) * t_1;
	} else if (t_4 <= -2e-308) {
		tmp = t_4;
	} else if (t_4 <= 0.0) {
		tmp = (t_2 / a) * (-1.0 / (z - (t / a)));
	} else if (t_4 <= 2e+306) {
		tmp = t_4;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 / (t_3 / y);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x / y) - z
	t_2 = x - (y * z)
	t_3 = t - (z * a)
	t_4 = t_2 / t_3
	tmp = 0
	if t_4 <= -math.inf:
		tmp = (y / t_3) * t_1
	elif t_4 <= -2e-308:
		tmp = t_4
	elif t_4 <= 0.0:
		tmp = (t_2 / a) * (-1.0 / (z - (t / a)))
	elif t_4 <= 2e+306:
		tmp = t_4
	elif t_4 <= math.inf:
		tmp = t_1 / (t_3 / y)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x / y) - z)
	t_2 = Float64(x - Float64(y * z))
	t_3 = Float64(t - Float64(z * a))
	t_4 = Float64(t_2 / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(y / t_3) * t_1);
	elseif (t_4 <= -2e-308)
		tmp = t_4;
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(t_2 / a) * Float64(-1.0 / Float64(z - Float64(t / a))));
	elseif (t_4 <= 2e+306)
		tmp = t_4;
	elseif (t_4 <= Inf)
		tmp = Float64(t_1 / Float64(t_3 / y));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x / y) - z;
	t_2 = x - (y * z);
	t_3 = t - (z * a);
	t_4 = t_2 / t_3;
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = (y / t_3) * t_1;
	elseif (t_4 <= -2e-308)
		tmp = t_4;
	elseif (t_4 <= 0.0)
		tmp = (t_2 / a) * (-1.0 / (z - (t / a)));
	elseif (t_4 <= 2e+306)
		tmp = t_4;
	elseif (t_4 <= Inf)
		tmp = t_1 / (t_3 / y);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(y / t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$4, -2e-308], t$95$4, If[LessEqual[t$95$4, 0.0], N[(N[(t$95$2 / a), $MachinePrecision] * N[(-1.0 / N[(z - N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+306], t$95$4, If[LessEqual[t$95$4, Infinity], N[(t$95$1 / N[(t$95$3 / y), $MachinePrecision]), $MachinePrecision], N[(y / a), $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 44.3%

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

      1. *-commutative 44.3%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in y around inf 44.3%

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

      1. div-inv 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.9999999999999998e-308 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.00000000000000003e306

   1. Initial program 99.7%

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

   if -1.9999999999999998e-308 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in a around inf 71.9%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 2.00000000000000003e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in y around inf 67.2%

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

      1. div-inv 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

   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 5 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{t - z \cdot a} \cdot \left(\frac{x}{y} - z\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{x - y \cdot z}{a} \cdot \frac{-1}{z - \frac{t}{a}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{\frac{x}{y} - z}{\frac{t - z \cdot a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} - z\\ t_2 := x - y \cdot z\\ t_3 := t - z \cdot a\\ t_4 := \frac{t\_2}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{y}{t\_3} \cdot t\_1\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{1}{a} \cdot \frac{t\_2}{\frac{t}{a} - z}\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{t\_1}{\frac{t\_3}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (/ x y) z))
        (t_2 (- x (* y z)))
        (t_3 (- t (* z a)))
        (t_4 (/ t_2 t_3)))
   (if (<= t_4 (- INFINITY))
     (* (/ y t_3) t_1)
     (if (<= t_4 -2e-308)
       t_4
       (if (<= t_4 0.0)
         (* (/ 1.0 a) (/ t_2 (- (/ t a) z)))
         (if (<= t_4 2e+306)
           t_4
           (if (<= t_4 INFINITY) (/ t_1 (/ t_3 y)) (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x / y) - z;
	double t_2 = x - (y * z);
	double t_3 = t - (z * a);
	double t_4 = t_2 / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (y / t_3) * t_1;
	} else if (t_4 <= -2e-308) {
		tmp = t_4;
	} else if (t_4 <= 0.0) {
		tmp = (1.0 / a) * (t_2 / ((t / a) - z));
	} else if (t_4 <= 2e+306) {
		tmp = t_4;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_1 / (t_3 / y);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x / y) - z;
	double t_2 = x - (y * z);
	double t_3 = t - (z * a);
	double t_4 = t_2 / t_3;
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = (y / t_3) * t_1;
	} else if (t_4 <= -2e-308) {
		tmp = t_4;
	} else if (t_4 <= 0.0) {
		tmp = (1.0 / a) * (t_2 / ((t / a) - z));
	} else if (t_4 <= 2e+306) {
		tmp = t_4;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 / (t_3 / y);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x / y) - z
	t_2 = x - (y * z)
	t_3 = t - (z * a)
	t_4 = t_2 / t_3
	tmp = 0
	if t_4 <= -math.inf:
		tmp = (y / t_3) * t_1
	elif t_4 <= -2e-308:
		tmp = t_4
	elif t_4 <= 0.0:
		tmp = (1.0 / a) * (t_2 / ((t / a) - z))
	elif t_4 <= 2e+306:
		tmp = t_4
	elif t_4 <= math.inf:
		tmp = t_1 / (t_3 / y)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x / y) - z)
	t_2 = Float64(x - Float64(y * z))
	t_3 = Float64(t - Float64(z * a))
	t_4 = Float64(t_2 / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(y / t_3) * t_1);
	elseif (t_4 <= -2e-308)
		tmp = t_4;
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(1.0 / a) * Float64(t_2 / Float64(Float64(t / a) - z)));
	elseif (t_4 <= 2e+306)
		tmp = t_4;
	elseif (t_4 <= Inf)
		tmp = Float64(t_1 / Float64(t_3 / y));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x / y) - z;
	t_2 = x - (y * z);
	t_3 = t - (z * a);
	t_4 = t_2 / t_3;
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = (y / t_3) * t_1;
	elseif (t_4 <= -2e-308)
		tmp = t_4;
	elseif (t_4 <= 0.0)
		tmp = (1.0 / a) * (t_2 / ((t / a) - z));
	elseif (t_4 <= 2e+306)
		tmp = t_4;
	elseif (t_4 <= Inf)
		tmp = t_1 / (t_3 / y);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] - z), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(y / t$95$3), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$4, -2e-308], t$95$4, If[LessEqual[t$95$4, 0.0], N[(N[(1.0 / a), $MachinePrecision] * N[(t$95$2 / N[(N[(t / a), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2e+306], t$95$4, If[LessEqual[t$95$4, Infinity], N[(t$95$1 / N[(t$95$3 / y), $MachinePrecision]), $MachinePrecision], N[(y / a), $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 44.3%

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

      1. *-commutative 44.3%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in y around inf 44.3%

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

      1. div-inv 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.9999999999999998e-308 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.00000000000000003e306

   1. Initial program 99.7%

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

   if -1.9999999999999998e-308 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in a around inf 71.9%

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

      1. *-un-lft-identity 71.9%

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

      2. times-frac 99.8%

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

   7. Applied egg-rr 99.8%

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

   if 2.00000000000000003e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in y around inf 67.2%

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

      1. div-inv 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

   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 5 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{t - z \cdot a} \cdot \left(\frac{x}{y} - z\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{1}{a} \cdot \frac{x - y \cdot z}{\frac{t}{a} - z}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{\frac{x}{y} - z}{\frac{t - z \cdot a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} - z\\ t_2 := t - z \cdot a\\ t_3 := \frac{x - y \cdot z}{t\_2}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{y}{t\_2} \cdot t\_1\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;y \cdot \frac{\frac{z}{a}}{z - \frac{t}{a}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{t\_1}{\frac{t\_2}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (/ x y) z)) (t_2 (- t (* z a))) (t_3 (/ (- x (* y z)) t_2)))
   (if (<= t_3 (- INFINITY))
     (* (/ y t_2) t_1)
     (if (<= t_3 -4e-311)
       t_3
       (if (<= t_3 0.0)
         (* y (/ (/ z a) (- z (/ t a))))
         (if (<= t_3 2e+306)
           t_3
           (if (<= t_3 INFINITY) (/ t_1 (/ t_2 y)) (/ y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x / y) - z;
	double t_2 = t - (z * a);
	double t_3 = (x - (y * z)) / t_2;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (y / t_2) * t_1;
	} else if (t_3 <= -4e-311) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = y * ((z / a) / (z - (t / a)));
	} else if (t_3 <= 2e+306) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1 / (t_2 / y);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x / y) - z;
	double t_2 = t - (z * a);
	double t_3 = (x - (y * z)) / t_2;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (y / t_2) * t_1;
	} else if (t_3 <= -4e-311) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = y * ((z / a) / (z - (t / a)));
	} else if (t_3 <= 2e+306) {
		tmp = t_3;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1 / (t_2 / y);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x / y) - z
	t_2 = t - (z * a)
	t_3 = (x - (y * z)) / t_2
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (y / t_2) * t_1
	elif t_3 <= -4e-311:
		tmp = t_3
	elif t_3 <= 0.0:
		tmp = y * ((z / a) / (z - (t / a)))
	elif t_3 <= 2e+306:
		tmp = t_3
	elif t_3 <= math.inf:
		tmp = t_1 / (t_2 / y)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x / y) - z)
	t_2 = Float64(t - Float64(z * a))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_2)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(y / t_2) * t_1);
	elseif (t_3 <= -4e-311)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = Float64(y * Float64(Float64(z / a) / Float64(z - Float64(t / a))));
	elseif (t_3 <= 2e+306)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = Float64(t_1 / Float64(t_2 / y));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x / y) - z;
	t_2 = t - (z * a);
	t_3 = (x - (y * z)) / t_2;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (y / t_2) * t_1;
	elseif (t_3 <= -4e-311)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = y * ((z / a) / (z - (t / a)));
	elseif (t_3 <= 2e+306)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_1 / (t_2 / y);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] - z), $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]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(y / t$95$2), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t$95$3, -4e-311], t$95$3, If[LessEqual[t$95$3, 0.0], N[(y * N[(N[(z / a), $MachinePrecision] / N[(z - N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+306], t$95$3, If[LessEqual[t$95$3, Infinity], N[(t$95$1 / N[(t$95$2 / y), $MachinePrecision]), $MachinePrecision], N[(y / a), $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 44.3%

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

      1. *-commutative 44.3%

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

   3. Simplified 44.3%

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

   5. Taylor expanded in y around inf 44.3%

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

      1. div-inv 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -3.99999999999979e-311 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.00000000000000003e306

   1. Initial program 99.7%

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

   if -3.99999999999979e-311 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 70.9%

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

      1. *-commutative 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in a around inf 70.9%

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

   6. Taylor expanded in x around 0 70.9%

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

      1. neg-mul-1 70.9%

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

      2. associate-/l* 70.9%

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

      3. distribute-rgt-neg-in 70.9%

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

      4. associate-/r* 90.3%

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

      5. distribute-neg-frac2 90.3%

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

   8. Simplified 90.3%

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

   if 2.00000000000000003e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in y around inf 67.2%

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

      1. div-inv 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

   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 5 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{t - z \cdot a} \cdot \left(\frac{x}{y} - z\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;y \cdot \frac{\frac{z}{a}}{z - \frac{t}{a}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{\frac{x}{y} - z}{\frac{t - z \cdot a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := \frac{y}{t\_1} \cdot \left(\frac{x}{y} - z\right)\\ t_3 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq -4 \cdot 10^{-311}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;y \cdot \frac{\frac{z}{a}}{z - \frac{t}{a}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a)))
        (t_2 (* (/ y t_1) (- (/ x y) z)))
        (t_3 (/ (- x (* y z)) t_1)))
   (if (<= t_3 (- INFINITY))
     t_2
     (if (<= t_3 -4e-311)
       t_3
       (if (<= t_3 0.0)
         (* y (/ (/ z a) (- z (/ t a))))
         (if (<= t_3 2e+306) 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 / t_1) * ((x / y) - z);
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_3 <= -4e-311) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = y * ((z / a) / (z - (t / a)));
	} else if (t_3 <= 2e+306) {
		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 / t_1) * ((x / y) - z);
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_3 <= -4e-311) {
		tmp = t_3;
	} else if (t_3 <= 0.0) {
		tmp = y * ((z / a) / (z - (t / a)));
	} else if (t_3 <= 2e+306) {
		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 / t_1) * ((x / y) - z)
	t_3 = (x - (y * z)) / t_1
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_2
	elif t_3 <= -4e-311:
		tmp = t_3
	elif t_3 <= 0.0:
		tmp = y * ((z / a) / (z - (t / a)))
	elif t_3 <= 2e+306:
		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(Float64(y / t_1) * Float64(Float64(x / y) - z))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_3 <= -4e-311)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = Float64(y * Float64(Float64(z / a) / Float64(z - Float64(t / a))));
	elseif (t_3 <= 2e+306)
		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 / t_1) * ((x / y) - z);
	t_3 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_2;
	elseif (t_3 <= -4e-311)
		tmp = t_3;
	elseif (t_3 <= 0.0)
		tmp = y * ((z / a) / (z - (t / a)));
	elseif (t_3 <= 2e+306)
		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[(N[(y / t$95$1), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$2, If[LessEqual[t$95$3, -4e-311], t$95$3, If[LessEqual[t$95$3, 0.0], N[(y * N[(N[(z / a), $MachinePrecision] / N[(z - N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+306], 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 2.00000000000000003e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 57.5%

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

      1. *-commutative 57.5%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in y around inf 57.5%

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

      1. div-inv 57.5%

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

      2. *-commutative 57.5%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -3.99999999999979e-311 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.00000000000000003e306

   1. Initial program 99.7%

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

   if -3.99999999999979e-311 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 70.9%

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

      1. *-commutative 70.9%

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

   3. Simplified 70.9%

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

   5. Taylor expanded in a around inf 70.9%

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

   6. Taylor expanded in x around 0 70.9%

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

      1. neg-mul-1 70.9%

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

      2. associate-/l* 70.9%

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

      3. distribute-rgt-neg-in 70.9%

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

      4. associate-/r* 90.3%

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

      5. distribute-neg-frac2 90.3%

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

   8. Simplified 90.3%

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

   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 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;\frac{y}{t - z \cdot a} \cdot \left(\frac{x}{y} - z\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -4 \cdot 10^{-311}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;y \cdot \frac{\frac{z}{a}}{z - \frac{t}{a}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{y}{t - z \cdot a} \cdot \left(\frac{x}{y} - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot a - t\\ t_2 := x - y \cdot z\\ t_3 := t - z \cdot a\\ t_4 := \frac{x}{t\_3}\\ t_5 := \frac{t\_2}{t\_3}\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;z \cdot \frac{y}{t\_1} + t\_4\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{t\_2}{a} \cdot \frac{-1}{z - \frac{t}{a}}\\ \mathbf{elif}\;t\_5 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{y \cdot z}{t\_1} + t\_4\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\frac{x}{y} - z}{\frac{t\_3}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* z a) t))
        (t_2 (- x (* y z)))
        (t_3 (- t (* z a)))
        (t_4 (/ x t_3))
        (t_5 (/ t_2 t_3)))
   (if (<= t_5 -2e-308)
     (+ (* z (/ y t_1)) t_4)
     (if (<= t_5 0.0)
       (* (/ t_2 a) (/ -1.0 (- z (/ t a))))
       (if (<= t_5 2e+306)
         (+ (/ (* y z) t_1) t_4)
         (if (<= t_5 INFINITY) (/ (- (/ x y) z) (/ t_3 y)) (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z * a) - t;
	double t_2 = x - (y * z);
	double t_3 = t - (z * a);
	double t_4 = x / t_3;
	double t_5 = t_2 / t_3;
	double tmp;
	if (t_5 <= -2e-308) {
		tmp = (z * (y / t_1)) + t_4;
	} else if (t_5 <= 0.0) {
		tmp = (t_2 / a) * (-1.0 / (z - (t / a)));
	} else if (t_5 <= 2e+306) {
		tmp = ((y * z) / t_1) + t_4;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = ((x / y) - z) / (t_3 / y);
	} else {
		tmp = y / a;
	}
	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 = x - (y * z);
	double t_3 = t - (z * a);
	double t_4 = x / t_3;
	double t_5 = t_2 / t_3;
	double tmp;
	if (t_5 <= -2e-308) {
		tmp = (z * (y / t_1)) + t_4;
	} else if (t_5 <= 0.0) {
		tmp = (t_2 / a) * (-1.0 / (z - (t / a)));
	} else if (t_5 <= 2e+306) {
		tmp = ((y * z) / t_1) + t_4;
	} else if (t_5 <= Double.POSITIVE_INFINITY) {
		tmp = ((x / y) - z) / (t_3 / y);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z * a) - t
	t_2 = x - (y * z)
	t_3 = t - (z * a)
	t_4 = x / t_3
	t_5 = t_2 / t_3
	tmp = 0
	if t_5 <= -2e-308:
		tmp = (z * (y / t_1)) + t_4
	elif t_5 <= 0.0:
		tmp = (t_2 / a) * (-1.0 / (z - (t / a)))
	elif t_5 <= 2e+306:
		tmp = ((y * z) / t_1) + t_4
	elif t_5 <= math.inf:
		tmp = ((x / y) - z) / (t_3 / y)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z * a) - t)
	t_2 = Float64(x - Float64(y * z))
	t_3 = Float64(t - Float64(z * a))
	t_4 = Float64(x / t_3)
	t_5 = Float64(t_2 / t_3)
	tmp = 0.0
	if (t_5 <= -2e-308)
		tmp = Float64(Float64(z * Float64(y / t_1)) + t_4);
	elseif (t_5 <= 0.0)
		tmp = Float64(Float64(t_2 / a) * Float64(-1.0 / Float64(z - Float64(t / a))));
	elseif (t_5 <= 2e+306)
		tmp = Float64(Float64(Float64(y * z) / t_1) + t_4);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(Float64(x / y) - z) / Float64(t_3 / y));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z * a) - t;
	t_2 = x - (y * z);
	t_3 = t - (z * a);
	t_4 = x / t_3;
	t_5 = t_2 / t_3;
	tmp = 0.0;
	if (t_5 <= -2e-308)
		tmp = (z * (y / t_1)) + t_4;
	elseif (t_5 <= 0.0)
		tmp = (t_2 / a) * (-1.0 / (z - (t / a)));
	elseif (t_5 <= 2e+306)
		tmp = ((y * z) / t_1) + t_4;
	elseif (t_5 <= Inf)
		tmp = ((x / y) - z) / (t_3 / y);
	else
		tmp = y / a;
	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[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(x / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$5, -2e-308], N[(N[(z * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[(t$95$2 / a), $MachinePrecision] * N[(-1.0 / N[(z - N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 2e+306], N[(N[(N[(y * z), $MachinePrecision] / t$95$1), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(N[(x / y), $MachinePrecision] - z), $MachinePrecision] / N[(t$95$3 / y), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.9999999999999998e-308

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in x around 0 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-/l* 97.8%

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

      3. *-commutative 97.8%

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

   7. Applied egg-rr 97.8%

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

   if -1.9999999999999998e-308 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in a around inf 71.9%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.00000000000000003e306

   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 2.00000000000000003e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in y around inf 67.2%

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

      1. div-inv 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

   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 5 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-308}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t} + \frac{x}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{x - y \cdot z}{a} \cdot \frac{-1}{z - \frac{t}{a}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{y \cdot z}{z \cdot a - t} + \frac{x}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{\frac{x}{y} - z}{\frac{t - z \cdot a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := t - z \cdot a\\ t_3 := \frac{t\_1}{t\_2}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t} + \frac{x}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{t\_1}{a} \cdot \frac{-1}{z - \frac{t}{a}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\frac{x}{y} - z}{\frac{t\_2}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z))) (t_2 (- t (* z a))) (t_3 (/ t_1 t_2)))
   (if (<= t_3 -2e-308)
     (+ (* z (/ y (- (* z a) t))) (/ x t_2))
     (if (<= t_3 0.0)
       (* (/ t_1 a) (/ -1.0 (- z (/ t a))))
       (if (<= t_3 2e+306)
         t_3
         (if (<= t_3 INFINITY) (/ (- (/ x y) z) (/ t_2 y)) (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (z * a);
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -2e-308) {
		tmp = (z * (y / ((z * a) - t))) + (x / t_2);
	} else if (t_3 <= 0.0) {
		tmp = (t_1 / a) * (-1.0 / (z - (t / a)));
	} else if (t_3 <= 2e+306) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = ((x / y) - z) / (t_2 / y);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (z * a);
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -2e-308) {
		tmp = (z * (y / ((z * a) - t))) + (x / t_2);
	} else if (t_3 <= 0.0) {
		tmp = (t_1 / a) * (-1.0 / (z - (t / a)));
	} else if (t_3 <= 2e+306) {
		tmp = t_3;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = ((x / y) - z) / (t_2 / y);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - (y * z)
	t_2 = t - (z * a)
	t_3 = t_1 / t_2
	tmp = 0
	if t_3 <= -2e-308:
		tmp = (z * (y / ((z * a) - t))) + (x / t_2)
	elif t_3 <= 0.0:
		tmp = (t_1 / a) * (-1.0 / (z - (t / a)))
	elif t_3 <= 2e+306:
		tmp = t_3
	elif t_3 <= math.inf:
		tmp = ((x / y) - z) / (t_2 / y)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(t - Float64(z * a))
	t_3 = Float64(t_1 / t_2)
	tmp = 0.0
	if (t_3 <= -2e-308)
		tmp = Float64(Float64(z * Float64(y / Float64(Float64(z * a) - t))) + Float64(x / t_2));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(t_1 / a) * Float64(-1.0 / Float64(z - Float64(t / a))));
	elseif (t_3 <= 2e+306)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(Float64(x / y) - z) / Float64(t_2 / y));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - (y * z);
	t_2 = t - (z * a);
	t_3 = t_1 / t_2;
	tmp = 0.0;
	if (t_3 <= -2e-308)
		tmp = (z * (y / ((z * a) - t))) + (x / t_2);
	elseif (t_3 <= 0.0)
		tmp = (t_1 / a) * (-1.0 / (z - (t / a)));
	elseif (t_3 <= 2e+306)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = ((x / y) - z) / (t_2 / y);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-308], N[(N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(t$95$1 / a), $MachinePrecision] * N[(-1.0 / N[(z - N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+306], t$95$3, If[LessEqual[t$95$3, Infinity], N[(N[(N[(x / y), $MachinePrecision] - z), $MachinePrecision] / N[(t$95$2 / y), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.9999999999999998e-308

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in x around 0 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-/l* 97.8%

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

      3. *-commutative 97.8%

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

   7. Applied egg-rr 97.8%

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

   if -1.9999999999999998e-308 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in a around inf 71.9%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.00000000000000003e306

   1. Initial program 99.7%

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

   if 2.00000000000000003e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in y around inf 67.2%

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

      1. div-inv 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

   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 5 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-308}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t} + \frac{x}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{x - y \cdot z}{a} \cdot \frac{-1}{z - \frac{t}{a}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{\frac{x}{y} - z}{\frac{t - z \cdot a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - y \cdot z\\ t_2 := t - z \cdot a\\ t_3 := \frac{t\_1}{t\_2}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{x}{t\_2} + \frac{z}{\frac{z \cdot a - t}{y}}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{t\_1}{a} \cdot \frac{-1}{z - \frac{t}{a}}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\frac{x}{y} - z}{\frac{t\_2}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* y z))) (t_2 (- t (* z a))) (t_3 (/ t_1 t_2)))
   (if (<= t_3 -2e-308)
     (+ (/ x t_2) (/ z (/ (- (* z a) t) y)))
     (if (<= t_3 0.0)
       (* (/ t_1 a) (/ -1.0 (- z (/ t a))))
       (if (<= t_3 2e+306)
         t_3
         (if (<= t_3 INFINITY) (/ (- (/ x y) z) (/ t_2 y)) (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (z * a);
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -2e-308) {
		tmp = (x / t_2) + (z / (((z * a) - t) / y));
	} else if (t_3 <= 0.0) {
		tmp = (t_1 / a) * (-1.0 / (z - (t / a)));
	} else if (t_3 <= 2e+306) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = ((x / y) - z) / (t_2 / y);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (y * z);
	double t_2 = t - (z * a);
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -2e-308) {
		tmp = (x / t_2) + (z / (((z * a) - t) / y));
	} else if (t_3 <= 0.0) {
		tmp = (t_1 / a) * (-1.0 / (z - (t / a)));
	} else if (t_3 <= 2e+306) {
		tmp = t_3;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = ((x / y) - z) / (t_2 / y);
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x - (y * z)
	t_2 = t - (z * a)
	t_3 = t_1 / t_2
	tmp = 0
	if t_3 <= -2e-308:
		tmp = (x / t_2) + (z / (((z * a) - t) / y))
	elif t_3 <= 0.0:
		tmp = (t_1 / a) * (-1.0 / (z - (t / a)))
	elif t_3 <= 2e+306:
		tmp = t_3
	elif t_3 <= math.inf:
		tmp = ((x / y) - z) / (t_2 / y)
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(y * z))
	t_2 = Float64(t - Float64(z * a))
	t_3 = Float64(t_1 / t_2)
	tmp = 0.0
	if (t_3 <= -2e-308)
		tmp = Float64(Float64(x / t_2) + Float64(z / Float64(Float64(Float64(z * a) - t) / y)));
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(t_1 / a) * Float64(-1.0 / Float64(z - Float64(t / a))));
	elseif (t_3 <= 2e+306)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(Float64(x / y) - z) / Float64(t_2 / y));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x - (y * z);
	t_2 = t - (z * a);
	t_3 = t_1 / t_2;
	tmp = 0.0;
	if (t_3 <= -2e-308)
		tmp = (x / t_2) + (z / (((z * a) - t) / y));
	elseif (t_3 <= 0.0)
		tmp = (t_1 / a) * (-1.0 / (z - (t / a)));
	elseif (t_3 <= 2e+306)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = ((x / y) - z) / (t_2 / y);
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-308], N[(N[(x / t$95$2), $MachinePrecision] + N[(z / N[(N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(t$95$1 / a), $MachinePrecision] * N[(-1.0 / N[(z - N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+306], t$95$3, If[LessEqual[t$95$3, Infinity], N[(N[(N[(x / y), $MachinePrecision] - z), $MachinePrecision] / N[(t$95$2 / y), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -1.9999999999999998e-308

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in x around 0 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-/l* 97.8%

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

      3. *-commutative 97.8%

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

   7. Applied egg-rr 97.8%

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

      1. +-commutative 97.8%

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

      2. mul-1-neg 97.8%

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

      3. unsub-neg 97.8%

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

      4. *-commutative 97.8%

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

      5. clear-num 97.7%

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

      6. un-div-inv 97.8%

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

   9. Applied egg-rr 97.8%

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

   if -1.9999999999999998e-308 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 71.9%

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

      1. *-commutative 71.9%

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

   3. Simplified 71.9%

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

   5. Taylor expanded in a around inf 71.9%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.9%

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

   7. Applied egg-rr 99.9%

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

   if 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 2.00000000000000003e306

   1. Initial program 99.7%

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

   if 2.00000000000000003e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in y around inf 67.2%

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

      1. div-inv 67.2%

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

      2. *-commutative 67.2%

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

      3. associate-*l* 99.7%

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

      4. div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. un-div-inv 99.9%

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

   9. Applied egg-rr 99.9%

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

   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 5 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-308}:\\ \;\;\;\;\frac{x}{t - z \cdot a} + \frac{z}{\frac{z \cdot a - t}{y}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{x - y \cdot z}{a} \cdot \frac{-1}{z - \frac{t}{a}}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 2 \cdot 10^{+306}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{\frac{x}{y} - z}{\frac{t - z \cdot a}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.36 \cdot 10^{+101}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+117}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{x}{z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.36e+101)
   (/ (- y (/ x z)) a)
   (if (<= z 1.75e+117)
     (/ (- x (* y z)) (- t (* z a)))
     (- (/ y a) (/ x (* z a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.36e+101) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1.75e+117) {
		tmp = (x - (y * z)) / (t - (z * a));
	} else {
		tmp = (y / a) - (x / (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 <= (-1.36d+101)) then
        tmp = (y - (x / z)) / a
    else if (z <= 1.75d+117) then
        tmp = (x - (y * z)) / (t - (z * a))
    else
        tmp = (y / a) - (x / (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 <= -1.36e+101) {
		tmp = (y - (x / z)) / a;
	} else if (z <= 1.75e+117) {
		tmp = (x - (y * z)) / (t - (z * a));
	} else {
		tmp = (y / a) - (x / (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1.36e+101:
		tmp = (y - (x / z)) / a
	elif z <= 1.75e+117:
		tmp = (x - (y * z)) / (t - (z * a))
	else:
		tmp = (y / a) - (x / (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.36e+101)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	elseif (z <= 1.75e+117)
		tmp = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)));
	else
		tmp = Float64(Float64(y / a) - Float64(x / Float64(z * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1.36e+101)
		tmp = (y - (x / z)) / a;
	elseif (z <= 1.75e+117)
		tmp = (x - (y * z)) / (t - (z * a));
	else
		tmp = (y / a) - (x / (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.36e+101], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[z, 1.75e+117], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / a), $MachinePrecision] - N[(x / N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35999999999999998e101

   1. Initial program 55.8%

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

      1. *-commutative 55.8%

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

   3. Simplified 55.8%

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

   5. Taylor expanded in x around 0 55.8%

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

      1. *-commutative 55.8%

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

      2. associate-/l* 74.1%

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

      3. *-commutative 74.1%

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

   7. Applied egg-rr 74.1%

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

   8. Taylor expanded in a around inf 79.8%

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

      1. mul-1-neg 79.8%

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

      2. unsub-neg 79.8%

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

   10. Simplified 79.8%

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

   if -1.35999999999999998e101 < z < 1.74999999999999991e117

   1. Initial program 95.7%

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

   if 1.74999999999999991e117 < z

   1. Initial program 50.0%

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

      1. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in x around 0 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-/l* 63.5%

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

      3. *-commutative 63.5%

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

   7. Applied egg-rr 63.5%

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

   8. Taylor expanded in t around 0 79.9%

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

      1. +-commutative 79.9%

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

      2. mul-1-neg 79.9%

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

      3. unsub-neg 79.9%

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

      4. *-commutative 79.9%

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

   10. Simplified 79.9%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.36 \cdot 10^{+101}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+117}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{x}{z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-48}:\\ \;\;\;\;\frac{x}{t} - y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 0.105:\\ \;\;\;\;\frac{y}{a} - \frac{x}{z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8e-48)
   (- (/ x t) (* y (/ z t)))
   (if (<= t 0.105) (- (/ y a) (/ x (* z a))) (/ (- x (* y z)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8e-48) {
		tmp = (x / t) - (y * (z / t));
	} else if (t <= 0.105) {
		tmp = (y / a) - (x / (z * a));
	} else {
		tmp = (x - (y * z)) / 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 (t <= (-8d-48)) then
        tmp = (x / t) - (y * (z / t))
    else if (t <= 0.105d0) then
        tmp = (y / a) - (x / (z * a))
    else
        tmp = (x - (y * z)) / 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 (t <= -8e-48) {
		tmp = (x / t) - (y * (z / t));
	} else if (t <= 0.105) {
		tmp = (y / a) - (x / (z * a));
	} else {
		tmp = (x - (y * z)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -8e-48:
		tmp = (x / t) - (y * (z / t))
	elif t <= 0.105:
		tmp = (y / a) - (x / (z * a))
	else:
		tmp = (x - (y * z)) / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8e-48)
		tmp = Float64(Float64(x / t) - Float64(y * Float64(z / t)));
	elseif (t <= 0.105)
		tmp = Float64(Float64(y / a) - Float64(x / Float64(z * a)));
	else
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -8e-48)
		tmp = (x / t) - (y * (z / t));
	elseif (t <= 0.105)
		tmp = (y / a) - (x / (z * a));
	else
		tmp = (x - (y * z)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8e-48], N[(N[(x / t), $MachinePrecision] - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.105], N[(N[(y / a), $MachinePrecision] - N[(x / N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.9999999999999998e-48

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in x around 0 83.6%

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

      1. *-commutative 83.6%

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

      2. associate-/l* 90.0%

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

      3. *-commutative 90.0%

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

   7. Applied egg-rr 90.0%

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

   8. Taylor expanded in a around 0 71.6%

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

      1. +-commutative 71.6%

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

      2. mul-1-neg 71.6%

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

      3. unsub-neg 71.6%

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

      4. associate-/l* 75.3%

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

   10. Simplified 75.3%

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

   if -7.9999999999999998e-48 < t < 0.104999999999999996

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

   3. Simplified 82.0%

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

   5. Taylor expanded in x around 0 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-/l* 84.9%

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

      3. *-commutative 84.9%

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

   7. Applied egg-rr 84.9%

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

   8. Taylor expanded in t around 0 74.1%

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

      1. +-commutative 74.1%

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

      2. mul-1-neg 74.1%

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

      3. unsub-neg 74.1%

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

      4. *-commutative 74.1%

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

   10. Simplified 74.1%

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

   if 0.104999999999999996 < t

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in t around inf 77.0%

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

Alternative 10: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-26} \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -1.85e-26) (not (<= t 0.23)))
   (/ (- x (* y z)) t)
   (/ (- y (/ x z)) a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -1.85e-26) || !(t <= 0.23)) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = (y - (x / 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 ((t <= (-1.85d-26)) .or. (.not. (t <= 0.23d0))) then
        tmp = (x - (y * z)) / t
    else
        tmp = (y - (x / 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 ((t <= -1.85e-26) || !(t <= 0.23)) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = (y - (x / z)) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -1.85e-26) or not (t <= 0.23):
		tmp = (x - (y * z)) / t
	else:
		tmp = (y - (x / z)) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -1.85e-26) || !(t <= 0.23))
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -1.85e-26) || ~((t <= 0.23)))
		tmp = (x - (y * z)) / t;
	else
		tmp = (y - (x / z)) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -1.85e-26], N[Not[LessEqual[t, 0.23]], $MachinePrecision]], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8499999999999999e-26 or 0.23000000000000001 < t

   1. Initial program 86.9%

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

      1. *-commutative 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in t around inf 74.8%

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

   if -1.8499999999999999e-26 < t < 0.23000000000000001

   1. Initial program 81.7%

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

      1. *-commutative 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in x around 0 81.7%

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

      1. *-commutative 81.7%

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

      2. associate-/l* 84.5%

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

      3. *-commutative 84.5%

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

   7. Applied egg-rr 84.5%

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

   8. Taylor expanded in a around inf 71.5%

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

      1. mul-1-neg 71.5%

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

      2. unsub-neg 71.5%

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

   10. Simplified 71.5%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{-26} \lor \neg \left(t \leq 0.23\right):\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+101} \lor \neg \left(z \leq 1.95 \cdot 10^{+90}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.4e+101) (not (<= z 1.95e+90))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.4e+101) || !(z <= 1.95e+90)) {
		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 <= (-1.4d+101)) .or. (.not. (z <= 1.95d+90))) 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 <= -1.4e+101) || !(z <= 1.95e+90)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.4e+101) or not (z <= 1.95e+90):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.4e+101) || !(z <= 1.95e+90))
		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 <= -1.4e+101) || ~((z <= 1.95e+90)))
		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, -1.4e+101], N[Not[LessEqual[z, 1.95e+90]], $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 < -1.39999999999999991e101 or 1.9500000000000001e90 < z

   1. Initial program 54.4%

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

      1. *-commutative 54.4%

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

   3. Simplified 54.4%

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

   5. Taylor expanded in z around inf 66.8%

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

   if -1.39999999999999991e101 < z < 1.9500000000000001e90

   1. Initial program 97.6%

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

      1. *-commutative 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in x around inf 68.6%

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

      1. *-commutative 68.6%

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

   7. Simplified 68.6%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+101} \lor \neg \left(z \leq 1.95 \cdot 10^{+90}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{t} - y \cdot \frac{z}{t}\\ \mathbf{elif}\;t \leq 0.7:\\ \;\;\;\;\frac{y - \frac{x}{z}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.4e-28)
   (- (/ x t) (* y (/ z t)))
   (if (<= t 0.7) (/ (- y (/ x z)) a) (/ (- x (* y z)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.4e-28) {
		tmp = (x / t) - (y * (z / t));
	} else if (t <= 0.7) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / 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 (t <= (-7.4d-28)) then
        tmp = (x / t) - (y * (z / t))
    else if (t <= 0.7d0) then
        tmp = (y - (x / z)) / a
    else
        tmp = (x - (y * z)) / 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 (t <= -7.4e-28) {
		tmp = (x / t) - (y * (z / t));
	} else if (t <= 0.7) {
		tmp = (y - (x / z)) / a;
	} else {
		tmp = (x - (y * z)) / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -7.4e-28:
		tmp = (x / t) - (y * (z / t))
	elif t <= 0.7:
		tmp = (y - (x / z)) / a
	else:
		tmp = (x - (y * z)) / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7.4e-28)
		tmp = Float64(Float64(x / t) - Float64(y * Float64(z / t)));
	elseif (t <= 0.7)
		tmp = Float64(Float64(y - Float64(x / z)) / a);
	else
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -7.4e-28)
		tmp = (x / t) - (y * (z / t));
	elseif (t <= 0.7)
		tmp = (y - (x / z)) / a;
	else
		tmp = (x - (y * z)) / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7.4e-28], N[(N[(x / t), $MachinePrecision] - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.7], N[(N[(y - N[(x / z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.40000000000000039e-28

   1. Initial program 84.1%

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

      1. *-commutative 84.1%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in x around 0 84.1%

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

      1. *-commutative 84.1%

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

      2. associate-/l* 90.8%

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

      3. *-commutative 90.8%

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

   7. Applied egg-rr 90.8%

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

   8. Taylor expanded in a around 0 72.7%

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

      1. +-commutative 72.7%

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

      2. mul-1-neg 72.7%

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

      3. unsub-neg 72.7%

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

      4. associate-/l* 76.6%

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

   10. Simplified 76.6%

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

   if -7.40000000000000039e-28 < t < 0.69999999999999996

   1. Initial program 81.7%

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

      1. *-commutative 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in x around 0 81.7%

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

      1. *-commutative 81.7%

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

      2. associate-/l* 84.5%

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

      3. *-commutative 84.5%

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

   7. Applied egg-rr 84.5%

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

   8. Taylor expanded in a around inf 71.5%

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

      1. mul-1-neg 71.5%

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

      2. unsub-neg 71.5%

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

   10. Simplified 71.5%

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

   if 0.69999999999999996 < t

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in t around inf 77.0%

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

Alternative 13: 59.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{-137}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-19}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -2.15e-137)
   (/ (- x (* y z)) t)
   (if (<= y 7.2e-19) (/ x (- t (* z a))) (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2.15e-137) {
		tmp = (x - (y * z)) / t;
	} else if (y <= 7.2e-19) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-2.15d-137)) then
        tmp = (x - (y * z)) / t
    else if (y <= 7.2d-19) then
        tmp = x / (t - (z * a))
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -2.15e-137) {
		tmp = (x - (y * z)) / t;
	} else if (y <= 7.2e-19) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -2.15e-137:
		tmp = (x - (y * z)) / t
	elif y <= 7.2e-19:
		tmp = x / (t - (z * a))
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -2.15e-137)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	elseif (y <= 7.2e-19)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -2.15e-137)
		tmp = (x - (y * z)) / t;
	elseif (y <= 7.2e-19)
		tmp = x / (t - (z * a));
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -2.15e-137], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[y, 7.2e-19], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.1499999999999999e-137

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

   3. Simplified 81.8%

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

   5. Taylor expanded in t around inf 58.9%

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

   if -2.1499999999999999e-137 < y < 7.2000000000000002e-19

   1. Initial program 97.1%

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

      1. *-commutative 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in x around inf 88.1%

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

      1. *-commutative 88.1%

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

   7. Simplified 88.1%

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

   if 7.2000000000000002e-19 < y

   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 z around inf 52.8%

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

Alternative 14: 53.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+87} \lor \neg \left(z \leq 2.2 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7.5e+87) (not (<= z 2.2e+68))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7.5e+87) || !(z <= 2.2e+68)) {
		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 <= (-7.5d+87)) .or. (.not. (z <= 2.2d+68))) 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 <= -7.5e+87) || !(z <= 2.2e+68)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -7.5e+87) or not (z <= 2.2e+68):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7.5e+87) || !(z <= 2.2e+68))
		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 <= -7.5e+87) || ~((z <= 2.2e+68)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7.5e+87], N[Not[LessEqual[z, 2.2e+68]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.50000000000000014e87 or 2.19999999999999987e68 < z

   1. Initial program 57.1%

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

      1. *-commutative 57.1%

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

   3. Simplified 57.1%

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

   5. Taylor expanded in z around inf 64.2%

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

   if -7.50000000000000014e87 < z < 2.19999999999999987e68

   1. Initial program 97.5%

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

      1. *-commutative 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in z around 0 52.8%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+87} \lor \neg \left(z \leq 2.2 \cdot 10^{+68}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.8% 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 84.8%

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

   1. *-commutative 84.8%

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

3. Simplified 84.8%

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

5. Taylor expanded in z around 0 40.1%

   \[\leadsto \color{blue}{\frac{x}{t}} \]
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 2024111 
(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))))