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

Percentage Accurate: 84.9% → 91.2%

Time: 11.4s

Alternatives: 9

Speedup: 0.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.9% 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: 91.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{-297}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y \cdot z - x}{a}\\ \mathbf{elif}\;t_1 \leq \infty:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 -2e-297)
     t_1
     (if (<= t_1 0.0)
       (* (/ 1.0 z) (/ (- (* y z) x) a))
       (if (<= t_1 INFINITY) t_1 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= -2e-297) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / z) * (((y * z) - x) / a);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} 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)) / (t - (z * a));
	double tmp;
	if (t_1 <= -2e-297) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = (1.0 / z) * (((y * z) - x) / a);
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if t_1 <= -2e-297:
		tmp = t_1
	elif t_1 <= 0.0:
		tmp = (1.0 / z) * (((y * z) - x) / a)
	elif t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - Float64(y * z)) / Float64(t - Float64(z * a)))
	tmp = 0.0
	if (t_1 <= -2e-297)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(Float64(y * z) - x) / a));
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x - (y * z)) / (t - (z * a));
	tmp = 0.0;
	if (t_1 <= -2e-297)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = (1.0 / z) * (((y * z) - x) / a);
	elseif (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-297], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], t$95$1, N[(y / a), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -2.00000000000000008e-297 or 0.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 94.1%

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

   if -2.00000000000000008e-297 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 0.0

   1. Initial program 51.9%

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

      1. *-commutative 51.9%

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

   3. Simplified 51.9%

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

   4. Taylor expanded in t around 0 23.8%

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

      1. *-commutative 23.8%

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

      2. associate-*r/ 23.8%

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

      3. neg-mul-1 23.8%

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

      4. neg-sub0 23.8%

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

      5. sub-neg 23.8%

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

      6. distribute-rgt-neg-out 23.8%

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

      7. +-commutative 23.8%

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

      8. associate--r+ 23.8%

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

      9. neg-sub0 23.8%

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

      10. distribute-rgt-neg-out 23.8%

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

      11. remove-double-neg 23.8%

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

   6. Simplified 23.8%

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

      1. *-un-lft-identity 23.8%

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

      2. times-frac 78.4%

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

   8. Applied egg-rr 78.4%

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

   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. Taylor expanded in z around inf 100.0%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -2 \cdot 10^{-297}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 0:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y \cdot z - x}{a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 2: 63.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-41} \lor \neg \left(t \leq 2.5 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y \cdot z - x}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -2.1e-41) (not (<= t 2.5e+36)))
   (/ (- x (* y z)) t)
   (* (/ 1.0 z) (/ (- (* y z) x) a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -2.1e-41) || !(t <= 2.5e+36)) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = (1.0 / z) * (((y * z) - x) / 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 <= (-2.1d-41)) .or. (.not. (t <= 2.5d+36))) then
        tmp = (x - (y * z)) / t
    else
        tmp = (1.0d0 / z) * (((y * z) - x) / 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 <= -2.1e-41) || !(t <= 2.5e+36)) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = (1.0 / z) * (((y * z) - x) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (t <= -2.1e-41) or not (t <= 2.5e+36):
		tmp = (x - (y * z)) / t
	else:
		tmp = (1.0 / z) * (((y * z) - x) / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -2.1e-41) || !(t <= 2.5e+36))
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(Float64(1.0 / z) * Float64(Float64(Float64(y * z) - x) / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((t <= -2.1e-41) || ~((t <= 2.5e+36)))
		tmp = (x - (y * z)) / t;
	else
		tmp = (1.0 / z) * (((y * z) - x) / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -2.1e-41], N[Not[LessEqual[t, 2.5e+36]], $MachinePrecision]], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(1.0 / z), $MachinePrecision] * N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.10000000000000013e-41 or 2.49999999999999988e36 < t

   1. Initial program 86.8%

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

      1. *-commutative 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in t around inf 74.1%

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

   if -2.10000000000000013e-41 < t < 2.49999999999999988e36

   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. Taylor expanded in t around 0 66.9%

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

      1. *-commutative 66.9%

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

      2. associate-*r/ 66.9%

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

      3. neg-mul-1 66.9%

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

      4. neg-sub0 66.9%

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

      5. sub-neg 66.9%

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

      6. distribute-rgt-neg-out 66.9%

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

      7. +-commutative 66.9%

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

      8. associate--r+ 66.9%

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

      9. neg-sub0 66.9%

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

      10. distribute-rgt-neg-out 66.9%

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

      11. remove-double-neg 66.9%

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

   6. Simplified 66.9%

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

      1. *-un-lft-identity 66.9%

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

      2. times-frac 73.4%

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

   8. Applied egg-rr 73.4%

      \[\leadsto \color{blue}{\frac{1}{z} \cdot \frac{y \cdot z - x}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-41} \lor \neg \left(t \leq 2.5 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{y \cdot z - x}{a}\\ \end{array} \]

Alternative 3: 64.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.76:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -0.76)
   (/ y a)
   (if (<= z -3.2e-55)
     (/ x (- t (* z a)))
     (if (<= z 5.8e+14) (/ (- x (* y z)) t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.76) {
		tmp = y / a;
	} else if (z <= -3.2e-55) {
		tmp = x / (t - (z * a));
	} else if (z <= 5.8e+14) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-0.76d0)) then
        tmp = y / a
    else if (z <= (-3.2d-55)) then
        tmp = x / (t - (z * a))
    else if (z <= 5.8d+14) then
        tmp = (x - (y * z)) / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.76) {
		tmp = y / a;
	} else if (z <= -3.2e-55) {
		tmp = x / (t - (z * a));
	} else if (z <= 5.8e+14) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -0.76:
		tmp = y / a
	elif z <= -3.2e-55:
		tmp = x / (t - (z * a))
	elif z <= 5.8e+14:
		tmp = (x - (y * z)) / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -0.76)
		tmp = Float64(y / a);
	elseif (z <= -3.2e-55)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 5.8e+14)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -0.76)
		tmp = y / a;
	elseif (z <= -3.2e-55)
		tmp = x / (t - (z * a));
	elseif (z <= 5.8e+14)
		tmp = (x - (y * z)) / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -0.76], N[(y / a), $MachinePrecision], If[LessEqual[z, -3.2e-55], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e+14], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.76000000000000001 or 5.8e14 < z

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in z around inf 60.2%

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

   if -0.76000000000000001 < z < -3.2000000000000001e-55

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 78.2%

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

      1. *-commutative 78.2%

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

   6. Simplified 78.2%

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

   if -3.2000000000000001e-55 < z < 5.8e14

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around inf 77.1%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.76:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]

Alternative 4: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-6} \lor \neg \left(y \leq 7.5 \cdot 10^{+39}\right):\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -3.6e-6) (not (<= y 7.5e+39)))
   (* z (/ y (- (* z a) t)))
   (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -3.6e-6) || !(y <= 7.5e+39)) {
		tmp = z * (y / ((z * a) - t));
	} 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 ((y <= (-3.6d-6)) .or. (.not. (y <= 7.5d+39))) then
        tmp = z * (y / ((z * a) - t))
    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 ((y <= -3.6e-6) || !(y <= 7.5e+39)) {
		tmp = z * (y / ((z * a) - t));
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (y <= -3.6e-6) or not (y <= 7.5e+39):
		tmp = z * (y / ((z * a) - t))
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -3.6e-6) || !(y <= 7.5e+39))
		tmp = Float64(z * Float64(y / Float64(Float64(z * a) - t)));
	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 ((y <= -3.6e-6) || ~((y <= 7.5e+39)))
		tmp = z * (y / ((z * a) - t));
	else
		tmp = x / (t - (z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -3.6e-6], N[Not[LessEqual[y, 7.5e+39]], $MachinePrecision]], N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.59999999999999984e-6 or 7.5000000000000005e39 < y

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

   3. Simplified 77.3%

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

   4. Taylor expanded in x around 0 57.1%

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

      1. associate-*r/ 57.1%

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

      2. associate-*r* 57.1%

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

      3. neg-mul-1 57.1%

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

      4. *-commutative 57.1%

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

   6. Simplified 57.1%

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

      1. frac-2neg 57.1%

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

      2. distribute-lft-neg-out 57.1%

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

      3. remove-double-neg 57.1%

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

      4. div-inv 57.1%

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

      5. *-commutative 57.1%

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

      6. neg-sub0 57.1%

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

      7. associate--r- 57.1%

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

      8. neg-sub0 57.1%

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

   8. Applied egg-rr 57.1%

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

      1. associate-*l* 65.7%

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

      2. associate-*r/ 65.7%

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

      3. *-rgt-identity 65.7%

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

      4. +-commutative 65.7%

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

      5. unsub-neg 65.7%

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

   10. Simplified 65.7%

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

   if -3.59999999999999984e-6 < y < 7.5000000000000005e39

   1. Initial program 94.3%

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

      1. *-commutative 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in x around inf 76.2%

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

      1. *-commutative 76.2%

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

   6. Simplified 76.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{-6} \lor \neg \left(y \leq 7.5 \cdot 10^{+39}\right):\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 5: 65.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ \mathbf{if}\;y \leq -0.000175:\\ \;\;\;\;\frac{-z}{\frac{t_1}{y}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{t_1}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a))))
   (if (<= y -0.000175)
     (/ (- z) (/ t_1 y))
     (if (<= y 3.1e+38) (/ x t_1) (* z (/ y (- (* z a) t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double tmp;
	if (y <= -0.000175) {
		tmp = -z / (t_1 / y);
	} else if (y <= 3.1e+38) {
		tmp = x / t_1;
	} else {
		tmp = z * (y / ((z * a) - 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) :: t_1
    real(8) :: tmp
    t_1 = t - (z * a)
    if (y <= (-0.000175d0)) then
        tmp = -z / (t_1 / y)
    else if (y <= 3.1d+38) then
        tmp = x / t_1
    else
        tmp = z * (y / ((z * a) - t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double tmp;
	if (y <= -0.000175) {
		tmp = -z / (t_1 / y);
	} else if (y <= 3.1e+38) {
		tmp = x / t_1;
	} else {
		tmp = z * (y / ((z * a) - t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	tmp = 0
	if y <= -0.000175:
		tmp = -z / (t_1 / y)
	elif y <= 3.1e+38:
		tmp = x / t_1
	else:
		tmp = z * (y / ((z * a) - t))
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	tmp = 0.0
	if (y <= -0.000175)
		tmp = Float64(Float64(-z) / Float64(t_1 / y));
	elseif (y <= 3.1e+38)
		tmp = Float64(x / t_1);
	else
		tmp = Float64(z * Float64(y / Float64(Float64(z * a) - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	tmp = 0.0;
	if (y <= -0.000175)
		tmp = -z / (t_1 / y);
	elseif (y <= 3.1e+38)
		tmp = x / t_1;
	else
		tmp = z * (y / ((z * a) - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -0.000175], N[((-z) / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+38], N[(x / t$95$1), $MachinePrecision], N[(z * N[(y / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.74999999999999998e-4

   1. Initial program 81.2%

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

      1. *-commutative 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in x around 0 56.6%

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

      1. associate-*r/ 56.6%

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

      2. mul-1-neg 56.6%

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

      3. distribute-rgt-neg-out 56.6%

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

      4. *-commutative 56.6%

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

      5. *-commutative 56.6%

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

      6. associate-/l* 63.6%

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

   6. Simplified 63.6%

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

   if -1.74999999999999998e-4 < y < 3.10000000000000018e38

   1. Initial program 94.3%

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

      1. *-commutative 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in x around inf 76.2%

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

      1. *-commutative 76.2%

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

   6. Simplified 76.2%

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

   if 3.10000000000000018e38 < y

   1. Initial program 73.2%

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

      1. *-commutative 73.2%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in x around 0 57.7%

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

      1. associate-*r/ 57.7%

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

      2. associate-*r* 57.7%

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

      3. neg-mul-1 57.7%

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

      4. *-commutative 57.7%

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

   6. Simplified 57.7%

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

      1. frac-2neg 57.7%

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

      2. distribute-lft-neg-out 57.7%

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

      3. remove-double-neg 57.7%

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

      4. div-inv 57.7%

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

      5. *-commutative 57.7%

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

      6. neg-sub0 57.7%

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

      7. associate--r- 57.7%

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

      8. neg-sub0 57.7%

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

   8. Applied egg-rr 57.7%

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

      1. associate-*l* 68.0%

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

      2. associate-*r/ 67.9%

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

      3. *-rgt-identity 67.9%

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

      4. +-commutative 67.9%

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

      5. unsub-neg 67.9%

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

   10. Simplified 67.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.000175:\\ \;\;\;\;\frac{-z}{\frac{t - z \cdot a}{y}}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{z \cdot a - t}\\ \end{array} \]

Alternative 6: 54.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-100}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-105} \lor \neg \left(z \leq 1.22 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.4e-8)
   (/ y a)
   (if (<= z -2.1e-100)
     (/ (- x) (* z a))
     (if (or (<= z -2.9e-105) (not (<= z 1.22e+14))) (/ y a) (/ x t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.4e-8) {
		tmp = y / a;
	} else if (z <= -2.1e-100) {
		tmp = -x / (z * a);
	} else if ((z <= -2.9e-105) || !(z <= 1.22e+14)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-3.4d-8)) then
        tmp = y / a
    else if (z <= (-2.1d-100)) then
        tmp = -x / (z * a)
    else if ((z <= (-2.9d-105)) .or. (.not. (z <= 1.22d+14))) then
        tmp = y / a
    else
        tmp = x / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.4e-8) {
		tmp = y / a;
	} else if (z <= -2.1e-100) {
		tmp = -x / (z * a);
	} else if ((z <= -2.9e-105) || !(z <= 1.22e+14)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.4e-8:
		tmp = y / a
	elif z <= -2.1e-100:
		tmp = -x / (z * a)
	elif (z <= -2.9e-105) or not (z <= 1.22e+14):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.4e-8)
		tmp = Float64(y / a);
	elseif (z <= -2.1e-100)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif ((z <= -2.9e-105) || !(z <= 1.22e+14))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.4e-8)
		tmp = y / a;
	elseif (z <= -2.1e-100)
		tmp = -x / (z * a);
	elseif ((z <= -2.9e-105) || ~((z <= 1.22e+14)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.4e-8], N[(y / a), $MachinePrecision], If[LessEqual[z, -2.1e-100], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -2.9e-105], N[Not[LessEqual[z, 1.22e+14]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.4e-8 or -2.10000000000000009e-100 < z < -2.90000000000000003e-105 or 1.22e14 < z

   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. Taylor expanded in z around inf 61.0%

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

   if -3.4e-8 < z < -2.10000000000000009e-100

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in t around 0 51.5%

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

      1. *-commutative 51.5%

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

      2. associate-*r/ 51.5%

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

      3. neg-mul-1 51.5%

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

      4. neg-sub0 51.5%

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

      5. sub-neg 51.5%

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

      6. distribute-rgt-neg-out 51.5%

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

      7. +-commutative 51.5%

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

      8. associate--r+ 51.5%

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

      9. neg-sub0 51.5%

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

      10. distribute-rgt-neg-out 51.5%

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

      11. remove-double-neg 51.5%

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

   6. Simplified 51.5%

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

   7. Taylor expanded in y around 0 43.6%

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

      1. mul-1-neg 43.6%

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

   9. Simplified 43.6%

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

   if -2.90000000000000003e-105 < z < 1.22e14

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 60.2%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-100}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq -2.9 \cdot 10^{-105} \lor \neg \left(z \leq 1.22 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 7: 64.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.8 \lor \neg \left(z \leq 2 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -0.8) (not (<= z 2e+66))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -0.8) || !(z <= 2e+66)) {
		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 <= (-0.8d0)) .or. (.not. (z <= 2d+66))) 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 <= -0.8) || !(z <= 2e+66)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -0.8) or not (z <= 2e+66):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -0.8) || !(z <= 2e+66))
		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 <= -0.8) || ~((z <= 2e+66)))
		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, -0.8], N[Not[LessEqual[z, 2e+66]], $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 < -0.80000000000000004 or 1.99999999999999989e66 < z

   1. Initial program 69.8%

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

      1. *-commutative 69.8%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in z around inf 62.8%

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

   if -0.80000000000000004 < z < 1.99999999999999989e66

   1. Initial program 97.2%

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

      1. *-commutative 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in x around inf 69.2%

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

      1. *-commutative 69.2%

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

   6. Simplified 69.2%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.8 \lor \neg \left(z \leq 2 \cdot 10^{+66}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]

Alternative 8: 54.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-105} \lor \neg \left(z \leq 68000000000000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.9e-105) (not (<= z 68000000000000.0))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.9e-105) || !(z <= 68000000000000.0)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-2.9d-105)) .or. (.not. (z <= 68000000000000.0d0))) then
        tmp = y / a
    else
        tmp = x / t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.9e-105) || !(z <= 68000000000000.0)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -2.9e-105) or not (z <= 68000000000000.0):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.9e-105) || !(z <= 68000000000000.0))
		tmp = Float64(y / a);
	else
		tmp = Float64(x / t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -2.9e-105) || ~((z <= 68000000000000.0)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.9e-105], N[Not[LessEqual[z, 68000000000000.0]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.90000000000000003e-105 or 6.8e13 < z

   1. Initial program 74.4%

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

      1. *-commutative 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in z around inf 56.6%

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

   if -2.90000000000000003e-105 < z < 6.8e13

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 60.2%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-105} \lor \neg \left(z \leq 68000000000000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]

Alternative 9: 35.6% 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 85.5%

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

   1. *-commutative 85.5%

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

3. Simplified 85.5%

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

4. Taylor expanded in z around 0 32.7%

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

5. Final simplification 32.7%

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

Developer target: 97.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023301 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

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

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