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

Percentage Accurate: 85.2% → 93.9%

Time: 10.6s

Alternatives: 9

Speedup: 0.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - z \cdot a\\ t_2 := y \cdot \left(\frac{z}{z \cdot a - t} + \frac{x}{y \cdot t\_1}\right)\\ t_3 := \frac{x - y \cdot z}{t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+223}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* z a)))
        (t_2 (* y (+ (/ z (- (* z a) t)) (/ x (* y t_1)))))
        (t_3 (/ (- x (* y z)) t_1)))
   (if (<= t_3 (- INFINITY))
     t_2
     (if (<= t_3 1e+223) t_3 (if (<= t_3 INFINITY) t_2 (/ y a))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = y * ((z / ((z * a) - t)) + (x / (y * t_1)));
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_3 <= 1e+223) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (z * a);
	double t_2 = y * ((z / ((z * a) - t)) + (x / (y * t_1)));
	double t_3 = (x - (y * z)) / t_1;
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_3 <= 1e+223) {
		tmp = t_3;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (z * a)
	t_2 = y * ((z / ((z * a) - t)) + (x / (y * t_1)))
	t_3 = (x - (y * z)) / t_1
	tmp = 0
	if t_3 <= -math.inf:
		tmp = t_2
	elif t_3 <= 1e+223:
		tmp = t_3
	elif t_3 <= math.inf:
		tmp = t_2
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(z * a))
	t_2 = Float64(y * Float64(Float64(z / Float64(Float64(z * a) - t)) + Float64(x / Float64(y * t_1))))
	t_3 = Float64(Float64(x - Float64(y * z)) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_3 <= 1e+223)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (z * a);
	t_2 = y * ((z / ((z * a) - t)) + (x / (y * t_1)));
	t_3 = (x - (y * z)) / t_1;
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = t_2;
	elseif (t_3 <= 1e+223)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_2;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + N[(x / N[(y * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$2, If[LessEqual[t$95$3, 1e+223], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$2, N[(y / a), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < -inf.0 or 1.00000000000000005e223 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < +inf.0

   1. Initial program 68.3%

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

      1. *-commutative 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in y around inf 99.8%

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

   if -inf.0 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.00000000000000005e223

   1. Initial program 92.6%

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

   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 3 regimes into one program.

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y \cdot z}{t - z \cdot a} \leq -\infty:\\ \;\;\;\;y \cdot \left(\frac{z}{z \cdot a - t} + \frac{x}{y \cdot \left(t - z \cdot a\right)}\right)\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq 10^{+223}:\\ \;\;\;\;\frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{elif}\;\frac{x - y \cdot z}{t - z \cdot a} \leq \infty:\\ \;\;\;\;y \cdot \left(\frac{z}{z \cdot a - t} + \frac{x}{y \cdot \left(t - z \cdot a\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t}\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{+227}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-31}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1150000:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) t)))
   (if (<= y -5.7e+227)
     (/ y a)
     (if (<= y -4.8e-31)
       t_1
       (if (<= y 1150000.0)
         (/ x (- t (* z a)))
         (if (<= y 1.6e+188) t_1 (/ y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / t;
	double tmp;
	if (y <= -5.7e+227) {
		tmp = y / a;
	} else if (y <= -4.8e-31) {
		tmp = t_1;
	} else if (y <= 1150000.0) {
		tmp = x / (t - (z * a));
	} else if (y <= 1.6e+188) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x - (y * z)) / t
    if (y <= (-5.7d+227)) then
        tmp = y / a
    else if (y <= (-4.8d-31)) then
        tmp = t_1
    else if (y <= 1150000.0d0) then
        tmp = x / (t - (z * a))
    else if (y <= 1.6d+188) then
        tmp = t_1
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - (y * z)) / t;
	double tmp;
	if (y <= -5.7e+227) {
		tmp = y / a;
	} else if (y <= -4.8e-31) {
		tmp = t_1;
	} else if (y <= 1150000.0) {
		tmp = x / (t - (z * a));
	} else if (y <= 1.6e+188) {
		tmp = t_1;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / t
	tmp = 0
	if y <= -5.7e+227:
		tmp = y / a
	elif y <= -4.8e-31:
		tmp = t_1
	elif y <= 1150000.0:
		tmp = x / (t - (z * a))
	elif y <= 1.6e+188:
		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)) / t)
	tmp = 0.0
	if (y <= -5.7e+227)
		tmp = Float64(y / a);
	elseif (y <= -4.8e-31)
		tmp = t_1;
	elseif (y <= 1150000.0)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (y <= 1.6e+188)
		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;
	tmp = 0.0;
	if (y <= -5.7e+227)
		tmp = y / a;
	elseif (y <= -4.8e-31)
		tmp = t_1;
	elseif (y <= 1150000.0)
		tmp = x / (t - (z * a));
	elseif (y <= 1.6e+188)
		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] / t), $MachinePrecision]}, If[LessEqual[y, -5.7e+227], N[(y / a), $MachinePrecision], If[LessEqual[y, -4.8e-31], t$95$1, If[LessEqual[y, 1150000.0], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+188], t$95$1, N[(y / a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.70000000000000011e227 or 1.59999999999999985e188 < y

   1. Initial program 60.8%

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

      1. *-commutative 60.8%

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

   3. Simplified 60.8%

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

   5. Taylor expanded in z around inf 62.4%

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

   if -5.70000000000000011e227 < y < -4.8e-31 or 1.15e6 < y < 1.59999999999999985e188

   1. Initial program 88.3%

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

      1. *-commutative 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in t around inf 66.3%

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

      1. *-commutative 66.3%

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

   7. Simplified 66.3%

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

   if -4.8e-31 < y < 1.15e6

   1. Initial program 91.0%

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

      1. *-commutative 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in x around inf 77.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+227}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-31}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{elif}\;y \leq 1150000:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+188}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - y \cdot z}{t - z \cdot a}\\ \mathbf{if}\;t\_1 \leq 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{z}}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x (* y z)) (- t (* z a)))))
   (if (<= t_1 1e+306) t_1 (- (/ y a) (/ (/ x z) 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 <= 1e+306) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (x - (y * z)) / (t - (z * a))
    if (t_1 <= 1d+306) then
        tmp = t_1
    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 t_1 = (x - (y * z)) / (t - (z * a));
	double tmp;
	if (t_1 <= 1e+306) {
		tmp = t_1;
	} else {
		tmp = (y / a) - ((x / z) / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x - (y * z)) / (t - (z * a))
	tmp = 0
	if t_1 <= 1e+306:
		tmp = t_1
	else:
		tmp = (y / a) - ((x / z) / 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 <= 1e+306)
		tmp = t_1;
	else
		tmp = Float64(Float64(y / a) - Float64(Float64(x / z) / 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 <= 1e+306)
		tmp = t_1;
	else
		tmp = (y / a) - ((x / z) / 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, 1e+306], t$95$1, N[(N[(y / a), $MachinePrecision] - N[(N[(x / z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z))) < 1.00000000000000002e306

   1. Initial program 90.6%

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

   if 1.00000000000000002e306 < (/.f64 (-.f64 x (*.f64 y z)) (-.f64 t (*.f64 a z)))

   1. Initial program 31.5%

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

      1. *-commutative 31.5%

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

   3. Simplified 31.5%

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

   5. Taylor expanded in t around 0 21.5%

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

      1. associate-*r/ 21.5%

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

      2. mul-1-neg 21.5%

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

      3. sub-neg 21.5%

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

      4. distribute-rgt-neg-out 21.5%

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

      5. +-commutative 21.5%

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

      6. fma-define 21.5%

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

      7. neg-sub0 21.5%

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

      8. fma-define 21.5%

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

      9. associate--r+ 21.5%

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

      10. neg-sub0 21.5%

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

      11. distribute-rgt-neg-out 21.5%

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

      12. remove-double-neg 21.5%

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

      13. *-commutative 21.5%

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

   7. Simplified 21.5%

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

   8. Taylor expanded in z around inf 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. sub-neg 83.5%

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

      4. *-commutative 83.5%

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

      5. associate-/r* 83.5%

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

   10. Simplified 83.5%

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

4. Final simplification 89.7%

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

Alternative 4: 71.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a} - \frac{\frac{x}{z}}{a}\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (/ y a) (/ (/ x z) a))))
   (if (<= z -3.5e-58)
     t_1
     (if (<= z 1.2e-98)
       (/ x (- t (* z a)))
       (if (<= z 4.2e+20) (/ (- x (* y z)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / a) - ((x / z) / a);
	double tmp;
	if (z <= -3.5e-58) {
		tmp = t_1;
	} else if (z <= 1.2e-98) {
		tmp = x / (t - (z * a));
	} else if (z <= 4.2e+20) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y / a) - ((x / z) / a)
    if (z <= (-3.5d-58)) then
        tmp = t_1
    else if (z <= 1.2d-98) then
        tmp = x / (t - (z * a))
    else if (z <= 4.2d+20) then
        tmp = (x - (y * z)) / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / a) - ((x / z) / a);
	double tmp;
	if (z <= -3.5e-58) {
		tmp = t_1;
	} else if (z <= 1.2e-98) {
		tmp = x / (t - (z * a));
	} else if (z <= 4.2e+20) {
		tmp = (x - (y * z)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / a) - ((x / z) / a)
	tmp = 0
	if z <= -3.5e-58:
		tmp = t_1
	elif z <= 1.2e-98:
		tmp = x / (t - (z * a))
	elif z <= 4.2e+20:
		tmp = (x - (y * z)) / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / a) - Float64(Float64(x / z) / a))
	tmp = 0.0
	if (z <= -3.5e-58)
		tmp = t_1;
	elseif (z <= 1.2e-98)
		tmp = Float64(x / Float64(t - Float64(z * a)));
	elseif (z <= 4.2e+20)
		tmp = Float64(Float64(x - Float64(y * z)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / a) - ((x / z) / a);
	tmp = 0.0;
	if (z <= -3.5e-58)
		tmp = t_1;
	elseif (z <= 1.2e-98)
		tmp = x / (t - (z * a));
	elseif (z <= 4.2e+20)
		tmp = (x - (y * z)) / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] - N[(N[(x / z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.5e-58], t$95$1, If[LessEqual[z, 1.2e-98], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.2e+20], N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.4999999999999999e-58 or 4.2e20 < z

   1. Initial program 71.2%

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

      1. *-commutative 71.2%

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

   3. Simplified 71.2%

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

   5. Taylor expanded in t around 0 48.2%

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

      1. associate-*r/ 48.2%

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

      2. mul-1-neg 48.2%

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

      3. sub-neg 48.2%

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

      4. distribute-rgt-neg-out 48.2%

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

      5. +-commutative 48.2%

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

      6. fma-define 48.2%

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

      7. neg-sub0 48.2%

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

      8. fma-define 48.2%

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

      9. associate--r+ 48.2%

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

      10. neg-sub0 48.2%

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

      11. distribute-rgt-neg-out 48.2%

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

      12. remove-double-neg 48.2%

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

      13. *-commutative 48.2%

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

   7. Simplified 48.2%

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

   8. Taylor expanded in z around inf 70.2%

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

      1. +-commutative 70.2%

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

      2. mul-1-neg 70.2%

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

      3. sub-neg 70.2%

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

      4. *-commutative 70.2%

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

      5. associate-/r* 73.4%

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

   10. Simplified 73.4%

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

   if -3.4999999999999999e-58 < z < 1.20000000000000002e-98

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 82.2%

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

   if 1.20000000000000002e-98 < z < 4.2e20

   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 t around inf 74.6%

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

      1. *-commutative 74.6%

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

   7. Simplified 74.6%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{z}}{a}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{x - y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a} - \frac{\frac{x}{z}}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-26} \lor \neg \left(x \leq 2.3 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= x -4.7e-26) (not (<= x 2.3e-26)))
   (/ x (- t (* z a)))
   (* y (/ z (- (* z a) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((x <= -4.7e-26) || !(x <= 2.3e-26)) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y * (z / ((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) :: tmp
    if ((x <= (-4.7d-26)) .or. (.not. (x <= 2.3d-26))) then
        tmp = x / (t - (z * a))
    else
        tmp = y * (z / ((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 tmp;
	if ((x <= -4.7e-26) || !(x <= 2.3e-26)) {
		tmp = x / (t - (z * a));
	} else {
		tmp = y * (z / ((z * a) - t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (x <= -4.7e-26) or not (x <= 2.3e-26):
		tmp = x / (t - (z * a))
	else:
		tmp = y * (z / ((z * a) - t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((x <= -4.7e-26) || !(x <= 2.3e-26))
		tmp = Float64(x / Float64(t - Float64(z * a)));
	else
		tmp = Float64(y * Float64(z / Float64(Float64(z * a) - t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((x <= -4.7e-26) || ~((x <= 2.3e-26)))
		tmp = x / (t - (z * a));
	else
		tmp = y * (z / ((z * a) - t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[x, -4.7e-26], N[Not[LessEqual[x, 2.3e-26]], $MachinePrecision]], N[(x / N[(t - N[(z * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(z / N[(N[(z * a), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.69999999999999989e-26 or 2.30000000000000009e-26 < x

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in x around inf 64.8%

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

   if -4.69999999999999989e-26 < x < 2.30000000000000009e-26

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in x around 0 72.1%

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

      1. mul-1-neg 72.1%

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

      2. associate-/l* 74.7%

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

      3. distribute-rgt-neg-in 74.7%

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

      4. sub-neg 74.7%

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

      5. mul-1-neg 74.7%

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

      6. +-commutative 74.7%

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

      7. mul-1-neg 74.7%

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

      8. distribute-rgt-neg-in 74.7%

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

      9. fma-undefine 74.7%

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

      10. distribute-neg-frac2 74.7%

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

      11. neg-sub0 74.7%

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

      12. fma-undefine 74.7%

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

      13. distribute-rgt-neg-in 74.7%

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

      14. mul-1-neg 74.7%

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

      15. associate-*r* 74.7%

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

      16. neg-mul-1 74.7%

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

      17. *-commutative 74.7%

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

      18. associate--r+ 74.7%

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

      19. neg-sub0 74.7%

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

      20. distribute-rgt-neg-out 74.7%

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

      21. remove-double-neg 74.7%

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

      22. *-commutative 74.7%

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

   7. Simplified 74.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-26} \lor \neg \left(x \leq 2.3 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z}{z \cdot a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{y}{a}\\ \mathbf{elif}\;z \leq -1.24 \cdot 10^{-141}:\\ \;\;\;\;\frac{-x}{z \cdot a}\\ \mathbf{elif}\;z \leq 6800000000:\\ \;\;\;\;\frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.6e-58)
   (/ y a)
   (if (<= z -1.24e-141)
     (/ (- x) (* z a))
     (if (<= z 6800000000.0) (/ x t) (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e-58) {
		tmp = y / a;
	} else if (z <= -1.24e-141) {
		tmp = -x / (z * a);
	} else if (z <= 6800000000.0) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.6d-58)) then
        tmp = y / a
    else if (z <= (-1.24d-141)) then
        tmp = -x / (z * a)
    else if (z <= 6800000000.0d0) then
        tmp = x / t
    else
        tmp = y / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.6e-58) {
		tmp = y / a;
	} else if (z <= -1.24e-141) {
		tmp = -x / (z * a);
	} else if (z <= 6800000000.0) {
		tmp = x / t;
	} else {
		tmp = y / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.6e-58:
		tmp = y / a
	elif z <= -1.24e-141:
		tmp = -x / (z * a)
	elif z <= 6800000000.0:
		tmp = x / t
	else:
		tmp = y / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.6e-58)
		tmp = Float64(y / a);
	elseif (z <= -1.24e-141)
		tmp = Float64(Float64(-x) / Float64(z * a));
	elseif (z <= 6800000000.0)
		tmp = Float64(x / t);
	else
		tmp = Float64(y / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.6e-58)
		tmp = y / a;
	elseif (z <= -1.24e-141)
		tmp = -x / (z * a);
	elseif (z <= 6800000000.0)
		tmp = x / t;
	else
		tmp = y / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.6e-58], N[(y / a), $MachinePrecision], If[LessEqual[z, -1.24e-141], N[((-x) / N[(z * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6800000000.0], N[(x / t), $MachinePrecision], N[(y / a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.60000000000000007e-58 or 6.8e9 < z

   1. Initial program 72.2%

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

      1. *-commutative 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in z around inf 56.6%

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

   if -2.60000000000000007e-58 < z < -1.24e-141

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 58.5%

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

      1. associate-*r/ 58.5%

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

      2. mul-1-neg 58.5%

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

      3. sub-neg 58.5%

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

      4. distribute-rgt-neg-out 58.5%

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

      5. +-commutative 58.5%

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

      6. fma-define 58.5%

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

      7. neg-sub0 58.5%

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

      8. fma-define 58.5%

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

      9. associate--r+ 58.5%

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

      10. neg-sub0 58.5%

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

      11. distribute-rgt-neg-out 58.5%

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

      12. remove-double-neg 58.5%

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

      13. *-commutative 58.5%

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

   7. Simplified 58.5%

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

   8. Taylor expanded in z around 0 49.3%

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

      1. associate-*r/ 49.3%

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

      2. neg-mul-1 49.3%

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

      3. *-commutative 49.3%

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

   10. Simplified 49.3%

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

   if -1.24e-141 < z < 6.8e9

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 66.8%

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

Alternative 7: 63.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-58} \lor \neg \left(z \leq 2.05 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3.7e-58) (not (<= z 2.05e+84))) (/ y a) (/ x (- t (* z a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3.7e-58) || !(z <= 2.05e+84)) {
		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 <= (-3.7d-58)) .or. (.not. (z <= 2.05d+84))) 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 <= -3.7e-58) || !(z <= 2.05e+84)) {
		tmp = y / a;
	} else {
		tmp = x / (t - (z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -3.7e-58) or not (z <= 2.05e+84):
		tmp = y / a
	else:
		tmp = x / (t - (z * a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3.7e-58) || !(z <= 2.05e+84))
		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 <= -3.7e-58) || ~((z <= 2.05e+84)))
		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, -3.7e-58], N[Not[LessEqual[z, 2.05e+84]], $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 < -3.7000000000000003e-58 or 2.05000000000000015e84 < z

   1. Initial program 70.6%

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

      1. *-commutative 70.6%

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

   3. Simplified 70.6%

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

   5. Taylor expanded in z around inf 59.6%

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

   if -3.7000000000000003e-58 < z < 2.05000000000000015e84

   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 x around inf 73.8%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{-58} \lor \neg \left(z \leq 2.05 \cdot 10^{+84}\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - z \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.24 \cdot 10^{-141} \lor \neg \left(z \leq 120000000000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.24e-141) (not (<= z 120000000000.0))) (/ y a) (/ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.24e-141) || !(z <= 120000000000.0)) {
		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 <= (-1.24d-141)) .or. (.not. (z <= 120000000000.0d0))) 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 <= -1.24e-141) || !(z <= 120000000000.0)) {
		tmp = y / a;
	} else {
		tmp = x / t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.24e-141) or not (z <= 120000000000.0):
		tmp = y / a
	else:
		tmp = x / t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.24e-141) || !(z <= 120000000000.0))
		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 <= -1.24e-141) || ~((z <= 120000000000.0)))
		tmp = y / a;
	else
		tmp = x / t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.24e-141], N[Not[LessEqual[z, 120000000000.0]], $MachinePrecision]], N[(y / a), $MachinePrecision], N[(x / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.24e-141 or 1.2e11 < z

   1. Initial program 75.3%

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

      1. *-commutative 75.3%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in z around inf 52.9%

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

   if -1.24e-141 < z < 1.2e11

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around 0 66.8%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.24 \cdot 10^{-141} \lor \neg \left(z \leq 120000000000\right):\\ \;\;\;\;\frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.9% 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 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 z around 0 30.6%

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

Developer Target 1: 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 2024165 
(FPCore (x y z t a)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -32113435955957344) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))) (if (< z 4392440296622287/125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (- x (* y z)) (/ 1 (- t (* a z)))) (- (/ x (- t (* a z))) (/ y (- (/ t z) a))))))

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