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

Percentage Accurate: 96.0% → 99.8%

Time: 11.1s

Alternatives: 5

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (* (/ -1.0 t) (/ x z))
   (if (<= (* z t) 2e+276) (/ x (- y (* z t))) (/ (/ x z) (- t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (-1.0 / t) * (x / z);
	} else if ((z * t) <= 2e+276) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / z) / -t;
	}
	return tmp;
}

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (-1.0 / t) * (x / z);
	} else if ((z * t) <= 2e+276) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / z) / -t;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (-1.0 / t) * (x / z)
	elif (z * t) <= 2e+276:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / z) / -t
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(-1.0 / t) * Float64(x / z));
	elseif (Float64(z * t) <= 2e+276)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / z) / Float64(-t));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = (-1.0 / t) * (x / z);
	elseif ((z * t) <= 2e+276)
		tmp = x / (y - (z * t));
	else
		tmp = (x / z) / -t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[(-1.0 / t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+276], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -inf.0

   1. Initial program 49.9%

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

   2. Taylor expanded in y around 0 49.9%

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

      1. associate-*r/ 49.9%

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

      2. neg-mul-1 49.9%

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

   4. Simplified 49.9%

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

      1. neg-mul-1 49.9%

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

      2. times-frac 99.6%

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

   6. Applied egg-rr 99.6%

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

   if -inf.0 < (*.f64 z t) < 2.0000000000000001e276

   1. Initial program 99.9%

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

   if 2.0000000000000001e276 < (*.f64 z t)

   1. Initial program 71.0%

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

      1. clear-num 71.0%

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

      2. associate-/r/ 71.0%

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

   3. Applied egg-rr 71.0%

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

   4. Taylor expanded in y around 0 71.0%

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

      1. associate-*l/ 71.0%

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

      2. neg-mul-1 71.0%

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

      3. add-sqr-sqrt 45.6%

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

      4. sqrt-unprod 71.0%

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

      5. sqr-neg 71.0%

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

      6. sqrt-unprod 25.4%

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

      7. add-sqr-sqrt 71.0%

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

      8. associate-/l/ 70.5%

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

      9. frac-2neg 70.5%

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

      10. distribute-neg-frac 70.5%

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

      11. add-sqr-sqrt 45.3%

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

      12. sqrt-unprod 76.5%

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

      13. sqr-neg 76.5%

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

      14. sqrt-unprod 31.3%

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

      15. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+276}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]

Alternative 2: 77.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+27} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e+27) (not (<= (* z t) 5e-35)))
   (/ (- x) (* z t))
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+27) || !((z * t) <= 5e-35)) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-5d+27)) .or. (.not. ((z * t) <= 5d-35))) then
        tmp = -x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+27) || !((z * t) <= 5e-35)) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+27) or not ((z * t) <= 5e-35):
		tmp = -x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+27) || !(Float64(z * t) <= 5e-35))
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -5e+27) || ~(((z * t) <= 5e-35)))
		tmp = -x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+27], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e-35]], $MachinePrecision]], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999979e27 or 4.99999999999999964e-35 < (*.f64 z t)

   1. Initial program 90.7%

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

   2. Taylor expanded in y around 0 74.3%

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

      1. associate-*r/ 74.3%

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

      2. neg-mul-1 74.3%

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

   4. Simplified 74.3%

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

   if -4.99999999999999979e27 < (*.f64 z t) < 4.99999999999999964e-35

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 84.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+27} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 79.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+27} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e+27) (not (<= (* z t) 2e+22)))
   (/ (- (/ x t)) z)
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+27) || !((z * t) <= 2e+22)) {
		tmp = -(x / t) / z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-5d+27)) .or. (.not. ((z * t) <= 2d+22))) then
        tmp = -(x / t) / z
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+27) || !((z * t) <= 2e+22)) {
		tmp = -(x / t) / z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+27) or not ((z * t) <= 2e+22):
		tmp = -(x / t) / z
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+27) || !(Float64(z * t) <= 2e+22))
		tmp = Float64(Float64(-Float64(x / t)) / z);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -5e+27) || ~(((z * t) <= 2e+22)))
		tmp = -(x / t) / z;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+27], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+22]], $MachinePrecision]], N[((-N[(x / t), $MachinePrecision]) / z), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999979e27 or 2e22 < (*.f64 z t)

   1. Initial program 90.2%

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

      1. clear-num 88.7%

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

      2. inv-pow 88.7%

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

   3. Applied egg-rr 88.7%

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

   4. Taylor expanded in y around 0 75.9%

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

      1. mul-1-neg 75.9%

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

      2. associate-/r* 82.1%

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

      3. distribute-neg-frac 82.1%

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

   6. Simplified 82.1%

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

   if -4.99999999999999979e27 < (*.f64 z t) < 2e22

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 82.6%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+27} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+22}\right):\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 63.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+149} \lor \neg \left(z \cdot t \leq 10^{+95}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -1e+149) (not (<= (* z t) 1e+95))) (/ x (* z t)) (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e+149) || !((z * t) <= 1e+95)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-1d+149)) .or. (.not. ((z * t) <= 1d+95))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e+149) || !((z * t) <= 1e+95)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -1e+149) or not ((z * t) <= 1e+95):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -1e+149) || !(Float64(z * t) <= 1e+95))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -1e+149) || ~(((z * t) <= 1e+95)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -1e+149], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+95]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000005e149 or 1.00000000000000002e95 < (*.f64 z t)

   1. Initial program 85.8%

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

      1. clear-num 84.7%

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

      2. associate-/r/ 85.7%

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

   3. Applied egg-rr 85.7%

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

   4. Taylor expanded in y around 0 81.6%

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

      1. associate-*l/ 81.6%

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

      2. neg-mul-1 81.6%

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

      3. add-sqr-sqrt 39.2%

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

      4. sqrt-unprod 57.5%

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

      5. sqr-neg 57.5%

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

      6. sqrt-unprod 26.5%

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

      7. add-sqr-sqrt 52.0%

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

      8. *-commutative 52.0%

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

   6. Applied egg-rr 52.0%

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

   if -1.00000000000000005e149 < (*.f64 z t) < 1.00000000000000002e95

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 74.3%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+149} \lor \neg \left(z \cdot t \leq 10^{+95}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 54.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ x y))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	return x / y;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x / y
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	return x / y;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	return x / y

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	return Float64(x / y)
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x / y;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

2. Taylor expanded in y around inf 55.7%

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

3. Final simplification 55.7%

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

Developer target: 96.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (- (/ y x) (* (/ z x) t)))))
   (if (< x -1.618195973607049e+50)
     t_1
     (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / ((y / x) - ((z / x) * t))
    if (x < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (x < 2.1378306434876444d+131) 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 t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / ((y / x) - ((z / x) * t))
	tmp = 0
	if x < -1.618195973607049e+50:
		tmp = t_1
	elif x < 2.1378306434876444e+131:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(Float64(y / x) - Float64(Float64(z / x) * t)))
	tmp = 0.0
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / ((y / x) - ((z / x) * t));
	tmp = 0.0;
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[x, -1.618195973607049e+50], t$95$1, If[Less[x, 2.1378306434876444e+131], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

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