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

Percentage Accurate: 95.7% → 99.8%

Time: 6.9s

Alternatives: 7

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: 95.7% 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, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+284}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-1}{t}\\ \end{array} \end{array} \]

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) -2e+284)
   (/ (/ (- x) z) t)
   (if (<= (* z t) 5e+292) (/ x (fma (- z) t y)) (* (/ x z) (/ -1.0 t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+284) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 5e+292) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = (x / z) * (-1.0 / t);
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+284)
		tmp = Float64(Float64(Float64(-x) / z) / t);
	elseif (Float64(z * t) <= 5e+292)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(x / z) * Float64(-1.0 / t));
	end
	return 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], -2e+284], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+292], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000016e284

   1. Initial program 70.2%

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

      1. clear-num 70.2%

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

      2. inv-pow 70.2%

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

   3. Applied egg-rr 70.2%

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

      1. frac-2neg 70.2%

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

      2. div-inv 70.2%

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

      3. sub-neg 70.2%

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

      4. +-commutative 70.2%

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

      5. distribute-neg-in 70.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. distribute-rgt-neg-in 0.0%

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

      8. distribute-lft-neg-out 0.0%

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

      9. sqr-neg 0.0%

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

      10. add-sqr-sqrt 70.2%

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

      11. fma-def 70.2%

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

   5. Applied egg-rr 70.2%

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

      1. fma-neg 70.2%

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

      2. *-commutative 70.2%

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

   7. Simplified 70.2%

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

   8. Taylor expanded in t around inf 70.2%

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

      1. neg-mul-1 70.2%

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

      2. associate-/l/ 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. distribute-neg-frac 99.8%

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

   10. Simplified 99.8%

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

   if -2.00000000000000016e284 < (*.f64 z t) < 4.9999999999999996e292

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   if 4.9999999999999996e292 < (*.f64 z t)

   1. Initial program 70.7%

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

      1. clear-num 70.7%

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

      2. inv-pow 70.7%

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

   3. Applied egg-rr 70.7%

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

   4. Taylor expanded in y around 0 70.7%

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

      1. mul-1-neg 70.7%

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

      2. associate-*l/ 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 70.7%

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

      1. mul-1-neg 70.7%

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

      2. associate-/l* 99.7%

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

      3. distribute-neg-frac 99.7%

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

   9. Simplified 99.7%

      \[\leadsto {\color{blue}{\left(\frac{-t}{\frac{x}{z}}\right)}}^{-1} \]
   10. Step-by-step derivation

       1. unpow-1 99.7%

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

       2. clear-num 99.8%

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

       3. add-sqr-sqrt 63.0%

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

       4. sqrt-unprod 75.5%

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

       5. sqr-neg 75.5%

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

       6. sqrt-unprod 31.8%

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

       7. add-sqr-sqrt 65.0%

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

       8. frac-2neg 65.0%

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

       9. div-inv 65.0%

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

       10. add-sqr-sqrt 33.2%

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

       11. sqrt-unprod 65.4%

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

       12. sqr-neg 65.4%

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

       13. sqrt-unprod 36.8%

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

       14. add-sqr-sqrt 99.8%

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

   11. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\left(-\frac{x}{z}\right) \cdot \frac{1}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+284}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-1}{t}\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+284}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+294}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \end{array} \]

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) -2e+284)
   (/ (/ (- x) z) t)
   (if (<= (* z t) 1e+294) (/ x (- y (* z t))) (* (/ -1.0 z) (/ x t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+284) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 1e+294) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / z) * (x / t);
	}
	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) <= (-2d+284)) then
        tmp = (-x / z) / t
    else if ((z * t) <= 1d+294) then
        tmp = x / (y - (z * t))
    else
        tmp = ((-1.0d0) / z) * (x / t)
    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) <= -2e+284) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 1e+294) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (-1.0 / z) * (x / t);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e+284:
		tmp = (-x / z) / t
	elif (z * t) <= 1e+294:
		tmp = x / (y - (z * t))
	else:
		tmp = (-1.0 / z) * (x / t)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+284)
		tmp = Float64(Float64(Float64(-x) / z) / t);
	elseif (Float64(z * t) <= 1e+294)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(-1.0 / z) * Float64(x / 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) <= -2e+284)
		tmp = (-x / z) / t;
	elseif ((z * t) <= 1e+294)
		tmp = x / (y - (z * t));
	else
		tmp = (-1.0 / z) * (x / 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], -2e+284], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+294], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / z), $MachinePrecision] * N[(x / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000016e284

   1. Initial program 70.2%

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

      1. clear-num 70.2%

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

      2. inv-pow 70.2%

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

   3. Applied egg-rr 70.2%

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

      1. frac-2neg 70.2%

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

      2. div-inv 70.2%

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

      3. sub-neg 70.2%

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

      4. +-commutative 70.2%

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

      5. distribute-neg-in 70.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. distribute-rgt-neg-in 0.0%

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

      8. distribute-lft-neg-out 0.0%

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

      9. sqr-neg 0.0%

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

      10. add-sqr-sqrt 70.2%

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

      11. fma-def 70.2%

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

   5. Applied egg-rr 70.2%

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

      1. fma-neg 70.2%

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

      2. *-commutative 70.2%

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

   7. Simplified 70.2%

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

   8. Taylor expanded in t around inf 70.2%

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

      1. neg-mul-1 70.2%

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

      2. associate-/l/ 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. distribute-neg-frac 99.8%

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

   10. Simplified 99.8%

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

   if -2.00000000000000016e284 < (*.f64 z t) < 1.00000000000000007e294

   1. Initial program 99.8%

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

   if 1.00000000000000007e294 < (*.f64 z t)

   1. Initial program 69.0%

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

      1. clear-num 69.0%

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

      2. inv-pow 69.0%

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

   3. Applied egg-rr 69.0%

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

   4. Taylor expanded in y around 0 69.0%

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

      1. mul-1-neg 69.0%

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

      2. associate-*l/ 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 69.0%

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

      1. mul-1-neg 69.0%

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

      2. associate-/l* 99.7%

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

      3. distribute-neg-frac 99.7%

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

   9. Simplified 99.7%

      \[\leadsto {\color{blue}{\left(\frac{-t}{\frac{x}{z}}\right)}}^{-1} \]
   10. Step-by-step derivation

       1. unpow-1 99.7%

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

       2. clear-num 99.8%

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

       3. *-un-lft-identity 99.8%

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

       4. neg-mul-1 99.8%

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

       5. times-frac 99.8%

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

       6. metadata-eval 99.8%

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

       7. metadata-eval 99.8%

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

       8. associate-/l/ 69.0%

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

       9. *-commutative 69.0%

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

       10. times-frac 69.0%

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

       11. *-un-lft-identity 69.0%

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

       12. times-frac 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+284}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 10^{+294}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z} \cdot \frac{x}{t}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+284}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-1}{t}\\ \end{array} \end{array} \]

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) -2e+284)
   (/ (/ (- x) z) t)
   (if (<= (* z t) 5e+292) (/ x (- y (* z t))) (* (/ x z) (/ -1.0 t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+284) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 5e+292) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / z) * (-1.0 / t);
	}
	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) <= (-2d+284)) then
        tmp = (-x / z) / t
    else if ((z * t) <= 5d+292) then
        tmp = x / (y - (z * t))
    else
        tmp = (x / z) * ((-1.0d0) / t)
    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) <= -2e+284) {
		tmp = (-x / z) / t;
	} else if ((z * t) <= 5e+292) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / z) * (-1.0 / t);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e+284:
		tmp = (-x / z) / t
	elif (z * t) <= 5e+292:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / z) * (-1.0 / t)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+284)
		tmp = Float64(Float64(Float64(-x) / z) / t);
	elseif (Float64(z * t) <= 5e+292)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / z) * Float64(-1.0 / 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) <= -2e+284)
		tmp = (-x / z) / t;
	elseif ((z * t) <= 5e+292)
		tmp = x / (y - (z * t));
	else
		tmp = (x / z) * (-1.0 / 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], -2e+284], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+292], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000016e284

   1. Initial program 70.2%

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

      1. clear-num 70.2%

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

      2. inv-pow 70.2%

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

   3. Applied egg-rr 70.2%

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

      1. frac-2neg 70.2%

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

      2. div-inv 70.2%

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

      3. sub-neg 70.2%

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

      4. +-commutative 70.2%

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

      5. distribute-neg-in 70.2%

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

      6. add-sqr-sqrt 0.0%

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

      7. distribute-rgt-neg-in 0.0%

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

      8. distribute-lft-neg-out 0.0%

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

      9. sqr-neg 0.0%

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

      10. add-sqr-sqrt 70.2%

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

      11. fma-def 70.2%

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

   5. Applied egg-rr 70.2%

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

      1. fma-neg 70.2%

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

      2. *-commutative 70.2%

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

   7. Simplified 70.2%

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

   8. Taylor expanded in t around inf 70.2%

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

      1. neg-mul-1 70.2%

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

      2. associate-/l/ 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. distribute-neg-frac 99.8%

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

   10. Simplified 99.8%

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

   if -2.00000000000000016e284 < (*.f64 z t) < 4.9999999999999996e292

   1. Initial program 99.8%

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

   if 4.9999999999999996e292 < (*.f64 z t)

   1. Initial program 70.7%

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

      1. clear-num 70.7%

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

      2. inv-pow 70.7%

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

   3. Applied egg-rr 70.7%

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

   4. Taylor expanded in y around 0 70.7%

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

      1. mul-1-neg 70.7%

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

      2. associate-*l/ 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in t around 0 70.7%

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

      1. mul-1-neg 70.7%

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

      2. associate-/l* 99.7%

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

      3. distribute-neg-frac 99.7%

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

   9. Simplified 99.7%

      \[\leadsto {\color{blue}{\left(\frac{-t}{\frac{x}{z}}\right)}}^{-1} \]
   10. Step-by-step derivation

       1. unpow-1 99.7%

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

       2. clear-num 99.8%

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

       3. add-sqr-sqrt 63.0%

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

       4. sqrt-unprod 75.5%

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

       5. sqr-neg 75.5%

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

       6. sqrt-unprod 31.8%

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

       7. add-sqr-sqrt 65.0%

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

       8. frac-2neg 65.0%

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

       9. div-inv 65.0%

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

       10. add-sqr-sqrt 33.2%

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

       11. sqrt-unprod 65.4%

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

       12. sqr-neg 65.4%

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

       13. sqrt-unprod 36.8%

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

       14. add-sqr-sqrt 99.8%

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

   11. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\left(-\frac{x}{z}\right) \cdot \frac{1}{t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+284}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-1}{t}\\ \end{array} \]

Alternative 4: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+84} \lor \neg \left(z \cdot t \leq 10^{+18}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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+84) (not (<= (* z t) 1e+18)))
   (/ (/ (- 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+84) || !((z * t) <= 1e+18)) {
		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+84)) .or. (.not. ((z * t) <= 1d+18))) 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+84) || !((z * t) <= 1e+18)) {
		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+84) or not ((z * t) <= 1e+18):
		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+84) || !(Float64(z * t) <= 1e+18))
		tmp = Float64(Float64(Float64(-x) / 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+84) || ~(((z * t) <= 1e+18)))
		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+84], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+18]], $MachinePrecision]], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -1.00000000000000006e84 or 1e18 < (*.f64 z t)

   1. Initial program 89.7%

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

      1. clear-num 89.3%

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

      2. inv-pow 89.3%

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

   3. Applied egg-rr 89.3%

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

      1. frac-2neg 89.3%

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

      2. div-inv 89.2%

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

      3. sub-neg 89.2%

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

      4. +-commutative 89.2%

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

      5. distribute-neg-in 89.2%

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

      6. add-sqr-sqrt 47.1%

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

      7. distribute-rgt-neg-in 47.1%

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

      8. distribute-lft-neg-out 47.1%

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

      9. sqr-neg 47.1%

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

      10. add-sqr-sqrt 89.2%

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

      11. fma-def 89.2%

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

   5. Applied egg-rr 89.2%

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

      1. fma-neg 89.2%

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

      2. *-commutative 89.2%

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

   7. Simplified 89.2%

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

   8. Taylor expanded in t around inf 81.0%

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

      1. neg-mul-1 81.0%

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

      2. associate-/l/ 86.5%

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

      3. distribute-frac-neg 86.5%

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

      4. distribute-neg-frac 86.5%

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

   10. Simplified 86.5%

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

   if -1.00000000000000006e84 < (*.f64 z t) < 1e18

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 74.2%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+84} \lor \neg \left(z \cdot t \leq 10^{+18}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 62.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+166} \lor \neg \left(z \cdot t \leq 10^{+71}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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+166) (not (<= (* z t) 1e+71))) (/ 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+166) || !((z * t) <= 1e+71)) {
		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+166)) .or. (.not. ((z * t) <= 1d+71))) 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+166) || !((z * t) <= 1e+71)) {
		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+166) or not ((z * t) <= 1e+71):
		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+166) || !(Float64(z * t) <= 1e+71))
		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) <= -5e+166) || ~(((z * t) <= 1e+71)))
		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+166], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+71]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5.0000000000000002e166 or 1e71 < (*.f64 z t)

   1. Initial program 87.1%

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

      1. clear-num 86.5%

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

      2. inv-pow 86.5%

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

   3. Applied egg-rr 86.5%

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

   4. Taylor expanded in y around 0 81.8%

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

      1. mul-1-neg 81.8%

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

      2. associate-*l/ 91.3%

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

      3. distribute-rgt-neg-in 91.3%

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

   6. Simplified 91.3%

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

      1. unpow-1 91.3%

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

      2. add-sqr-sqrt 51.7%

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

      3. sqrt-unprod 67.1%

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

      4. sqr-neg 67.1%

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

      5. sqrt-unprod 24.7%

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

      6. add-sqr-sqrt 52.6%

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

      7. associate-/r/ 52.5%

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

      8. clear-num 52.5%

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

      9. associate-/l/ 52.7%

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

      10. *-commutative 52.7%

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

   8. Applied egg-rr 52.7%

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

   if -5.0000000000000002e166 < (*.f64 z t) < 1e71

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 69.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+166} \lor \neg \left(z \cdot t \leq 10^{+71}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 72.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+31}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.6e+31) (/ x y) (if (<= y 1.15e+31) (/ (- x) (* z t)) (/ x y))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.6e+31) {
		tmp = x / y;
	} else if (y <= 1.15e+31) {
		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 (y <= (-2.6d+31)) then
        tmp = x / y
    else if (y <= 1.15d+31) 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 (y <= -2.6e+31) {
		tmp = x / y;
	} else if (y <= 1.15e+31) {
		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 y <= -2.6e+31:
		tmp = x / y
	elif y <= 1.15e+31:
		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 (y <= -2.6e+31)
		tmp = Float64(x / y);
	elseif (y <= 1.15e+31)
		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 (y <= -2.6e+31)
		tmp = x / y;
	elseif (y <= 1.15e+31)
		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[LessEqual[y, -2.6e+31], N[(x / y), $MachinePrecision], If[LessEqual[y, 1.15e+31], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.6e31 or 1.15e31 < y

   1. Initial program 93.7%

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

   2. Taylor expanded in y around inf 80.8%

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

   if -2.6e31 < y < 1.15e31

   1. Initial program 97.0%

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

   2. Taylor expanded in y around 0 78.1%

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

      1. associate-*r/ 78.1%

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

      2. neg-mul-1 78.1%

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

   4. Simplified 78.1%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{+31}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+31}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 54.6% 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 52.7%

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

3. Final simplification 52.7%

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

Developer target: 96.5% 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 2023213 
(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))))