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

Percentage Accurate: 95.8% → 97.6%

Time: 11.8s

Alternatives: 8

Speedup: 0.6×

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: 95.8% 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: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+280}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 2e+280) (/ x (- y (* z t))) (/ -1.0 (* t (/ z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 2e+280) {
		tmp = x / (y - (z * t));
	} else {
		tmp = -1.0 / (t * (z / x));
	}
	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) :: tmp
    if ((z * t) <= 2d+280) then
        tmp = x / (y - (z * t))
    else
        tmp = (-1.0d0) / (t * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 2e+280) {
		tmp = x / (y - (z * t));
	} else {
		tmp = -1.0 / (t * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z * t) <= 2e+280:
		tmp = x / (y - (z * t))
	else:
		tmp = -1.0 / (t * (z / x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 2e+280)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(-1.0 / Float64(t * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= 2e+280)
		tmp = x / (y - (z * t));
	else
		tmp = -1.0 / (t * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], 2e+280], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(t * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 2.0000000000000001e280

   1. Initial program 98.2%

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

   if 2.0000000000000001e280 < (*.f64 z t)

   1. Initial program 68.0%

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

      1. flip3-- 0.0%

         \[\leadsto \frac{x}{\color{blue}{\frac{{y}^{3} - {\left(z \cdot t\right)}^{3}}{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}}} \]

      2. clear-num 0.0%

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}{{y}^{3} - {\left(z \cdot t\right)}^{3}}}}} \]

   3. Applied egg-rr 0.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \left(z \cdot t\right) \cdot \mathsf{fma}\left(z, t, y\right)\right)}{{y}^{3} - {\left(z \cdot t\right)}^{3}}}}} \]

   4. Taylor expanded in y around 0 68.0%

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

      1. associate-/r* 68.0%

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

   6. Simplified 68.0%

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

      1. clear-num 67.9%

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

      2. inv-pow 67.9%

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

      3. associate-/r/ 68.0%

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

      4. *-un-lft-identity 68.0%

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

      5. times-frac 99.8%

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

      6. frac-2neg 99.8%

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

      7. metadata-eval 99.8%

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

      8. remove-double-div 99.9%

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

      9. metadata-eval 99.9%

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

      10. frac-2neg 99.9%

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

      11. clear-num 99.8%

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

      12. frac-2neg 99.8%

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

      13. metadata-eval 99.8%

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

      14. remove-double-div 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. associate-*r/ 68.0%

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

      3. clear-num 68.0%

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

      4. remove-double-neg 68.0%

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

      5. distribute-lft-neg-in 68.0%

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

      6. frac-2neg 68.0%

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

      7. neg-mul-1 68.0%

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

      8. associate-/l* 68.0%

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

      9. associate-*r/ 99.9%

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

   10. Applied egg-rr 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+280}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 2: 72.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-14} \lor \neg \left(y \leq -3.6 \cdot 10^{-38} \lor \neg \left(y \leq -1.2 \cdot 10^{-87}\right) \land y \leq 3.4 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.6e-14)
         (not (or (<= y -3.6e-38) (and (not (<= y -1.2e-87)) (<= y 3.4e-45)))))
   (/ x y)
   (/ (- x) (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.6e-14) || !((y <= -3.6e-38) || (!(y <= -1.2e-87) && (y <= 3.4e-45)))) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	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) :: tmp
    if ((y <= (-5.6d-14)) .or. (.not. (y <= (-3.6d-38)) .or. (.not. (y <= (-1.2d-87))) .and. (y <= 3.4d-45))) then
        tmp = x / y
    else
        tmp = -x / (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.6e-14) || !((y <= -3.6e-38) || (!(y <= -1.2e-87) && (y <= 3.4e-45)))) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -5.6e-14) or not ((y <= -3.6e-38) or (not (y <= -1.2e-87) and (y <= 3.4e-45))):
		tmp = x / y
	else:
		tmp = -x / (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.6e-14) || !((y <= -3.6e-38) || (!(y <= -1.2e-87) && (y <= 3.4e-45))))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(-x) / Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -5.6e-14) || ~(((y <= -3.6e-38) || (~((y <= -1.2e-87)) && (y <= 3.4e-45)))))
		tmp = x / y;
	else
		tmp = -x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.6e-14], N[Not[Or[LessEqual[y, -3.6e-38], And[N[Not[LessEqual[y, -1.2e-87]], $MachinePrecision], LessEqual[y, 3.4e-45]]]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.6000000000000001e-14 or -3.6000000000000001e-38 < y < -1.2e-87 or 3.40000000000000004e-45 < y

   1. Initial program 94.2%

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

   2. Taylor expanded in y around inf 76.2%

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

   if -5.6000000000000001e-14 < y < -3.6000000000000001e-38 or -1.2e-87 < y < 3.40000000000000004e-45

   1. Initial program 96.8%

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

   2. Taylor expanded in y around 0 79.8%

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

      1. associate-*r/ 79.8%

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

      2. neg-mul-1 79.8%

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

   4. Simplified 79.8%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-14} \lor \neg \left(y \leq -3.6 \cdot 10^{-38} \lor \neg \left(y \leq -1.2 \cdot 10^{-87}\right) \land y \leq 3.4 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 3: 68.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t \cdot \frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.65e-164)
   (/ (/ (- x) t) z)
   (if (<= t 1.1e+61) (/ x y) (/ -1.0 (* t (/ z x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.65e-164) {
		tmp = (-x / t) / z;
	} else if (t <= 1.1e+61) {
		tmp = x / y;
	} else {
		tmp = -1.0 / (t * (z / x));
	}
	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) :: tmp
    if (t <= (-1.65d-164)) then
        tmp = (-x / t) / z
    else if (t <= 1.1d+61) then
        tmp = x / y
    else
        tmp = (-1.0d0) / (t * (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.65e-164) {
		tmp = (-x / t) / z;
	} else if (t <= 1.1e+61) {
		tmp = x / y;
	} else {
		tmp = -1.0 / (t * (z / x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.65e-164:
		tmp = (-x / t) / z
	elif t <= 1.1e+61:
		tmp = x / y
	else:
		tmp = -1.0 / (t * (z / x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.65e-164)
		tmp = Float64(Float64(Float64(-x) / t) / z);
	elseif (t <= 1.1e+61)
		tmp = Float64(x / y);
	else
		tmp = Float64(-1.0 / Float64(t * Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.65e-164)
		tmp = (-x / t) / z;
	elseif (t <= 1.1e+61)
		tmp = x / y;
	else
		tmp = -1.0 / (t * (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.65e-164], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, 1.1e+61], N[(x / y), $MachinePrecision], N[(-1.0 / N[(t * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.65e-164

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0 57.2%

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

      1. mul-1-neg 57.2%

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

      2. associate-/r* 58.1%

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

      3. distribute-neg-frac 58.1%

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

   4. Simplified 58.1%

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

   if -1.65e-164 < t < 1.1e61

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 77.7%

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

   if 1.1e61 < t

   1. Initial program 84.8%

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

      1. flip3-- 18.2%

         \[\leadsto \frac{x}{\color{blue}{\frac{{y}^{3} - {\left(z \cdot t\right)}^{3}}{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}}} \]

      2. clear-num 18.1%

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}{{y}^{3} - {\left(z \cdot t\right)}^{3}}}}} \]

   3. Applied egg-rr 18.1%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \left(z \cdot t\right) \cdot \mathsf{fma}\left(z, t, y\right)\right)}{{y}^{3} - {\left(z \cdot t\right)}^{3}}}}} \]

   4. Taylor expanded in y around 0 57.2%

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

      1. associate-/r* 57.2%

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

   6. Simplified 57.2%

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

      1. clear-num 57.1%

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

      2. inv-pow 57.1%

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

      3. associate-/r/ 57.1%

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

      4. *-un-lft-identity 57.1%

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

      5. times-frac 72.0%

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

      6. frac-2neg 72.0%

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

      7. metadata-eval 72.0%

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

      8. remove-double-div 72.0%

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

      9. metadata-eval 72.0%

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

      10. frac-2neg 72.0%

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

      11. clear-num 72.0%

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

      12. frac-2neg 72.0%

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

      13. metadata-eval 72.0%

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

      14. remove-double-div 72.0%

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

   8. Applied egg-rr 72.0%

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

      1. unpow-1 72.0%

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

      2. associate-*r/ 57.1%

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

      3. clear-num 57.2%

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

      4. remove-double-neg 57.2%

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

      5. distribute-lft-neg-in 57.2%

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

      6. frac-2neg 57.2%

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

      7. neg-mul-1 57.2%

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

      8. associate-/l* 57.1%

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

      9. associate-*r/ 72.0%

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

   10. Applied egg-rr 72.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+61}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{t \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 4: 68.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-164} \lor \neg \left(t \leq 1.6 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.65e-164) (not (<= t 1.6e+61))) (/ (/ (- x) t) z) (/ x y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.65e-164) || !(t <= 1.6e+61)) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / y;
	}
	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) :: tmp
    if ((t <= (-1.65d-164)) .or. (.not. (t <= 1.6d+61))) then
        tmp = (-x / t) / z
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.65e-164) || !(t <= 1.6e+61)) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.65e-164) or not (t <= 1.6e+61):
		tmp = (-x / t) / z
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.65e-164) || !(t <= 1.6e+61))
		tmp = Float64(Float64(Float64(-x) / t) / z);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.65e-164) || ~((t <= 1.6e+61)))
		tmp = (-x / t) / z;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.65e-164], N[Not[LessEqual[t, 1.6e+61]], $MachinePrecision]], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.65e-164 or 1.5999999999999999e61 < t

   1. Initial program 91.5%

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

   2. Taylor expanded in y around 0 56.9%

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

      1. mul-1-neg 56.9%

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

      2. associate-/r* 63.5%

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

      3. distribute-neg-frac 63.5%

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

   4. Simplified 63.5%

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

   if -1.65e-164 < t < 1.5999999999999999e61

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 77.1%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-164} \lor \neg \left(t \leq 1.6 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 68.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.65e-164)
   (/ (/ (- x) t) z)
   (if (<= t 2.7e+60) (/ x y) (/ (/ x z) (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.65e-164) {
		tmp = (-x / t) / z;
	} else if (t <= 2.7e+60) {
		tmp = x / y;
	} else {
		tmp = (x / z) / -t;
	}
	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) :: tmp
    if (t <= (-1.65d-164)) then
        tmp = (-x / t) / z
    else if (t <= 2.7d+60) then
        tmp = x / y
    else
        tmp = (x / z) / -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.65e-164) {
		tmp = (-x / t) / z;
	} else if (t <= 2.7e+60) {
		tmp = x / y;
	} else {
		tmp = (x / z) / -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.65e-164:
		tmp = (-x / t) / z
	elif t <= 2.7e+60:
		tmp = x / y
	else:
		tmp = (x / z) / -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.65e-164)
		tmp = Float64(Float64(Float64(-x) / t) / z);
	elseif (t <= 2.7e+60)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / z) / Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.65e-164)
		tmp = (-x / t) / z;
	elseif (t <= 2.7e+60)
		tmp = x / y;
	else
		tmp = (x / z) / -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.65e-164], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, 2.7e+60], N[(x / y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.65e-164

   1. Initial program 95.8%

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

   2. Taylor expanded in y around 0 57.2%

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

      1. mul-1-neg 57.2%

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

      2. associate-/r* 58.1%

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

      3. distribute-neg-frac 58.1%

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

   4. Simplified 58.1%

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

   if -1.65e-164 < t < 2.6999999999999999e60

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 77.7%

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

   if 2.6999999999999999e60 < t

   1. Initial program 84.8%

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

      1. *-un-lft-identity 84.8%

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

      2. *-commutative 84.8%

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

      3. add-sqr-sqrt 84.7%

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

      4. sqrt-unprod 53.2%

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

      5. sqr-neg 53.2%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 54.5%

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

      8. distribute-rgt-neg-in 54.5%

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

      9. neg-mul-1 54.5%

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

      10. add-sqr-sqrt 54.5%

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

      11. sqrt-unprod 42.8%

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

      12. sqr-neg 42.8%

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

      13. sqrt-unprod 0.0%

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

      14. add-sqr-sqrt 84.8%

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

      15. distribute-rgt-neg-in 84.8%

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

      16. prod-diff 69.5%

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

   3. Applied egg-rr 69.5%

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

      1. fma-udef 69.5%

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

      2. distribute-rgt-neg-in 69.5%

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

      3. metadata-eval 69.5%

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

      4. distribute-rgt-in 69.5%

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

      5. +-commutative 69.5%

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

      6. fma-def 69.5%

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

      7. *-lft-identity 69.5%

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

      8. fma-def 69.5%

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

      9. *-commutative 69.5%

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

      10. fma-def 69.5%

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

      11. fma-udef 69.5%

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

      12. distribute-lft-out 69.5%

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

      13. *-commutative 69.5%

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

      14. metadata-eval 69.5%

          \[\leadsto \frac{x}{\mathsf{fma}\left(t, z, y\right) + \left(t \cdot z\right) \cdot \color{blue}{-2}} \]

   5. Simplified 69.5%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, z, y\right) + \left(t \cdot z\right) \cdot -2}} \]
   6. Step-by-step derivation

      1. +-commutative 69.5%

         \[\leadsto \frac{x}{\color{blue}{\left(t \cdot z\right) \cdot -2 + \mathsf{fma}\left(t, z, y\right)}} \]

      2. associate-*l* 69.5%

         \[\leadsto \frac{x}{\color{blue}{t \cdot \left(z \cdot -2\right)} + \mathsf{fma}\left(t, z, y\right)} \]

      3. fma-def 84.7%

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, z \cdot -2, \mathsf{fma}\left(t, z, y\right)\right)}} \]

   7. Applied egg-rr 84.7%

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(t, z \cdot -2, \mathsf{fma}\left(t, z, y\right)\right)}} \]

   8. Taylor expanded in t around inf 57.2%

      \[\leadsto \color{blue}{\frac{x}{t \cdot \left(z + -2 \cdot z\right)}} \]
   9. Step-by-step derivation

      1. distribute-rgt1-in 57.2%

         \[\leadsto \frac{x}{t \cdot \color{blue}{\left(\left(-2 + 1\right) \cdot z\right)}} \]

      2. metadata-eval 57.2%

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

      3. neg-mul-1 57.2%

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

      4. distribute-rgt-neg-in 57.2%

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

      5. *-commutative 57.2%

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

      6. distribute-rgt-neg-out 57.2%

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

      7. associate-/r* 72.1%

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

   10. Simplified 72.1%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]

Alternative 6: 55.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -395000:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+252}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -395000.0)
   (/ x (* z t))
   (if (<= t 3.3e+252) (/ x y) (/ (/ x t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -395000.0) {
		tmp = x / (z * t);
	} else if (t <= 3.3e+252) {
		tmp = x / y;
	} else {
		tmp = (x / t) / z;
	}
	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) :: tmp
    if (t <= (-395000.0d0)) then
        tmp = x / (z * t)
    else if (t <= 3.3d+252) then
        tmp = x / y
    else
        tmp = (x / t) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -395000.0) {
		tmp = x / (z * t);
	} else if (t <= 3.3e+252) {
		tmp = x / y;
	} else {
		tmp = (x / t) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -395000.0:
		tmp = x / (z * t)
	elif t <= 3.3e+252:
		tmp = x / y
	else:
		tmp = (x / t) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -395000.0)
		tmp = Float64(x / Float64(z * t));
	elseif (t <= 3.3e+252)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -395000.0)
		tmp = x / (z * t);
	elseif (t <= 3.3e+252)
		tmp = x / y;
	else
		tmp = (x / t) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -395000.0], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e+252], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -395000

   1. Initial program 94.1%

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

   2. Taylor expanded in y around 0 62.5%

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

      1. mul-1-neg 62.5%

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

      2. associate-/r* 63.8%

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

      3. distribute-neg-frac 63.8%

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

   4. Simplified 63.8%

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

      1. expm1-log1p-u 60.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{x}{t}}{z}\right)\right)} \]

      2. expm1-udef 38.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{x}{t}}{z}\right)} - 1} \]

      3. add-sqr-sqrt 27.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-\frac{x}{t}} \cdot \sqrt{-\frac{x}{t}}}}{z}\right)} - 1 \]

      4. sqrt-unprod 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-\frac{x}{t}\right) \cdot \left(-\frac{x}{t}\right)}}}{z}\right)} - 1 \]

      5. sqr-neg 33.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\frac{x}{t} \cdot \frac{x}{t}}}}{z}\right)} - 1 \]

      6. sqrt-unprod 25.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\frac{x}{t}} \cdot \sqrt{\frac{x}{t}}}}{z}\right)} - 1 \]

      7. add-sqr-sqrt 32.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{x}{t}}}{z}\right)} - 1 \]

   6. Applied egg-rr 32.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 29.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)\right)} \]

      2. expm1-log1p 29.8%

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

      3. associate-/r* 28.2%

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

   8. Simplified 28.2%

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

   if -395000 < t < 3.3000000000000001e252

   1. Initial program 97.2%

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

   2. Taylor expanded in y around inf 65.7%

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

   if 3.3000000000000001e252 < t

   1. Initial program 79.0%

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

      1. flip3-- 6.1%

         \[\leadsto \frac{x}{\color{blue}{\frac{{y}^{3} - {\left(z \cdot t\right)}^{3}}{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}}} \]

      2. clear-num 6.0%

         \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{y \cdot y + \left(\left(z \cdot t\right) \cdot \left(z \cdot t\right) + y \cdot \left(z \cdot t\right)\right)}{{y}^{3} - {\left(z \cdot t\right)}^{3}}}}} \]

   3. Applied egg-rr 6.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, y, \left(z \cdot t\right) \cdot \mathsf{fma}\left(z, t, y\right)\right)}{{y}^{3} - {\left(z \cdot t\right)}^{3}}}}} \]

   4. Taylor expanded in y around 0 59.6%

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

      1. associate-/r* 59.7%

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

   6. Simplified 59.7%

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

      1. clear-num 59.6%

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

      2. inv-pow 59.6%

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

      3. associate-/r/ 59.6%

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

      4. *-un-lft-identity 59.6%

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

      5. times-frac 80.4%

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

      6. frac-2neg 80.4%

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

      7. metadata-eval 80.4%

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

      8. remove-double-div 80.4%

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

      9. metadata-eval 80.4%

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

      10. frac-2neg 80.4%

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

      11. clear-num 80.4%

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

      12. frac-2neg 80.4%

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

      13. metadata-eval 80.4%

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

      14. remove-double-div 80.4%

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

   8. Applied egg-rr 80.4%

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

      1. unpow-1 80.4%

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

      2. associate-*r/ 59.6%

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

      3. clear-num 59.7%

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

      4. associate-/r* 85.6%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 36.4%

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

      7. sqr-neg 36.4%

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

      8. sqrt-unprod 35.4%

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

      9. add-sqr-sqrt 35.4%

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

   10. Applied egg-rr 35.4%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -395000:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+252}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 7: 54.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.8e+20) (/ x y) (/ x (* z t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.8e+20) {
		tmp = x / y;
	} else {
		tmp = x / (z * t);
	}
	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) :: tmp
    if (z <= 3.8d+20) then
        tmp = x / y
    else
        tmp = x / (z * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.8e+20) {
		tmp = x / y;
	} else {
		tmp = x / (z * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 3.8e+20:
		tmp = x / y
	else:
		tmp = x / (z * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.8e+20)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(z * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.8e+20)
		tmp = x / y;
	else
		tmp = x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 3.8e+20], N[(x / y), $MachinePrecision], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.8e20

   1. Initial program 98.0%

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

   2. Taylor expanded in y around inf 64.2%

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

   if 3.8e20 < z

   1. Initial program 87.0%

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

   2. Taylor expanded in y around 0 61.0%

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

      1. mul-1-neg 61.0%

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

      2. associate-/r* 73.8%

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

      3. distribute-neg-frac 73.8%

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

   4. Simplified 73.8%

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

      1. expm1-log1p-u 70.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-\frac{x}{t}}{z}\right)\right)} \]

      2. expm1-udef 37.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-\frac{x}{t}}{z}\right)} - 1} \]

      3. add-sqr-sqrt 22.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-\frac{x}{t}} \cdot \sqrt{-\frac{x}{t}}}}{z}\right)} - 1 \]

      4. sqrt-unprod 36.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-\frac{x}{t}\right) \cdot \left(-\frac{x}{t}\right)}}}{z}\right)} - 1 \]

      5. sqr-neg 36.4%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{\frac{x}{t} \cdot \frac{x}{t}}}}{z}\right)} - 1 \]

      6. sqrt-unprod 26.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\frac{x}{t}} \cdot \sqrt{\frac{x}{t}}}}{z}\right)} - 1 \]

      7. add-sqr-sqrt 34.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\frac{x}{t}}}{z}\right)} - 1 \]

   6. Applied egg-rr 34.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 29.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)\right)} \]

      2. expm1-log1p 29.3%

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

      3. associate-/r* 29.6%

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

   8. Simplified 29.6%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 8: 55.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.1%

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

2. Taylor expanded in y around inf 56.6%

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

3. Final simplification 56.6%

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

Developer target: 96.7% 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 2023318 
(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))))