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

Percentage Accurate: 95.9% → 97.8%

Time: 8.2s

Alternatives: 5

Speedup: 1.0×

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: 95.9% 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: 97.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+271}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \frac{t}{-x}}\\ \end{array} \end{array} \]

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) 1e+271) (/ x (- y (* z t))) (/ 1.0 (* z (/ t (- x))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 1e+271) {
		tmp = x / (y - (z * t));
	} else {
		tmp = 1.0 / (z * (t / -x));
	}
	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+271) then
        tmp = x / (y - (z * t))
    else
        tmp = 1.0d0 / (z * (t / -x))
    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+271) {
		tmp = x / (y - (z * t));
	} else {
		tmp = 1.0 / (z * (t / -x));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= 1e+271:
		tmp = x / (y - (z * t))
	else:
		tmp = 1.0 / (z * (t / -x))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 1e+271)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(1.0 / Float64(z * Float64(t / Float64(-x))));
	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+271)
		tmp = x / (y - (z * t));
	else
		tmp = 1.0 / (z * (t / -x));
	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], 1e+271], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z * N[(t / (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 9.99999999999999953e270

   1. Initial program 98.3%

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

   if 9.99999999999999953e270 < (*.f64 z t)

   1. Initial program 65.2%

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

      1. clear-num 65.2%

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

      2. associate-/r/ 65.2%

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

   3. Applied egg-rr 65.2%

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

      1. associate-/r/ 65.2%

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

   5. Applied egg-rr 65.2%

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

   6. Taylor expanded in y around 0 65.2%

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

      1. associate-*r/ 65.2%

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

      2. *-commutative 65.2%

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

      3. neg-mul-1 65.2%

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

      4. distribute-neg-frac 65.2%

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

      5. associate-*l/ 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

   8. Simplified 99.6%

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

      1. associate-*l/ 65.2%

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

      2. add-sqr-sqrt 43.3%

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

      3. sqrt-unprod 60.1%

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

      4. sqr-neg 60.1%

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

      5. sqrt-unprod 16.7%

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

      6. add-sqr-sqrt 60.0%

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

      7. add-sqr-sqrt 16.7%

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

      8. sqrt-unprod 60.0%

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

      9. sqr-neg 60.0%

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

      10. sqrt-unprod 43.3%

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

      11. add-sqr-sqrt 65.2%

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

      12. add-sqr-sqrt 26.9%

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

      13. frac-times 36.8%

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

      14. clear-num 36.8%

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

      15. frac-2neg 36.8%

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

      16. frac-times 36.8%

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

      17. *-un-lft-identity 36.8%

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

      18. add-sqr-sqrt 15.7%

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

      19. sqrt-unprod 31.6%

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

      20. sqr-neg 31.6%

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

      21. sqrt-unprod 15.9%

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

      22. add-sqr-sqrt 26.7%

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

   10. Applied egg-rr 36.8%

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

       1. associate-*l/ 36.7%

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

       2. associate-/l* 26.9%

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

       3. distribute-rgt-neg-out 26.9%

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

       4. rem-square-sqrt 65.2%

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

       5. associate-/l* 99.6%

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

       6. associate-/r/ 99.5%

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

   12. Simplified 99.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+271}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{z \cdot \frac{t}{-x}}\\ \end{array} \]

Alternative 2: 73.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-41}:\\ \;\;\;\;-\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -6.5e-9) (/ x y) (if (<= y 3.8e-41) (- (/ x (* z t))) (/ x y))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.5e-9) {
		tmp = x / y;
	} else if (y <= 3.8e-41) {
		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 <= (-6.5d-9)) then
        tmp = x / y
    else if (y <= 3.8d-41) 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 <= -6.5e-9) {
		tmp = x / y;
	} else if (y <= 3.8e-41) {
		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 <= -6.5e-9:
		tmp = x / y
	elif y <= 3.8e-41:
		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 <= -6.5e-9)
		tmp = Float64(x / y);
	elseif (y <= 3.8e-41)
		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 <= -6.5e-9)
		tmp = x / y;
	elseif (y <= 3.8e-41)
		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, -6.5e-9], N[(x / y), $MachinePrecision], If[LessEqual[y, 3.8e-41], (-N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]), N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5000000000000003e-9 or 3.79999999999999979e-41 < y

   1. Initial program 96.5%

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

   2. Taylor expanded in y around inf 77.3%

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

   if -6.5000000000000003e-9 < y < 3.79999999999999979e-41

   1. Initial program 95.0%

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

   2. Taylor expanded in y around 0 78.8%

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

      1. associate-*r/ 78.8%

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

      2. neg-mul-1 78.8%

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

   4. Simplified 78.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-41}:\\ \;\;\;\;-\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 58.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-51} \lor \neg \left(t \leq 1.35 \cdot 10^{+159}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -8.5e-51) (not (<= t 1.35e+159))) (/ x (* z t)) (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -8.5e-51) || !(t <= 1.35e+159)) {
		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 ((t <= (-8.5d-51)) .or. (.not. (t <= 1.35d+159))) 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 ((t <= -8.5e-51) || !(t <= 1.35e+159)) {
		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 (t <= -8.5e-51) or not (t <= 1.35e+159):
		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 ((t <= -8.5e-51) || !(t <= 1.35e+159))
		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 ((t <= -8.5e-51) || ~((t <= 1.35e+159)))
		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[t, -8.5e-51], N[Not[LessEqual[t, 1.35e+159]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.50000000000000036e-51 or 1.35000000000000004e159 < t

   1. Initial program 93.7%

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

      1. clear-num 91.8%

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

      2. associate-/r/ 93.6%

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

   3. Applied egg-rr 93.6%

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

      1. associate-/r/ 91.8%

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

   5. Applied egg-rr 91.8%

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

   6. Taylor expanded in y around 0 66.4%

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

      1. associate-*r/ 66.4%

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

      2. *-commutative 66.4%

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

      3. neg-mul-1 66.4%

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

      4. distribute-neg-frac 66.4%

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

      5. associate-*l/ 66.2%

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

      6. distribute-rgt-neg-in 66.2%

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

   8. Simplified 66.2%

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

      1. expm1-log1p-u 61.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{z}{x} \cdot \left(-t\right)}\right)\right)} \]

      2. expm1-udef 46.9%

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

      3. *-commutative 46.9%

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

      4. add-sqr-sqrt 27.3%

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

      5. sqrt-unprod 45.9%

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

      6. sqr-neg 45.9%

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

      7. sqrt-unprod 18.6%

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

      8. add-sqr-sqrt 40.7%

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

      9. associate-*r/ 40.8%

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

   10. Applied egg-rr 40.8%

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

       1. expm1-def 36.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{t \cdot z}{x}}\right)\right)} \]

       2. expm1-log1p 36.7%

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

       3. associate-/r/ 36.7%

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

       4. associate-*l/ 36.7%

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

       5. *-lft-identity 36.7%

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

   12. Simplified 36.7%

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

   if -8.50000000000000036e-51 < t < 1.35000000000000004e159

   1. Initial program 97.4%

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

   2. Taylor expanded in y around inf 63.0%

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

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-51} \lor \neg \left(t \leq 1.35 \cdot 10^{+159}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 95.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - (z * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.8%

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

2. Final simplification 95.8%

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

Alternative 5: 54.2% 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 95.8%

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

2. Taylor expanded in y around inf 52.3%

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

3. Final simplification 52.3%

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

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 2023240 
(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))))