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

Percentage Accurate: 95.9% → 98.0%

Time: 8.0s

Alternatives: 6

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: 95.9% 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: 98.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.9%

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

      1. clear-num 59.9%

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

      2. associate-/r/ 59.9%

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

   3. Applied egg-rr 59.9%

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

   4. Taylor expanded in y around 0 59.9%

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

      1. associate-/r* 64.1%

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

   6. Simplified 64.1%

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

      1. add-sqr-sqrt 64.1%

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

      2. sqrt-unprod 59.9%

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

      3. pow2 59.9%

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

      4. *-commutative 59.9%

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

      5. div-inv 59.9%

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

      6. associate-*r* 70.8%

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

      7. frac-2neg 70.8%

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

      8. metadata-eval 70.8%

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

      9. div-inv 70.8%

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

      10. clear-num 70.8%

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

      11. unpow-1 70.8%

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

      12. div-inv 70.9%

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

      13. unpow-1 70.9%

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

      14. clear-num 70.9%

          \[\leadsto \sqrt{{\left(\frac{\color{blue}{\frac{x}{-t}}}{z}\right)}^{2}} \]

      15. add-sqr-sqrt 30.1%

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

      16. sqrt-unprod 59.8%

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

      17. sqr-neg 59.8%

          \[\leadsto \sqrt{{\left(\frac{\frac{x}{\sqrt{\color{blue}{t \cdot t}}}}{z}\right)}^{2}} \]

      18. sqrt-unprod 40.8%

          \[\leadsto \sqrt{{\left(\frac{\frac{x}{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}}{z}\right)}^{2}} \]

      19. add-sqr-sqrt 70.9%

          \[\leadsto \sqrt{{\left(\frac{\frac{x}{\color{blue}{t}}}{z}\right)}^{2}} \]

   8. Applied egg-rr 70.9%

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

      1. unpow2 70.9%

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

      2. sqrt-prod 53.6%

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

      3. add-sqr-sqrt 59.1%

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

      4. associate-/r* 59.9%

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

      5. add-sqr-sqrt 35.4%

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

      6. sqrt-unprod 47.5%

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

      7. sqr-neg 47.5%

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

      8. sqrt-unprod 24.4%

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

      9. add-sqr-sqrt 59.9%

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

      10. associate-/r* 99.8%

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

      11. frac-2neg 99.8%

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

      12. frac-2neg 99.8%

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

      13. remove-double-neg 99.8%

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

      14. distribute-frac-neg 99.8%

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

      15. frac-2neg 99.8%

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

   10. Applied egg-rr 99.8%

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

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

   1. Initial program 98.3%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 2: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-67} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -1e-67) (not (<= (* z t) 5e-52)))
   (/ (- 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-67) || !((z * t) <= 5e-52)) {
		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-67)) .or. (.not. ((z * t) <= 5d-52))) 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-67) || !((z * t) <= 5e-52)) {
		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-67) or not ((z * t) <= 5e-52):
		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-67) || !(Float64(z * t) <= 5e-52))
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -1e-67) || ~(((z * t) <= 5e-52)))
		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-67], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e-52]], $MachinePrecision]], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.99999999999999943e-68 or 5e-52 < (*.f64 z t)

   1. Initial program 92.7%

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

   2. Taylor expanded in y around 0 72.5%

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

      1. associate-*r/ 72.5%

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

      2. neg-mul-1 72.5%

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

   4. Simplified 72.5%

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

   if -9.99999999999999943e-68 < (*.f64 z t) < 5e-52

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 85.6%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-67} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 78.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-67} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -1e-67) (not (<= (* z t) 5e-52)))
   (/ (/ x t) (- z))
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e-67) || !((z * t) <= 5e-52)) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-1d-67)) .or. (.not. ((z * t) <= 5d-52))) then
        tmp = (x / t) / -z
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -1e-67) || !((z * t) <= 5e-52)) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -1e-67) or not ((z * t) <= 5e-52):
		tmp = (x / t) / -z
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -1e-67) || !(Float64(z * t) <= 5e-52))
		tmp = Float64(Float64(x / t) / Float64(-z));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -1e-67) || ~(((z * t) <= 5e-52)))
		tmp = (x / t) / -z;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -1e-67], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e-52]], $MachinePrecision]], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.99999999999999943e-68 or 5e-52 < (*.f64 z t)

   1. Initial program 92.7%

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

      1. clear-num 91.8%

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

      2. associate-/r/ 92.5%

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

   3. Applied egg-rr 92.5%

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

   4. Taylor expanded in y around 0 72.4%

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

      1. associate-/r* 72.9%

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

   6. Simplified 72.9%

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

      1. add-sqr-sqrt 50.5%

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

      2. sqrt-unprod 49.7%

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

      3. pow2 49.7%

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

      4. *-commutative 49.7%

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

      5. div-inv 49.7%

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

      6. associate-*r* 51.6%

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

      7. frac-2neg 51.6%

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

      8. metadata-eval 51.6%

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

      9. div-inv 51.6%

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

      10. clear-num 51.6%

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

      11. unpow-1 51.6%

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

      12. div-inv 51.7%

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

      13. unpow-1 51.7%

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

      14. clear-num 51.7%

          \[\leadsto \sqrt{{\left(\frac{\color{blue}{\frac{x}{-t}}}{z}\right)}^{2}} \]

      15. add-sqr-sqrt 21.9%

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

      16. sqrt-unprod 45.0%

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

      17. sqr-neg 45.0%

          \[\leadsto \sqrt{{\left(\frac{\frac{x}{\sqrt{\color{blue}{t \cdot t}}}}{z}\right)}^{2}} \]

      18. sqrt-unprod 29.7%

          \[\leadsto \sqrt{{\left(\frac{\frac{x}{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}}{z}\right)}^{2}} \]

      19. add-sqr-sqrt 51.7%

          \[\leadsto \sqrt{{\left(\frac{\frac{x}{\color{blue}{t}}}{z}\right)}^{2}} \]

   8. Applied egg-rr 51.7%

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

      1. unpow2 51.7%

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

      2. sqrt-prod 32.1%

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

      3. add-sqr-sqrt 33.9%

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

      4. associate-/r* 32.0%

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

      5. add-sqr-sqrt 17.9%

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

      6. sqrt-unprod 41.5%

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

      7. sqr-neg 41.5%

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

      8. sqrt-unprod 32.4%

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

      9. add-sqr-sqrt 72.5%

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

      10. associate-/r* 76.0%

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

      11. frac-2neg 76.0%

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

      12. frac-2neg 76.0%

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

      13. remove-double-neg 76.0%

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

      14. distribute-frac-neg 76.0%

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

      15. frac-2neg 76.0%

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

   10. Applied egg-rr 76.0%

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

   if -9.99999999999999943e-68 < (*.f64 z t) < 5e-52

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 85.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-67} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 78.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

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-67)
   (/ (/ x z) (- t))
   (if (<= (* z t) 5e-52) (/ x y) (/ (/ x t) (- z)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -1e-67) {
		tmp = (x / z) / -t;
	} else if ((z * t) <= 5e-52) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	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-67)) then
        tmp = (x / z) / -t
    else if ((z * t) <= 5d-52) then
        tmp = x / y
    else
        tmp = (x / t) / -z
    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-67) {
		tmp = (x / z) / -t;
	} else if ((z * t) <= 5e-52) {
		tmp = x / y;
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -1e-67:
		tmp = (x / z) / -t
	elif (z * t) <= 5e-52:
		tmp = x / y
	else:
		tmp = (x / t) / -z
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -1e-67)
		tmp = Float64(Float64(x / z) / Float64(-t));
	elseif (Float64(z * t) <= 5e-52)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	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-67)
		tmp = (x / z) / -t;
	elseif ((z * t) <= 5e-52)
		tmp = x / y;
	else
		tmp = (x / t) / -z;
	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-67], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-52], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.99999999999999943e-68

   1. Initial program 90.9%

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

   2. Taylor expanded in y around 0 70.5%

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

      1. associate-*r/ 70.5%

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

      2. neg-mul-1 70.5%

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

   4. Simplified 70.5%

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

      1. frac-2neg 70.5%

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

      2. *-commutative 70.5%

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

      3. div-inv 70.4%

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

      4. remove-double-neg 70.4%

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

      5. distribute-rgt-neg-in 70.4%

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

   6. Applied egg-rr 70.4%

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

      1. associate-*r/ 70.5%

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

      2. *-rgt-identity 70.5%

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

      3. associate-/r* 73.2%

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

   8. Simplified 73.2%

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

   if -9.99999999999999943e-68 < (*.f64 z t) < 5e-52

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 85.6%

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

   if 5e-52 < (*.f64 z t)

   1. Initial program 94.6%

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

      1. clear-num 94.5%

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

      2. associate-/r/ 94.5%

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

   3. Applied egg-rr 94.5%

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

   4. Taylor expanded in y around 0 74.6%

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

      1. associate-/r* 74.6%

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

   6. Simplified 74.6%

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

      1. add-sqr-sqrt 48.9%

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

      2. sqrt-unprod 50.2%

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

      3. pow2 50.2%

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

      4. *-commutative 50.2%

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

      5. div-inv 50.3%

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

      6. associate-*r* 51.5%

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

      7. frac-2neg 51.5%

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

      8. metadata-eval 51.5%

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

      9. div-inv 51.6%

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

      10. clear-num 51.6%

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

      11. unpow-1 51.6%

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

      12. div-inv 51.6%

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

      13. unpow-1 51.6%

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

      14. clear-num 51.7%

          \[\leadsto \sqrt{{\left(\frac{\color{blue}{\frac{x}{-t}}}{z}\right)}^{2}} \]

      15. add-sqr-sqrt 20.7%

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

      16. sqrt-unprod 43.3%

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

      17. sqr-neg 43.3%

          \[\leadsto \sqrt{{\left(\frac{\frac{x}{\sqrt{\color{blue}{t \cdot t}}}}{z}\right)}^{2}} \]

      18. sqrt-unprod 30.8%

          \[\leadsto \sqrt{{\left(\frac{\frac{x}{\color{blue}{\sqrt{t} \cdot \sqrt{t}}}}{z}\right)}^{2}} \]

      19. add-sqr-sqrt 51.7%

          \[\leadsto \sqrt{{\left(\frac{\frac{x}{\color{blue}{t}}}{z}\right)}^{2}} \]

   8. Applied egg-rr 51.7%

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

      1. unpow2 51.7%

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

      2. sqrt-prod 30.7%

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

      3. add-sqr-sqrt 32.1%

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

      4. associate-/r* 30.8%

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

      5. add-sqr-sqrt 16.4%

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

      6. sqrt-unprod 41.1%

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

      7. sqr-neg 41.1%

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

      8. sqrt-unprod 33.4%

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

      9. add-sqr-sqrt 74.6%

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

      10. associate-/r* 73.4%

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

      11. frac-2neg 73.4%

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

      12. frac-2neg 73.4%

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

      13. remove-double-neg 73.4%

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

      14. distribute-frac-neg 73.4%

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

      15. frac-2neg 73.4%

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

   10. Applied egg-rr 73.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]

Alternative 5: 63.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+126} \lor \neg \left(z \cdot t \leq 10^{+122}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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+126) (not (<= (* z t) 1e+122)))
   (/ 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+126) || !((z * t) <= 1e+122)) {
		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+126)) .or. (.not. ((z * t) <= 1d+122))) 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+126) || !((z * t) <= 1e+122)) {
		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+126) or not ((z * t) <= 1e+122):
		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+126) || !(Float64(z * t) <= 1e+122))
		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+126) || ~(((z * t) <= 1e+122)))
		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+126], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+122]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999977e126 or 1.00000000000000001e122 < (*.f64 z t)

   1. Initial program 85.9%

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

      1. clear-num 85.3%

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

      2. associate-/r/ 85.9%

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

   3. Applied egg-rr 85.9%

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

   4. Taylor expanded in y around 0 80.6%

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

      1. associate-/r* 81.6%

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

   6. Simplified 81.6%

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

      1. *-commutative 81.6%

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

      2. div-inv 81.5%

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

      3. associate-*r* 93.0%

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

      4. frac-2neg 93.0%

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

      5. metadata-eval 93.0%

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

      6. div-inv 93.1%

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

      7. clear-num 93.2%

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

      8. unpow-1 93.2%

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

      9. inv-pow 93.2%

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

      10. pow-prod-down 92.4%

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

      11. expm1-log1p-u 88.2%

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

      12. expm1-udef 60.7%

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

   8. Applied egg-rr 55.8%

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

      1. expm1-def 54.0%

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

      2. expm1-log1p 54.1%

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

      3. associate-/r* 54.3%

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

   10. Simplified 54.3%

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

   if -4.99999999999999977e126 < (*.f64 z t) < 1.00000000000000001e122

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 68.0%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+126} \lor \neg \left(z \cdot t \leq 10^{+122}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 54.9% 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 54.8%

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

3. Final simplification 54.8%

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

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 2023199 
(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))))