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

Percentage Accurate: 96.0% → 99.4%

Time: 6.8s

Alternatives: 8

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+251}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+251)
   (* (/ -1.0 t) (/ x z))
   (if (<= (* z t) 2e+214) (/ x (- y (* z t))) (/ (/ x z) (- t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+251) {
		tmp = (-1.0 / t) * (x / z);
	} else if ((z * t) <= 2e+214) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / z) / -t;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * t) <= (-2d+251)) then
        tmp = ((-1.0d0) / t) * (x / z)
    else if ((z * t) <= 2d+214) then
        tmp = x / (y - (z * t))
    else
        tmp = (x / z) / -t
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+251) {
		tmp = (-1.0 / t) * (x / z);
	} else if ((z * t) <= 2e+214) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / z) / -t;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e+251:
		tmp = (-1.0 / t) * (x / z)
	elif (z * t) <= 2e+214:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / z) / -t
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+251)
		tmp = Float64(Float64(-1.0 / t) * Float64(x / z));
	elseif (Float64(z * t) <= 2e+214)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / z) / Float64(-t));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -2e+251)
		tmp = (-1.0 / t) * (x / z);
	elseif ((z * t) <= 2e+214)
		tmp = x / (y - (z * t));
	else
		tmp = (x / z) / -t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+251], N[(N[(-1.0 / t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+214], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.0000000000000001e251

   1. Initial program 79.2%

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

   2. Taylor expanded in y around 0 79.2%

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

      1. associate-*r/ 79.2%

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

      2. neg-mul-1 79.2%

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

   4. Simplified 79.2%

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

      1. neg-mul-1 79.2%

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

      2. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

   if -2.0000000000000001e251 < (*.f64 z t) < 1.9999999999999999e214

   1. Initial program 99.9%

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

   if 1.9999999999999999e214 < (*.f64 z t)

   1. Initial program 76.5%

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

   2. Taylor expanded in y around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. neg-mul-1 76.5%

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

   4. Simplified 76.5%

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

      1. neg-mul-1 76.5%

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

      2. *-commutative 76.5%

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

      3. times-frac 97.4%

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

   6. Applied egg-rr 97.4%

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

      1. *-commutative 97.4%

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

      2. frac-2neg 97.4%

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

      3. associate-*l/ 97.4%

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

      4. add-sqr-sqrt 66.8%

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

      5. sqrt-unprod 72.0%

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

      6. sqr-neg 72.0%

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

      7. sqrt-unprod 26.2%

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

      8. add-sqr-sqrt 59.2%

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

      9. *-commutative 59.2%

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

      10. associate-*l/ 59.2%

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

      11. neg-mul-1 59.2%

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

      12. add-sqr-sqrt 33.0%

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

      13. sqrt-unprod 58.7%

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

      14. sqr-neg 58.7%

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

      15. sqrt-unprod 30.4%

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

      16. add-sqr-sqrt 97.5%

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

   8. Applied egg-rr 97.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+251}:\\ \;\;\;\;\frac{-1}{t} \cdot \frac{x}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]

Alternative 2: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := -\frac{\frac{x}{t}}{z}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq -0.001:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ (/ x t) z))))
   (if (<= (* z t) -1e+120)
     t_1
     (if (<= (* z t) -0.001)
       (/ x y)
       (if (<= (* z t) -1e-31)
         (/ (- x) (* z t))
         (if (<= (* z t) 5e-77) (/ x y) t_1))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double t_1 = -((x / t) / z);
	double tmp;
	if ((z * t) <= -1e+120) {
		tmp = t_1;
	} else if ((z * t) <= -0.001) {
		tmp = x / y;
	} else if ((z * t) <= -1e-31) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 5e-77) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = -((x / t) / z)
    if ((z * t) <= (-1d+120)) then
        tmp = t_1
    else if ((z * t) <= (-0.001d0)) then
        tmp = x / y
    else if ((z * t) <= (-1d-31)) then
        tmp = -x / (z * t)
    else if ((z * t) <= 5d-77) then
        tmp = x / y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = -((x / t) / z);
	double tmp;
	if ((z * t) <= -1e+120) {
		tmp = t_1;
	} else if ((z * t) <= -0.001) {
		tmp = x / y;
	} else if ((z * t) <= -1e-31) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 5e-77) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	t_1 = -((x / t) / z)
	tmp = 0
	if (z * t) <= -1e+120:
		tmp = t_1
	elif (z * t) <= -0.001:
		tmp = x / y
	elif (z * t) <= -1e-31:
		tmp = -x / (z * t)
	elif (z * t) <= 5e-77:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	t_1 = Float64(-Float64(Float64(x / t) / z))
	tmp = 0.0
	if (Float64(z * t) <= -1e+120)
		tmp = t_1;
	elseif (Float64(z * t) <= -0.001)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= -1e-31)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (Float64(z * t) <= 5e-77)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = -((x / t) / z);
	tmp = 0.0;
	if ((z * t) <= -1e+120)
		tmp = t_1;
	elseif ((z * t) <= -0.001)
		tmp = x / y;
	elseif ((z * t) <= -1e-31)
		tmp = -x / (z * t);
	elseif ((z * t) <= 5e-77)
		tmp = x / y;
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = (-N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision])}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+120], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -0.001], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-31], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-77], N[(x / y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -9.9999999999999998e119 or 4.99999999999999963e-77 < (*.f64 z t)

   1. Initial program 91.5%

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

   2. Taylor expanded in y around 0 73.5%

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

      1. mul-1-neg 73.5%

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

      2. associate-/r* 76.6%

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

   4. Simplified 76.6%

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

   if -9.9999999999999998e119 < (*.f64 z t) < -1e-3 or -1e-31 < (*.f64 z t) < 4.99999999999999963e-77

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 85.8%

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

   if -1e-3 < (*.f64 z t) < -1e-31

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 99.7%

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

      1. associate-*r/ 99.7%

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

      2. neg-mul-1 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+120}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq -0.001:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 3: 76.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+120}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \mathbf{elif}\;z \cdot t \leq -0.001:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

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+120)
   (* (/ x t) (/ -1.0 z))
   (if (<= (* z t) -0.001)
     (/ x y)
     (if (<= (* z t) -1e-31)
       (/ (- x) (* z t))
       (if (<= (* z t) 5e-77) (/ 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+120) {
		tmp = (x / t) * (-1.0 / z);
	} else if ((z * t) <= -0.001) {
		tmp = x / y;
	} else if ((z * t) <= -1e-31) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 5e-77) {
		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+120)) then
        tmp = (x / t) * ((-1.0d0) / z)
    else if ((z * t) <= (-0.001d0)) then
        tmp = x / y
    else if ((z * t) <= (-1d-31)) then
        tmp = -x / (z * t)
    else if ((z * t) <= 5d-77) 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+120) {
		tmp = (x / t) * (-1.0 / z);
	} else if ((z * t) <= -0.001) {
		tmp = x / y;
	} else if ((z * t) <= -1e-31) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 5e-77) {
		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+120:
		tmp = (x / t) * (-1.0 / z)
	elif (z * t) <= -0.001:
		tmp = x / y
	elif (z * t) <= -1e-31:
		tmp = -x / (z * t)
	elif (z * t) <= 5e-77:
		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+120)
		tmp = Float64(Float64(x / t) * Float64(-1.0 / z));
	elseif (Float64(z * t) <= -0.001)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= -1e-31)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (Float64(z * t) <= 5e-77)
		tmp = Float64(x / y);
	else
		tmp = Float64(-Float64(Float64(x / t) / 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+120)
		tmp = (x / t) * (-1.0 / z);
	elseif ((z * t) <= -0.001)
		tmp = x / y;
	elseif ((z * t) <= -1e-31)
		tmp = -x / (z * t);
	elseif ((z * t) <= 5e-77)
		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+120], N[(N[(x / t), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -0.001], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-31], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-77], N[(x / y), $MachinePrecision], (-N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision])]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -9.9999999999999998e119

   1. Initial program 89.7%

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

   2. Taylor expanded in y around 0 79.7%

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

      1. associate-*r/ 79.7%

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

      2. neg-mul-1 79.7%

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

   4. Simplified 79.7%

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

      1. neg-mul-1 79.7%

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

      2. *-commutative 79.7%

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

      3. times-frac 85.6%

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

   6. Applied egg-rr 85.6%

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

   if -9.9999999999999998e119 < (*.f64 z t) < -1e-3 or -1e-31 < (*.f64 z t) < 4.99999999999999963e-77

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 85.8%

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

   if -1e-3 < (*.f64 z t) < -1e-31

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 99.7%

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

      1. associate-*r/ 99.7%

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

      2. neg-mul-1 99.7%

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

   4. Simplified 99.7%

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

   if 4.99999999999999963e-77 < (*.f64 z t)

   1. Initial program 92.7%

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

   2. Taylor expanded in y around 0 69.6%

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

      1. mul-1-neg 69.6%

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

      2. associate-/r* 71.0%

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

   4. Simplified 71.0%

      \[\leadsto \color{blue}{-\frac{\frac{x}{t}}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+120}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \mathbf{elif}\;z \cdot t \leq -0.001:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 4: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+120}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \mathbf{elif}\;z \cdot t \leq -0.001:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

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+120)
   (/ -1.0 (* z (/ t x)))
   (if (<= (* z t) -0.001)
     (/ x y)
     (if (<= (* z t) -1e-31)
       (/ (- x) (* z t))
       (if (<= (* z t) 5e-77) (/ 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+120) {
		tmp = -1.0 / (z * (t / x));
	} else if ((z * t) <= -0.001) {
		tmp = x / y;
	} else if ((z * t) <= -1e-31) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 5e-77) {
		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+120)) then
        tmp = (-1.0d0) / (z * (t / x))
    else if ((z * t) <= (-0.001d0)) then
        tmp = x / y
    else if ((z * t) <= (-1d-31)) then
        tmp = -x / (z * t)
    else if ((z * t) <= 5d-77) 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+120) {
		tmp = -1.0 / (z * (t / x));
	} else if ((z * t) <= -0.001) {
		tmp = x / y;
	} else if ((z * t) <= -1e-31) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 5e-77) {
		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+120:
		tmp = -1.0 / (z * (t / x))
	elif (z * t) <= -0.001:
		tmp = x / y
	elif (z * t) <= -1e-31:
		tmp = -x / (z * t)
	elif (z * t) <= 5e-77:
		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+120)
		tmp = Float64(-1.0 / Float64(z * Float64(t / x)));
	elseif (Float64(z * t) <= -0.001)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= -1e-31)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (Float64(z * t) <= 5e-77)
		tmp = Float64(x / y);
	else
		tmp = Float64(-Float64(Float64(x / t) / 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+120)
		tmp = -1.0 / (z * (t / x));
	elseif ((z * t) <= -0.001)
		tmp = x / y;
	elseif ((z * t) <= -1e-31)
		tmp = -x / (z * t);
	elseif ((z * t) <= 5e-77)
		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+120], N[(-1.0 / N[(z * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -0.001], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-31], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-77], N[(x / y), $MachinePrecision], (-N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision])]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -9.9999999999999998e119

   1. Initial program 89.7%

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

   2. Taylor expanded in y around 0 79.7%

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

      1. associate-*r/ 79.7%

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

      2. neg-mul-1 79.7%

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

   4. Simplified 79.7%

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

      1. neg-mul-1 79.7%

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

      2. *-commutative 79.7%

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

      3. times-frac 85.6%

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

   6. Applied egg-rr 85.6%

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

      1. *-commutative 85.6%

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

      2. clear-num 85.6%

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

      3. frac-times 85.6%

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

      4. metadata-eval 85.6%

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

   8. Applied egg-rr 85.6%

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

   if -9.9999999999999998e119 < (*.f64 z t) < -1e-3 or -1e-31 < (*.f64 z t) < 4.99999999999999963e-77

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 85.8%

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

   if -1e-3 < (*.f64 z t) < -1e-31

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 99.7%

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

      1. associate-*r/ 99.7%

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

      2. neg-mul-1 99.7%

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

   4. Simplified 99.7%

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

   if 4.99999999999999963e-77 < (*.f64 z t)

   1. Initial program 92.7%

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

   2. Taylor expanded in y around 0 69.6%

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

      1. mul-1-neg 69.6%

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

      2. associate-/r* 71.0%

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

   4. Simplified 71.0%

      \[\leadsto \color{blue}{-\frac{\frac{x}{t}}{z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+120}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \mathbf{elif}\;z \cdot t \leq -0.001:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-77}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 5: 77.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+120} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{-77}\right):\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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+120) (not (<= (* z t) 5e-77)))
   (- (/ (/ 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+120) || !((z * t) <= 5e-77)) {
		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+120)) .or. (.not. ((z * t) <= 5d-77))) 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+120) || !((z * t) <= 5e-77)) {
		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+120) or not ((z * t) <= 5e-77):
		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+120) || !(Float64(z * t) <= 5e-77))
		tmp = Float64(-Float64(Float64(x / t) / z));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -1e+120) || ~(((z * t) <= 5e-77)))
		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+120], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e-77]], $MachinePrecision]], (-N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]), N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -9.9999999999999998e119 or 4.99999999999999963e-77 < (*.f64 z t)

   1. Initial program 91.5%

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

   2. Taylor expanded in y around 0 73.5%

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

      1. mul-1-neg 73.5%

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

      2. associate-/r* 76.6%

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

   4. Simplified 76.6%

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

   if -9.9999999999999998e119 < (*.f64 z t) < 4.99999999999999963e-77

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 82.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+120} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{-77}\right):\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 63.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+156} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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) -2e+156) (not (<= (* z t) 5e+217)))
   (/ x (* z t))
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e+156) || !((z * t) <= 5e+217)) {
		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) <= (-2d+156)) .or. (.not. ((z * t) <= 5d+217))) 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) <= -2e+156) || !((z * t) <= 5e+217)) {
		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) <= -2e+156) or not ((z * t) <= 5e+217):
		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) <= -2e+156) || !(Float64(z * t) <= 5e+217))
		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) <= -2e+156) || ~(((z * t) <= 5e+217)))
		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], -2e+156], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5e+217]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e156 or 5.00000000000000041e217 < (*.f64 z t)

   1. Initial program 83.3%

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

   2. Taylor expanded in y around 0 78.7%

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

      1. associate-*r/ 78.7%

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

      2. neg-mul-1 78.7%

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

   4. Simplified 78.7%

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

      1. expm1-log1p-u 78.6%

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

      2. expm1-udef 63.0%

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

      3. add-sqr-sqrt 31.4%

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

      4. sqrt-unprod 59.2%

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

      5. sqr-neg 59.2%

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

      6. sqrt-unprod 30.0%

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

      7. add-sqr-sqrt 59.9%

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

      8. *-commutative 59.9%

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

   6. Applied egg-rr 59.9%

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

      1. expm1-def 58.4%

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

      2. expm1-log1p 58.5%

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

      3. *-commutative 58.5%

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

   8. Simplified 58.5%

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

   if -2e156 < (*.f64 z t) < 5.00000000000000041e217

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 68.3%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+156} \lor \neg \left(z \cdot t \leq 5 \cdot 10^{+217}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 63.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \end{array} \]

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) -2e+156)
   (/ (/ x t) z)
   (if (<= (* z t) 5e+217) (/ x y) (/ x (* z t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+156) {
		tmp = (x / t) / z;
	} else if ((z * t) <= 5e+217) {
		tmp = x / y;
	} else {
		tmp = x / (z * t);
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z * t) <= (-2d+156)) then
        tmp = (x / t) / z
    else if ((z * t) <= 5d+217) then
        tmp = x / y
    else
        tmp = x / (z * t)
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+156) {
		tmp = (x / t) / z;
	} else if ((z * t) <= 5e+217) {
		tmp = x / y;
	} else {
		tmp = x / (z * t);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e+156:
		tmp = (x / t) / z
	elif (z * t) <= 5e+217:
		tmp = x / y
	else:
		tmp = x / (z * t)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+156)
		tmp = Float64(Float64(x / t) / z);
	elseif (Float64(z * t) <= 5e+217)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / 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) <= -2e+156)
		tmp = (x / t) / z;
	elseif ((z * t) <= 5e+217)
		tmp = x / y;
	else
		tmp = x / (z * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e+156], N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+217], N[(x / y), $MachinePrecision], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2e156

   1. Initial program 87.6%

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

      1. clear-num 87.7%

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

      2. associate-/r/ 87.6%

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

   3. Applied egg-rr 87.6%

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

   4. Taylor expanded in y around 0 80.5%

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

      1. associate-*r/ 80.5%

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

      2. rem-square-sqrt 0.0%

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

      3. unpow2 0.0%

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

      4. *-commutative 0.0%

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

      5. associate-/r* 0.0%

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

      6. *-commutative 0.0%

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

      7. unpow2 0.0%

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

      8. rem-square-sqrt 90.2%

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

      9. neg-mul-1 90.2%

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

   6. Simplified 90.2%

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

      1. associate-/r* 80.5%

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

      2. expm1-log1p-u 80.4%

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

      3. expm1-udef 60.8%

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

      4. add-sqr-sqrt 26.9%

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

      5. sqrt-unprod 57.9%

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

      6. sqr-neg 57.9%

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

      7. sqrt-unprod 31.5%

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

      8. add-sqr-sqrt 58.6%

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

      9. *-commutative 58.6%

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

   8. Applied egg-rr 58.6%

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

      1. expm1-def 56.4%

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

      2. expm1-log1p 56.4%

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

      3. associate-/l/ 56.1%

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

   10. Simplified 56.1%

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

   if -2e156 < (*.f64 z t) < 5.00000000000000041e217

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 68.3%

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

   if 5.00000000000000041e217 < (*.f64 z t)

   1. Initial program 75.5%

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

   2. Taylor expanded in y around 0 75.5%

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

      1. associate-*r/ 75.5%

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

      2. neg-mul-1 75.5%

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

   4. Simplified 75.5%

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

      1. expm1-log1p-u 75.5%

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

      2. expm1-udef 66.8%

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

      3. add-sqr-sqrt 39.3%

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

      4. sqrt-unprod 61.5%

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

      5. sqr-neg 61.5%

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

      6. sqrt-unprod 27.5%

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

      7. add-sqr-sqrt 62.3%

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

      8. *-commutative 62.3%

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

   6. Applied egg-rr 62.3%

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

      1. expm1-def 62.1%

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

      2. expm1-log1p 62.2%

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

      3. *-commutative 62.2%

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

   8. Simplified 62.2%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+217}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 8: 55.3% 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 95.9%

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

2. Taylor expanded in y around inf 55.3%

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

3. Final simplification 55.3%

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

Developer target: 96.8% 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 2023292 
(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))))