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

Percentage Accurate: 95.8% → 99.6%

Time: 8.7s

Alternatives: 9

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 95.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+200}\right):\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \end{array} \]

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) (- INFINITY)) (not (<= (* z t) 2e+200)))
   (- (/ (/ x t) z))
   (/ x (fma z (- t) y))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 2e+200)) {
		tmp = -((x / t) / z);
	} else {
		tmp = x / fma(z, -t, y);
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= Float64(-Inf)) || !(Float64(z * t) <= 2e+200))
		tmp = Float64(-Float64(Float64(x / t) / z));
	else
		tmp = Float64(x / fma(z, Float64(-t), y));
	end
	return tmp
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+200]], $MachinePrecision]], (-N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]), N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 1.9999999999999999e200 < (*.f64 z t)

   1. Initial program 76.4%

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

   2. Taylor expanded in y around 0 76.4%

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

      1. mul-1-neg 76.4%

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

      2. associate-/r* 99.8%

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

      3. distribute-neg-frac 99.8%

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

   4. Simplified 99.8%

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

   if -inf.0 < (*.f64 z t) < 1.9999999999999999e200

   1. Initial program 99.8%

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

      1. cancel-sign-sub-inv 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-out 99.8%

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

      4. distribute-rgt-neg-out 99.8%

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

      5. fma-def 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(z, -t, y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+200}\right):\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \end{array} \]

Alternative 2: 74.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ t_2 := -\frac{\frac{x}{t}}{z}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

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) (* z t))) (t_2 (- (/ (/ x t) z))))
   (if (<= (* z t) -1e+54)
     t_2
     (if (<= (* z t) -2e-34)
       (/ x y)
       (if (<= (* z t) -1e-61)
         t_1
         (if (<= (* z t) 5e-141)
           (/ x y)
           (if (<= (* z t) 5e-85)
             t_1
             (if (<= (* z t) 5e-55)
               (/ x y)
               (if (<= (* z t) 5e+76)
                 t_1
                 (if (<= (* z t) 2e+200) (/ x y) t_2))))))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double t_1 = -x / (z * t);
	double t_2 = -((x / t) / z);
	double tmp;
	if ((z * t) <= -1e+54) {
		tmp = t_2;
	} else if ((z * t) <= -2e-34) {
		tmp = x / y;
	} else if ((z * t) <= -1e-61) {
		tmp = t_1;
	} else if ((z * t) <= 5e-141) {
		tmp = x / y;
	} else if ((z * t) <= 5e-85) {
		tmp = t_1;
	} else if ((z * t) <= 5e-55) {
		tmp = x / y;
	} else if ((z * t) <= 5e+76) {
		tmp = t_1;
	} else if ((z * t) <= 2e+200) {
		tmp = x / y;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = -x / (z * t)
    t_2 = -((x / t) / z)
    if ((z * t) <= (-1d+54)) then
        tmp = t_2
    else if ((z * t) <= (-2d-34)) then
        tmp = x / y
    else if ((z * t) <= (-1d-61)) then
        tmp = t_1
    else if ((z * t) <= 5d-141) then
        tmp = x / y
    else if ((z * t) <= 5d-85) then
        tmp = t_1
    else if ((z * t) <= 5d-55) then
        tmp = x / y
    else if ((z * t) <= 5d+76) then
        tmp = t_1
    else if ((z * t) <= 2d+200) then
        tmp = x / y
    else
        tmp = t_2
    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 / (z * t);
	double t_2 = -((x / t) / z);
	double tmp;
	if ((z * t) <= -1e+54) {
		tmp = t_2;
	} else if ((z * t) <= -2e-34) {
		tmp = x / y;
	} else if ((z * t) <= -1e-61) {
		tmp = t_1;
	} else if ((z * t) <= 5e-141) {
		tmp = x / y;
	} else if ((z * t) <= 5e-85) {
		tmp = t_1;
	} else if ((z * t) <= 5e-55) {
		tmp = x / y;
	} else if ((z * t) <= 5e+76) {
		tmp = t_1;
	} else if ((z * t) <= 2e+200) {
		tmp = x / y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	t_1 = -x / (z * t)
	t_2 = -((x / t) / z)
	tmp = 0
	if (z * t) <= -1e+54:
		tmp = t_2
	elif (z * t) <= -2e-34:
		tmp = x / y
	elif (z * t) <= -1e-61:
		tmp = t_1
	elif (z * t) <= 5e-141:
		tmp = x / y
	elif (z * t) <= 5e-85:
		tmp = t_1
	elif (z * t) <= 5e-55:
		tmp = x / y
	elif (z * t) <= 5e+76:
		tmp = t_1
	elif (z * t) <= 2e+200:
		tmp = x / y
	else:
		tmp = t_2
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(-x) / Float64(z * t))
	t_2 = Float64(-Float64(Float64(x / t) / z))
	tmp = 0.0
	if (Float64(z * t) <= -1e+54)
		tmp = t_2;
	elseif (Float64(z * t) <= -2e-34)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= -1e-61)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-141)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 5e-85)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-55)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 5e+76)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+200)
		tmp = Float64(x / y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = -x / (z * t);
	t_2 = -((x / t) / z);
	tmp = 0.0;
	if ((z * t) <= -1e+54)
		tmp = t_2;
	elseif ((z * t) <= -2e-34)
		tmp = x / y;
	elseif ((z * t) <= -1e-61)
		tmp = t_1;
	elseif ((z * t) <= 5e-141)
		tmp = x / y;
	elseif ((z * t) <= 5e-85)
		tmp = t_1;
	elseif ((z * t) <= 5e-55)
		tmp = x / y;
	elseif ((z * t) <= 5e+76)
		tmp = t_1;
	elseif ((z * t) <= 2e+200)
		tmp = x / y;
	else
		tmp = t_2;
	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[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision])}, If[LessEqual[N[(z * t), $MachinePrecision], -1e+54], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], -2e-34], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-61], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-141], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-85], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-55], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+76], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+200], N[(x / y), $MachinePrecision], t$95$2]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.0000000000000001e54 or 1.9999999999999999e200 < (*.f64 z t)

   1. Initial program 86.5%

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

   2. Taylor expanded in y around 0 81.3%

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

      1. mul-1-neg 81.3%

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

      2. associate-/r* 90.6%

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

      3. distribute-neg-frac 90.6%

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

   4. Simplified 90.6%

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

   if -1.0000000000000001e54 < (*.f64 z t) < -1.99999999999999986e-34 or -1e-61 < (*.f64 z t) < 4.9999999999999999e-141 or 5.0000000000000002e-85 < (*.f64 z t) < 5.0000000000000002e-55 or 4.99999999999999991e76 < (*.f64 z t) < 1.9999999999999999e200

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 84.2%

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

   if -1.99999999999999986e-34 < (*.f64 z t) < -1e-61 or 4.9999999999999999e-141 < (*.f64 z t) < 5.0000000000000002e-85 or 5.0000000000000002e-55 < (*.f64 z t) < 4.99999999999999991e76

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 67.4%

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

      1. associate-*r/ 67.4%

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

      2. neg-mul-1 67.4%

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

   4. Simplified 67.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+54}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 3: 73.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ t_2 := \frac{\frac{x}{z}}{-t}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-85}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x) (* z t))) (t_2 (/ (/ x z) (- t))))
   (if (<= (* z t) -2e+64)
     t_2
     (if (<= (* z t) -2e-34)
       (/ x y)
       (if (<= (* z t) -1e-61)
         t_1
         (if (<= (* z t) 5e-141)
           (/ x y)
           (if (<= (* z t) 5e-85)
             t_2
             (if (<= (* z t) 5e-55)
               (/ x y)
               (if (<= (* z t) 5e+76)
                 t_1
                 (if (<= (* z t) 2e+200) (/ x y) (- (/ (/ x t) z))))))))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double t_1 = -x / (z * t);
	double t_2 = (x / z) / -t;
	double tmp;
	if ((z * t) <= -2e+64) {
		tmp = t_2;
	} else if ((z * t) <= -2e-34) {
		tmp = x / y;
	} else if ((z * t) <= -1e-61) {
		tmp = t_1;
	} else if ((z * t) <= 5e-141) {
		tmp = x / y;
	} else if ((z * t) <= 5e-85) {
		tmp = t_2;
	} else if ((z * t) <= 5e-55) {
		tmp = x / y;
	} else if ((z * t) <= 5e+76) {
		tmp = t_1;
	} else if ((z * t) <= 2e+200) {
		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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = -x / (z * t)
    t_2 = (x / z) / -t
    if ((z * t) <= (-2d+64)) then
        tmp = t_2
    else if ((z * t) <= (-2d-34)) then
        tmp = x / y
    else if ((z * t) <= (-1d-61)) then
        tmp = t_1
    else if ((z * t) <= 5d-141) then
        tmp = x / y
    else if ((z * t) <= 5d-85) then
        tmp = t_2
    else if ((z * t) <= 5d-55) then
        tmp = x / y
    else if ((z * t) <= 5d+76) then
        tmp = t_1
    else if ((z * t) <= 2d+200) 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 t_1 = -x / (z * t);
	double t_2 = (x / z) / -t;
	double tmp;
	if ((z * t) <= -2e+64) {
		tmp = t_2;
	} else if ((z * t) <= -2e-34) {
		tmp = x / y;
	} else if ((z * t) <= -1e-61) {
		tmp = t_1;
	} else if ((z * t) <= 5e-141) {
		tmp = x / y;
	} else if ((z * t) <= 5e-85) {
		tmp = t_2;
	} else if ((z * t) <= 5e-55) {
		tmp = x / y;
	} else if ((z * t) <= 5e+76) {
		tmp = t_1;
	} else if ((z * t) <= 2e+200) {
		tmp = x / y;
	} else {
		tmp = -((x / t) / z);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	t_1 = -x / (z * t)
	t_2 = (x / z) / -t
	tmp = 0
	if (z * t) <= -2e+64:
		tmp = t_2
	elif (z * t) <= -2e-34:
		tmp = x / y
	elif (z * t) <= -1e-61:
		tmp = t_1
	elif (z * t) <= 5e-141:
		tmp = x / y
	elif (z * t) <= 5e-85:
		tmp = t_2
	elif (z * t) <= 5e-55:
		tmp = x / y
	elif (z * t) <= 5e+76:
		tmp = t_1
	elif (z * t) <= 2e+200:
		tmp = x / y
	else:
		tmp = -((x / t) / z)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(-x) / Float64(z * t))
	t_2 = Float64(Float64(x / z) / Float64(-t))
	tmp = 0.0
	if (Float64(z * t) <= -2e+64)
		tmp = t_2;
	elseif (Float64(z * t) <= -2e-34)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= -1e-61)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-141)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 5e-85)
		tmp = t_2;
	elseif (Float64(z * t) <= 5e-55)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 5e+76)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e+200)
		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)
	t_1 = -x / (z * t);
	t_2 = (x / z) / -t;
	tmp = 0.0;
	if ((z * t) <= -2e+64)
		tmp = t_2;
	elseif ((z * t) <= -2e-34)
		tmp = x / y;
	elseif ((z * t) <= -1e-61)
		tmp = t_1;
	elseif ((z * t) <= 5e-141)
		tmp = x / y;
	elseif ((z * t) <= 5e-85)
		tmp = t_2;
	elseif ((z * t) <= 5e-55)
		tmp = x / y;
	elseif ((z * t) <= 5e+76)
		tmp = t_1;
	elseif ((z * t) <= 2e+200)
		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_] := Block[{t$95$1 = N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+64], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], -2e-34], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-61], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-141], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-85], t$95$2, If[LessEqual[N[(z * t), $MachinePrecision], 5e-55], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+76], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e+200], N[(x / y), $MachinePrecision], (-N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision])]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -2.00000000000000004e64 or 4.9999999999999999e-141 < (*.f64 z t) < 5.0000000000000002e-85

   1. Initial program 91.5%

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

      1. *-un-lft-identity 91.5%

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

      2. *-commutative 91.5%

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

      3. add-sqr-sqrt 51.6%

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

      4. sqrt-unprod 61.3%

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

      5. sqr-neg 61.3%

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

      6. sqrt-unprod 14.8%

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

      7. add-sqr-sqrt 40.8%

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

      8. distribute-rgt-neg-in 40.8%

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

      9. neg-mul-1 40.8%

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

      10. add-sqr-sqrt 26.0%

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

      11. sqrt-unprod 54.1%

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

      12. sqr-neg 54.1%

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

      13. sqrt-unprod 39.6%

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

      14. add-sqr-sqrt 91.5%

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

      15. distribute-rgt-neg-in 91.5%

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

      16. prod-diff 73.3%

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

   3. Applied egg-rr 73.3%

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

      1. fma-udef 73.3%

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

      2. distribute-rgt-neg-in 73.3%

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

      3. metadata-eval 73.3%

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

      4. distribute-rgt-in 73.3%

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

      5. +-commutative 73.3%

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

      6. fma-def 73.3%

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

      7. *-lft-identity 73.3%

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

      8. fma-def 73.3%

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

      9. *-commutative 73.3%

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

      10. fma-def 73.3%

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

      11. fma-udef 73.3%

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

      12. distribute-lft-out 73.3%

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

      13. *-commutative 73.3%

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

      14. metadata-eval 73.3%

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

   5. Simplified 73.3%

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

      1. +-commutative 73.3%

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

      2. associate-*l* 73.3%

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

      3. fma-def 91.4%

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

   7. Applied egg-rr 91.4%

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

   8. Taylor expanded in t around inf 83.8%

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

      1. distribute-rgt-in 65.6%

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

      2. *-commutative 65.6%

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

      3. associate-*l* 65.6%

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

      4. distribute-lft-in 83.8%

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

      5. associate-/r* 91.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{t + -2 \cdot t}} \]

      6. distribute-rgt1-in 91.1%

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

      7. metadata-eval 91.1%

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

      8. mul-1-neg 91.1%

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

   10. Simplified 91.1%

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

   if -2.00000000000000004e64 < (*.f64 z t) < -1.99999999999999986e-34 or -1e-61 < (*.f64 z t) < 4.9999999999999999e-141 or 5.0000000000000002e-85 < (*.f64 z t) < 5.0000000000000002e-55 or 4.99999999999999991e76 < (*.f64 z t) < 1.9999999999999999e200

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 83.8%

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

   if -1.99999999999999986e-34 < (*.f64 z t) < -1e-61 or 5.0000000000000002e-55 < (*.f64 z t) < 4.99999999999999991e76

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 64.1%

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

      1. associate-*r/ 64.1%

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

      2. neg-mul-1 64.1%

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

   4. Simplified 64.1%

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

   if 1.9999999999999999e200 < (*.f64 z t)

   1. Initial program 80.0%

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

   2. Taylor expanded in y around 0 80.0%

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

      1. mul-1-neg 80.0%

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

      2. associate-/r* 99.9%

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

      3. distribute-neg-frac 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 4: 73.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x}{z}}{-t}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x z) (- t))))
   (if (<= (* z t) -2e+64)
     t_1
     (if (<= (* z t) -2e-34)
       (/ x y)
       (if (<= (* z t) -1e-61)
         (* x (/ (/ -1.0 t) z))
         (if (<= (* z t) 5e-141)
           (/ x y)
           (if (<= (* z t) 5e-85)
             t_1
             (if (<= (* z t) 5e-55)
               (/ x y)
               (if (<= (* z t) 5e+76)
                 (/ (- x) (* z t))
                 (if (<= (* z t) 2e+200) (/ x y) (- (/ (/ x t) z))))))))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (x / z) / -t;
	double tmp;
	if ((z * t) <= -2e+64) {
		tmp = t_1;
	} else if ((z * t) <= -2e-34) {
		tmp = x / y;
	} else if ((z * t) <= -1e-61) {
		tmp = x * ((-1.0 / t) / z);
	} else if ((z * t) <= 5e-141) {
		tmp = x / y;
	} else if ((z * t) <= 5e-85) {
		tmp = t_1;
	} else if ((z * t) <= 5e-55) {
		tmp = x / y;
	} else if ((z * t) <= 5e+76) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e+200) {
		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) :: t_1
    real(8) :: tmp
    t_1 = (x / z) / -t
    if ((z * t) <= (-2d+64)) then
        tmp = t_1
    else if ((z * t) <= (-2d-34)) then
        tmp = x / y
    else if ((z * t) <= (-1d-61)) then
        tmp = x * (((-1.0d0) / t) / z)
    else if ((z * t) <= 5d-141) then
        tmp = x / y
    else if ((z * t) <= 5d-85) then
        tmp = t_1
    else if ((z * t) <= 5d-55) then
        tmp = x / y
    else if ((z * t) <= 5d+76) then
        tmp = -x / (z * t)
    else if ((z * t) <= 2d+200) 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 t_1 = (x / z) / -t;
	double tmp;
	if ((z * t) <= -2e+64) {
		tmp = t_1;
	} else if ((z * t) <= -2e-34) {
		tmp = x / y;
	} else if ((z * t) <= -1e-61) {
		tmp = x * ((-1.0 / t) / z);
	} else if ((z * t) <= 5e-141) {
		tmp = x / y;
	} else if ((z * t) <= 5e-85) {
		tmp = t_1;
	} else if ((z * t) <= 5e-55) {
		tmp = x / y;
	} else if ((z * t) <= 5e+76) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 2e+200) {
		tmp = x / y;
	} else {
		tmp = -((x / t) / z);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	t_1 = (x / z) / -t
	tmp = 0
	if (z * t) <= -2e+64:
		tmp = t_1
	elif (z * t) <= -2e-34:
		tmp = x / y
	elif (z * t) <= -1e-61:
		tmp = x * ((-1.0 / t) / z)
	elif (z * t) <= 5e-141:
		tmp = x / y
	elif (z * t) <= 5e-85:
		tmp = t_1
	elif (z * t) <= 5e-55:
		tmp = x / y
	elif (z * t) <= 5e+76:
		tmp = -x / (z * t)
	elif (z * t) <= 2e+200:
		tmp = x / y
	else:
		tmp = -((x / t) / z)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) / Float64(-t))
	tmp = 0.0
	if (Float64(z * t) <= -2e+64)
		tmp = t_1;
	elseif (Float64(z * t) <= -2e-34)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= -1e-61)
		tmp = Float64(x * Float64(Float64(-1.0 / t) / z));
	elseif (Float64(z * t) <= 5e-141)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 5e-85)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-55)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 5e+76)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (Float64(z * t) <= 2e+200)
		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)
	t_1 = (x / z) / -t;
	tmp = 0.0;
	if ((z * t) <= -2e+64)
		tmp = t_1;
	elseif ((z * t) <= -2e-34)
		tmp = x / y;
	elseif ((z * t) <= -1e-61)
		tmp = x * ((-1.0 / t) / z);
	elseif ((z * t) <= 5e-141)
		tmp = x / y;
	elseif ((z * t) <= 5e-85)
		tmp = t_1;
	elseif ((z * t) <= 5e-55)
		tmp = x / y;
	elseif ((z * t) <= 5e+76)
		tmp = -x / (z * t);
	elseif ((z * t) <= 2e+200)
		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_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e+64], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], -2e-34], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], -1e-61], N[(x * N[(N[(-1.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-141], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e-85], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-55], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+76], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+200], N[(x / y), $MachinePrecision], (-N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision])]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 z t) < -2.00000000000000004e64 or 4.9999999999999999e-141 < (*.f64 z t) < 5.0000000000000002e-85

   1. Initial program 91.5%

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

      1. *-un-lft-identity 91.5%

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

      2. *-commutative 91.5%

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

      3. add-sqr-sqrt 51.6%

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

      4. sqrt-unprod 61.3%

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

      5. sqr-neg 61.3%

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

      6. sqrt-unprod 14.8%

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

      7. add-sqr-sqrt 40.8%

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

      8. distribute-rgt-neg-in 40.8%

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

      9. neg-mul-1 40.8%

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

      10. add-sqr-sqrt 26.0%

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

      11. sqrt-unprod 54.1%

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

      12. sqr-neg 54.1%

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

      13. sqrt-unprod 39.6%

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

      14. add-sqr-sqrt 91.5%

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

      15. distribute-rgt-neg-in 91.5%

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

      16. prod-diff 73.3%

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

   3. Applied egg-rr 73.3%

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

      1. fma-udef 73.3%

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

      2. distribute-rgt-neg-in 73.3%

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

      3. metadata-eval 73.3%

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

      4. distribute-rgt-in 73.3%

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

      5. +-commutative 73.3%

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

      6. fma-def 73.3%

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

      7. *-lft-identity 73.3%

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

      8. fma-def 73.3%

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

      9. *-commutative 73.3%

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

      10. fma-def 73.3%

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

      11. fma-udef 73.3%

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

      12. distribute-lft-out 73.3%

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

      13. *-commutative 73.3%

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

      14. metadata-eval 73.3%

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

   5. Simplified 73.3%

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

      1. +-commutative 73.3%

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

      2. associate-*l* 73.3%

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

      3. fma-def 91.4%

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

   7. Applied egg-rr 91.4%

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

   8. Taylor expanded in t around inf 83.8%

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

      1. distribute-rgt-in 65.6%

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

      2. *-commutative 65.6%

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

      3. associate-*l* 65.6%

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

      4. distribute-lft-in 83.8%

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

      5. associate-/r* 91.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{t + -2 \cdot t}} \]

      6. distribute-rgt1-in 91.1%

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

      7. metadata-eval 91.1%

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

      8. mul-1-neg 91.1%

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

   10. Simplified 91.1%

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

   if -2.00000000000000004e64 < (*.f64 z t) < -1.99999999999999986e-34 or -1e-61 < (*.f64 z t) < 4.9999999999999999e-141 or 5.0000000000000002e-85 < (*.f64 z t) < 5.0000000000000002e-55 or 4.99999999999999991e76 < (*.f64 z t) < 1.9999999999999999e200

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 83.8%

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

   if -1.99999999999999986e-34 < (*.f64 z t) < -1e-61

   1. Initial program 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. *-commutative 99.8%

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

      3. add-sqr-sqrt 42.6%

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

      4. sqrt-unprod 18.8%

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

      5. sqr-neg 18.8%

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

      6. sqrt-unprod 1.4%

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

      7. add-sqr-sqrt 16.1%

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

      8. distribute-rgt-neg-in 16.1%

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

      9. neg-mul-1 16.1%

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

      10. add-sqr-sqrt 14.7%

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

      11. sqrt-unprod 32.5%

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

      12. sqr-neg 32.5%

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

      13. sqrt-unprod 56.7%

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

      14. add-sqr-sqrt 99.8%

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

      15. distribute-rgt-neg-in 99.8%

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

      16. prod-diff 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. fma-udef 99.8%

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

      2. distribute-rgt-neg-in 99.8%

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

      3. metadata-eval 99.8%

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

      4. distribute-rgt-in 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. *-lft-identity 99.8%

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

      8. fma-def 99.8%

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

      9. *-commutative 99.8%

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

      10. fma-def 99.8%

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

      11. fma-udef 99.8%

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

      12. distribute-lft-out 99.8%

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

      13. *-commutative 99.8%

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

      14. metadata-eval 99.8%

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

   5. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. fma-def 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.3%

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

      3. fma-udef 99.3%

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

      4. *-commutative 99.3%

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

      5. fma-def 99.3%

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

   9. Applied egg-rr 99.3%

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

   10. Taylor expanded in t around -inf 77.2%

       \[\leadsto \color{blue}{\frac{-1}{t \cdot \left(-1 \cdot z + 2 \cdot z\right)}} \cdot x \]
   11. Step-by-step derivation

       1. associate-/r* 76.6%

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

       2. distribute-rgt-out 76.6%

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

       3. metadata-eval 76.6%

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

       4. *-rgt-identity 76.6%

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

   12. Simplified 76.6%

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

   if 5.0000000000000002e-55 < (*.f64 z t) < 4.99999999999999991e76

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0 61.2%

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

      1. associate-*r/ 61.2%

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

      2. neg-mul-1 61.2%

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

   4. Simplified 61.2%

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

   if 1.9999999999999999e200 < (*.f64 z t)

   1. Initial program 80.0%

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

   2. Taylor expanded in y around 0 80.0%

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

      1. mul-1-neg 80.0%

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

      2. associate-/r* 99.9%

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

      3. distribute-neg-frac 99.9%

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

   4. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{-\frac{x}{t}}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq -2 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq -1 \cdot 10^{-61}:\\ \;\;\;\;x \cdot \frac{\frac{-1}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+76}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+200}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 5: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+200}\right):\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 2e+200)))
   (- (/ (/ 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)) || !((z * t) <= 2e+200)) {
		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) || !((z * t) <= 2e+200)) {
		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) or not ((z * t) <= 2e+200):
		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)) || !(Float64(z * t) <= 2e+200))
		tmp = Float64(-Float64(Float64(x / t) / 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) || ~(((z * t) <= 2e+200)))
		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[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+200]], $MachinePrecision]], (-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 or 1.9999999999999999e200 < (*.f64 z t)

   1. Initial program 76.4%

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

   2. Taylor expanded in y around 0 76.4%

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

      1. mul-1-neg 76.4%

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

      2. associate-/r* 99.8%

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

      3. distribute-neg-frac 99.8%

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

   4. Simplified 99.8%

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

   if -inf.0 < (*.f64 z t) < 1.9999999999999999e200

   1. Initial program 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+200}\right):\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]

Alternative 6: 72.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1060000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- x) (* z t))))
   (if (<= y -5.2e+85)
     (/ x y)
     (if (<= y -1.65e+21)
       t_1
       (if (<= y -8.5e-37)
         (/ x y)
         (if (<= y 1060000000.0) t_1 (/ 1.0 (/ y x))))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double t_1 = -x / (z * t);
	double tmp;
	if (y <= -5.2e+85) {
		tmp = x / y;
	} else if (y <= -1.65e+21) {
		tmp = t_1;
	} else if (y <= -8.5e-37) {
		tmp = x / y;
	} else if (y <= 1060000000.0) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (y / 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) :: t_1
    real(8) :: tmp
    t_1 = -x / (z * t)
    if (y <= (-5.2d+85)) then
        tmp = x / y
    else if (y <= (-1.65d+21)) then
        tmp = t_1
    else if (y <= (-8.5d-37)) then
        tmp = x / y
    else if (y <= 1060000000.0d0) then
        tmp = t_1
    else
        tmp = 1.0d0 / (y / x)
    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 / (z * t);
	double tmp;
	if (y <= -5.2e+85) {
		tmp = x / y;
	} else if (y <= -1.65e+21) {
		tmp = t_1;
	} else if (y <= -8.5e-37) {
		tmp = x / y;
	} else if (y <= 1060000000.0) {
		tmp = t_1;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	t_1 = -x / (z * t)
	tmp = 0
	if y <= -5.2e+85:
		tmp = x / y
	elif y <= -1.65e+21:
		tmp = t_1
	elif y <= -8.5e-37:
		tmp = x / y
	elif y <= 1060000000.0:
		tmp = t_1
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(-x) / Float64(z * t))
	tmp = 0.0
	if (y <= -5.2e+85)
		tmp = Float64(x / y);
	elseif (y <= -1.65e+21)
		tmp = t_1;
	elseif (y <= -8.5e-37)
		tmp = Float64(x / y);
	elseif (y <= 1060000000.0)
		tmp = t_1;
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = -x / (z * t);
	tmp = 0.0;
	if (y <= -5.2e+85)
		tmp = x / y;
	elseif (y <= -1.65e+21)
		tmp = t_1;
	elseif (y <= -8.5e-37)
		tmp = x / y;
	elseif (y <= 1060000000.0)
		tmp = t_1;
	else
		tmp = 1.0 / (y / 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_] := Block[{t$95$1 = N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.2e+85], N[(x / y), $MachinePrecision], If[LessEqual[y, -1.65e+21], t$95$1, If[LessEqual[y, -8.5e-37], N[(x / y), $MachinePrecision], If[LessEqual[y, 1060000000.0], t$95$1, N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.20000000000000021e85 or -1.65e21 < y < -8.5000000000000007e-37

   1. Initial program 96.9%

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

   2. Taylor expanded in y around inf 80.1%

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

   if -5.20000000000000021e85 < y < -1.65e21 or -8.5000000000000007e-37 < y < 1.06e9

   1. Initial program 95.3%

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

   2. Taylor expanded in y around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. neg-mul-1 71.4%

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

   4. Simplified 71.4%

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

   if 1.06e9 < y

   1. Initial program 97.0%

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

      1. *-un-lft-identity 97.0%

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

      2. *-commutative 97.0%

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

      3. add-sqr-sqrt 39.1%

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

      4. sqrt-unprod 88.1%

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

      5. sqr-neg 88.1%

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

      6. sqrt-unprod 56.4%

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

      7. add-sqr-sqrt 90.2%

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

      8. distribute-rgt-neg-in 90.2%

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

      9. neg-mul-1 90.2%

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

      10. add-sqr-sqrt 33.8%

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

      11. sqrt-unprod 85.8%

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

      12. sqr-neg 85.8%

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

      13. sqrt-unprod 57.9%

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

      14. add-sqr-sqrt 97.0%

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

      15. distribute-rgt-neg-in 97.0%

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

      16. prod-diff 90.6%

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

   3. Applied egg-rr 90.6%

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

      1. fma-udef 90.6%

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

      2. distribute-rgt-neg-in 90.6%

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

      3. metadata-eval 90.6%

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

      4. distribute-rgt-in 90.6%

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

      5. +-commutative 90.6%

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

      6. fma-def 90.6%

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

      7. *-lft-identity 90.6%

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

      8. fma-def 90.6%

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

      9. *-commutative 90.6%

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

      10. fma-def 90.6%

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

      11. fma-udef 90.6%

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

      12. distribute-lft-out 90.6%

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

      13. *-commutative 90.6%

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

      14. metadata-eval 90.6%

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

   5. Simplified 90.6%

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

      1. clear-num 90.4%

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

      2. inv-pow 90.4%

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

      3. +-commutative 90.4%

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

      4. associate-*l* 90.4%

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

      5. fma-def 96.8%

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

   7. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(t, z \cdot -2, \mathsf{fma}\left(t, z, y\right)\right)}{x}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 96.8%

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

      2. fma-udef 90.4%

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

      3. fma-udef 90.4%

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

      4. associate-+r+ 90.3%

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

      5. associate-*r* 90.3%

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

      6. *-commutative 90.3%

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

      7. associate-*r* 90.3%

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

      8. distribute-rgt-in 96.8%

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

      9. +-commutative 96.8%

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

      10. fma-def 96.8%

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

      11. distribute-rgt1-in 96.8%

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

      12. metadata-eval 96.8%

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

      13. mul-1-neg 96.8%

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

   9. Simplified 96.8%

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

   10. Taylor expanded in z around 0 87.0%

       \[\leadsto \frac{1}{\color{blue}{\frac{y}{x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+85}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{+21}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 1060000000:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \]

Alternative 7: 63.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+182} \lor \neg \left(z \cdot t \leq 4 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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+182) (not (<= (* z t) 4e+138)))
   (/ 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+182) || !((z * t) <= 4e+138)) {
		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+182)) .or. (.not. ((z * t) <= 4d+138))) 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+182) || !((z * t) <= 4e+138)) {
		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+182) or not ((z * t) <= 4e+138):
		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+182) || !(Float64(z * t) <= 4e+138))
		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+182) || ~(((z * t) <= 4e+138)))
		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+182], N[Not[LessEqual[N[(z * t), $MachinePrecision], 4e+138]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999973e182 or 4.0000000000000001e138 < (*.f64 z t)

   1. Initial program 82.4%

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

      1. flip3-- 0.0%

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

      2. clear-num 0.0%

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

   3. Applied egg-rr 0.0%

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

   4. Taylor expanded in y around 0 80.6%

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

      1. associate-/r* 80.5%

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

   6. Simplified 80.5%

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

      1. clear-num 80.6%

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

      2. div-inv 80.6%

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

      3. frac-2neg 80.6%

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

      4. metadata-eval 80.6%

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

      5. remove-double-div 80.6%

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

      6. add-sqr-sqrt 44.7%

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

      7. sqrt-unprod 66.7%

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

      8. sqr-neg 66.7%

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

      9. sqrt-unprod 30.6%

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

      10. add-sqr-sqrt 57.7%

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

   8. Applied egg-rr 57.7%

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

   if -4.99999999999999973e182 < (*.f64 z t) < 4.0000000000000001e138

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 67.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+182} \lor \neg \left(z \cdot t \leq 4 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 8: 53.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \frac{1}{\frac{y}{x}} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ 1.0 (/ y x)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	return 1.0 / (y / x);
}

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 = 1.0d0 / (y / x)
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	return 1.0 / (y / x);
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	return 1.0 / (y / x)

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	return Float64(1.0 / Float64(y / x))
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t)
	tmp = 1.0 / (y / x);
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.1%

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

   1. *-un-lft-identity 96.1%

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

   2. *-commutative 96.1%

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

   3. add-sqr-sqrt 46.2%

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

   4. sqrt-unprod 72.4%

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

   5. sqr-neg 72.4%

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

   6. sqrt-unprod 33.5%

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

   7. add-sqr-sqrt 64.2%

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

   8. distribute-rgt-neg-in 64.2%

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

   9. neg-mul-1 64.2%

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

   10. add-sqr-sqrt 30.7%

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

   11. sqrt-unprod 71.3%

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

   12. sqr-neg 71.3%

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

   13. sqrt-unprod 49.7%

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

   14. add-sqr-sqrt 96.1%

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

   15. distribute-rgt-neg-in 96.1%

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

   16. prod-diff 87.7%

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

3. Applied egg-rr 87.7%

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

   1. fma-udef 87.7%

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

   2. distribute-rgt-neg-in 87.7%

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

   3. metadata-eval 87.7%

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

   4. distribute-rgt-in 87.7%

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

   5. +-commutative 87.7%

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

   6. fma-def 87.7%

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

   7. *-lft-identity 87.7%

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

   8. fma-def 87.7%

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

   9. *-commutative 87.7%

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

   10. fma-def 87.7%

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

   11. fma-udef 87.7%

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

   12. distribute-lft-out 87.7%

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

   13. *-commutative 87.7%

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

   14. metadata-eval 87.7%

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

5. Simplified 87.7%

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

   1. clear-num 87.3%

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

   2. inv-pow 87.3%

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

   3. +-commutative 87.3%

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

   4. associate-*l* 87.3%

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

   5. fma-def 95.6%

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

7. Applied egg-rr 95.6%

   \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(t, z \cdot -2, \mathsf{fma}\left(t, z, y\right)\right)}{x}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 95.6%

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

   2. fma-udef 87.3%

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

   3. fma-udef 87.3%

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

   4. associate-+r+ 87.3%

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

   5. associate-*r* 87.3%

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

   6. *-commutative 87.3%

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

   7. associate-*r* 87.3%

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

   8. distribute-rgt-in 95.7%

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

   9. +-commutative 95.7%

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

   10. fma-def 95.7%

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

   11. distribute-rgt1-in 95.7%

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

   12. metadata-eval 95.7%

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

   13. mul-1-neg 95.7%

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

9. Simplified 95.7%

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

10. Taylor expanded in z around 0 55.8%

    \[\leadsto \frac{1}{\color{blue}{\frac{y}{x}}} \]

11. Final simplification 55.8%

    \[\leadsto \frac{1}{\frac{y}{x}} \]

Alternative 9: 54.3% 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 96.1%

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

2. Taylor expanded in y around inf 55.7%

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

3. Final simplification 55.7%

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

Developer target: 96.4% 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 2023299 
(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))))