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

Percentage Accurate: 95.7% → 97.1%

Time: 6.4s

Alternatives: 9

Speedup: 0.6×

• Unsound rule application detected in e-graph. Results may not be sound.

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: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 0.1× speedup

Math

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

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) 5e+182) (/ x (fma (- z) t y)) (/ (/ x t) (- z))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 5e+182) {
		tmp = x / fma(-z, t, 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) <= 5e+182)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	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[LessEqual[N[(z * t), $MachinePrecision], 5e+182], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 4.99999999999999973e182

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. distribute-lft-neg-in 99.5%

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

      4. fma-def 99.5%

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

   3. Applied egg-rr 99.5%

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

   if 4.99999999999999973e182 < (*.f64 z t)

   1. Initial program 80.6%

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

      1. clear-num 80.6%

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

      2. associate-/r/ 80.6%

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

   3. Applied egg-rr 80.6%

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

   4. Taylor expanded in y around 0 80.6%

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

      1. expm1-log1p-u 80.6%

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

      2. expm1-udef 58.0%

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

      3. *-commutative 58.0%

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

   6. Applied egg-rr 58.0%

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

      1. expm1-def 80.6%

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

      2. expm1-log1p 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-/r* 80.5%

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

   8. Simplified 80.5%

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

      1. associate-*l/ 96.6%

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

      2. frac-2neg 96.6%

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

      3. associate-*l/ 96.7%

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

      4. neg-mul-1 96.7%

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

      5. add-sqr-sqrt 41.2%

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

      6. sqrt-unprod 66.8%

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

      7. sqr-neg 66.8%

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

      8. sqrt-unprod 36.0%

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

      9. add-sqr-sqrt 57.4%

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

      10. distribute-frac-neg 57.4%

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

      11. add-sqr-sqrt 21.4%

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

      12. sqrt-unprod 66.6%

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

      13. sqr-neg 66.6%

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

      14. sqrt-unprod 55.3%

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

      15. add-sqr-sqrt 96.7%

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

   10. Applied egg-rr 96.7%

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

4. Final simplification 99.2%

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

Alternative 2: 78.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{-x}{z \cdot t}\\ \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 10^{-75}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+145}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \end{array} \]

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) -1e-61)
     t_1
     (if (<= (* z t) 1e-75)
       (/ x y)
       (if (<= (* z t) 2e+145) t_1 (/ (/ 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) <= -1e-61) {
		tmp = t_1;
	} else if ((z * t) <= 1e-75) {
		tmp = x / y;
	} else if ((z * t) <= 2e+145) {
		tmp = t_1;
	} 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) <= (-1d-61)) then
        tmp = t_1
    else if ((z * t) <= 1d-75) then
        tmp = x / y
    else if ((z * t) <= 2d+145) then
        tmp = t_1
    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) <= -1e-61) {
		tmp = t_1;
	} else if ((z * t) <= 1e-75) {
		tmp = x / y;
	} else if ((z * t) <= 2e+145) {
		tmp = t_1;
	} 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) <= -1e-61:
		tmp = t_1
	elif (z * t) <= 1e-75:
		tmp = x / y
	elif (z * t) <= 2e+145:
		tmp = t_1
	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))
	tmp = 0.0
	if (Float64(z * t) <= -1e-61)
		tmp = t_1;
	elseif (Float64(z * t) <= 1e-75)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 2e+145)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = -x / (z * t);
	tmp = 0.0;
	if ((z * t) <= -1e-61)
		tmp = t_1;
	elseif ((z * t) <= 1e-75)
		tmp = x / y;
	elseif ((z * t) <= 2e+145)
		tmp = t_1;
	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]}, If[LessEqual[N[(z * t), $MachinePrecision], -1e-61], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 1e-75], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+145], t$95$1, N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1e-61 or 9.9999999999999996e-76 < (*.f64 z t) < 2e145

   1. Initial program 99.0%

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

   2. Taylor expanded in y around 0 71.3%

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

      1. associate-*r/ 71.3%

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

      2. neg-mul-1 71.3%

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

   4. Simplified 71.3%

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

   if -1e-61 < (*.f64 z t) < 9.9999999999999996e-76

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 81.9%

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

   if 2e145 < (*.f64 z t)

   1. Initial program 84.3%

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

      1. clear-num 84.3%

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

      2. associate-/r/ 84.3%

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

   3. Applied egg-rr 84.3%

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

   4. Taylor expanded in y around 0 77.0%

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

      1. expm1-log1p-u 77.0%

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

      2. expm1-udef 50.1%

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

      3. *-commutative 50.1%

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

   6. Applied egg-rr 50.1%

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

      1. expm1-def 77.0%

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

      2. expm1-log1p 77.0%

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

      3. *-commutative 77.0%

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

      4. associate-/r* 76.9%

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

   8. Simplified 76.9%

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

      1. associate-*l/ 89.8%

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

      2. frac-2neg 89.8%

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

      3. associate-*l/ 89.9%

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

      4. neg-mul-1 89.9%

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

      5. add-sqr-sqrt 33.2%

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

      6. sqrt-unprod 57.1%

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

      7. sqr-neg 57.1%

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

      8. sqrt-unprod 32.2%

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

      9. add-sqr-sqrt 49.7%

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

      10. distribute-frac-neg 49.7%

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

      11. add-sqr-sqrt 17.5%

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

      12. sqrt-unprod 63.1%

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

      13. sqr-neg 63.1%

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

      14. sqrt-unprod 56.5%

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

      15. add-sqr-sqrt 89.9%

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

   10. Applied egg-rr 89.9%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 10^{-75}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+145}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]

Alternative 3: 70.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-75} \lor \neg \left(y \leq -5.6 \cdot 10^{-139} \lor \neg \left(y \leq -4.3 \cdot 10^{-148}\right) \land y \leq 10^{+87}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.4e-75)
         (not (or (<= y -5.6e-139) (and (not (<= y -4.3e-148)) (<= y 1e+87)))))
   (/ x y)
   (/ (- x) (* z t))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.4e-75) || !((y <= -5.6e-139) || (!(y <= -4.3e-148) && (y <= 1e+87)))) {
		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 ((y <= (-3.4d-75)) .or. (.not. (y <= (-5.6d-139)) .or. (.not. (y <= (-4.3d-148))) .and. (y <= 1d+87))) 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 ((y <= -3.4e-75) || !((y <= -5.6e-139) || (!(y <= -4.3e-148) && (y <= 1e+87)))) {
		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 (y <= -3.4e-75) or not ((y <= -5.6e-139) or (not (y <= -4.3e-148) and (y <= 1e+87))):
		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 ((y <= -3.4e-75) || !((y <= -5.6e-139) || (!(y <= -4.3e-148) && (y <= 1e+87))))
		tmp = Float64(x / y);
	else
		tmp = Float64(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 ((y <= -3.4e-75) || ~(((y <= -5.6e-139) || (~((y <= -4.3e-148)) && (y <= 1e+87)))))
		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[Or[LessEqual[y, -3.4e-75], N[Not[Or[LessEqual[y, -5.6e-139], And[N[Not[LessEqual[y, -4.3e-148]], $MachinePrecision], LessEqual[y, 1e+87]]]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.40000000000000015e-75 or -5.5999999999999997e-139 < y < -4.2999999999999998e-148 or 9.9999999999999996e86 < y

   1. Initial program 96.1%

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

   2. Taylor expanded in y around inf 81.0%

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

   if -3.40000000000000015e-75 < y < -5.5999999999999997e-139 or -4.2999999999999998e-148 < y < 9.9999999999999996e86

   1. Initial program 98.5%

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

   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%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-75} \lor \neg \left(y \leq -5.6 \cdot 10^{-139} \lor \neg \left(y \leq -4.3 \cdot 10^{-148}\right) \land y \leq 10^{+87}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 4: 77.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 10^{-75}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \end{array} \]

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) -1e-61)
   (/ (- x) (* z t))
   (if (<= (* z t) 1e-75) (/ x y) (/ (/ x z) (- t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -1e-61) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 1e-75) {
		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) <= (-1d-61)) then
        tmp = -x / (z * t)
    else if ((z * t) <= 1d-75) 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) <= -1e-61) {
		tmp = -x / (z * t);
	} else if ((z * t) <= 1e-75) {
		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) <= -1e-61:
		tmp = -x / (z * t)
	elif (z * t) <= 1e-75:
		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) <= -1e-61)
		tmp = Float64(Float64(-x) / Float64(z * t));
	elseif (Float64(z * t) <= 1e-75)
		tmp = Float64(x / y);
	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) <= -1e-61)
		tmp = -x / (z * t);
	elseif ((z * t) <= 1e-75)
		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], -1e-61], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e-75], N[(x / y), $MachinePrecision], N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 98.4%

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

   2. Taylor expanded in y around 0 77.3%

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

      1. associate-*r/ 77.3%

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

      2. neg-mul-1 77.3%

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

   4. Simplified 77.3%

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

   if -1e-61 < (*.f64 z t) < 9.9999999999999996e-76

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 81.9%

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

   if 9.9999999999999996e-76 < (*.f64 z t)

   1. Initial program 93.2%

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

      1. clear-num 91.4%

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

      2. associate-/r/ 93.0%

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

   3. Applied egg-rr 93.0%

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

   4. Taylor expanded in y around 0 68.9%

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

      1. expm1-log1p-u 55.8%

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

      2. expm1-udef 30.8%

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

      3. *-commutative 30.8%

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

   6. Applied egg-rr 30.8%

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

      1. expm1-def 55.8%

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

      2. expm1-log1p 68.9%

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

      3. *-commutative 68.9%

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

      4. associate-/r* 68.9%

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

   8. Simplified 68.9%

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

      1. associate-/l/ 68.9%

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

      2. associate-*l/ 69.0%

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

      3. neg-mul-1 69.0%

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

      4. associate-/r* 71.5%

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

      5. frac-2neg 71.5%

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

      6. add-sqr-sqrt 28.2%

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

      7. sqrt-unprod 39.6%

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

      8. sqr-neg 39.6%

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

      9. sqrt-unprod 17.4%

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

      10. add-sqr-sqrt 27.2%

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

      11. distribute-frac-neg 27.2%

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

      12. add-sqr-sqrt 9.8%

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

      13. sqrt-unprod 42.5%

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

      14. sqr-neg 42.5%

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

      15. sqrt-unprod 43.0%

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

      16. add-sqr-sqrt 71.5%

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

   10. Applied egg-rr 71.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 10^{-75}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]

Alternative 5: 62.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -8.5 \cdot 10^{+68} \lor \neg \left(z \cdot t \leq 2.35 \cdot 10^{+166}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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) -8.5e+68) (not (<= (* z t) 2.35e+166)))
   (/ x (* z t))
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -8.5e+68) || !((z * t) <= 2.35e+166)) {
		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) <= (-8.5d+68)) .or. (.not. ((z * t) <= 2.35d+166))) 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) <= -8.5e+68) || !((z * t) <= 2.35e+166)) {
		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) <= -8.5e+68) or not ((z * t) <= 2.35e+166):
		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) <= -8.5e+68) || !(Float64(z * t) <= 2.35e+166))
		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) <= -8.5e+68) || ~(((z * t) <= 2.35e+166)))
		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], -8.5e+68], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2.35e+166]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -8.49999999999999966e68 or 2.35e166 < (*.f64 z t)

   1. Initial program 90.3%

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

      1. clear-num 89.3%

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

      2. associate-/r/ 90.2%

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

   3. Applied egg-rr 90.2%

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

   4. Taylor expanded in y around 0 86.1%

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

      1. expm1-log1p-u 86.1%

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

      2. expm1-udef 52.1%

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

      3. *-commutative 52.1%

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

   6. Applied egg-rr 52.1%

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

      1. expm1-def 86.1%

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

      2. expm1-log1p 86.1%

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

      3. *-commutative 86.1%

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

      4. associate-/r* 86.1%

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

   8. Simplified 86.1%

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

      1. associate-/l/ 86.1%

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

      2. associate-*l/ 86.1%

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

      3. neg-mul-1 86.1%

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

      4. add-sqr-sqrt 38.4%

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

      5. sqrt-unprod 59.5%

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

      6. sqr-neg 59.5%

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

      7. sqrt-unprod 29.8%

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

      8. add-sqr-sqrt 51.5%

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

   10. Applied egg-rr 51.5%

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

   if -8.49999999999999966e68 < (*.f64 z t) < 2.35e166

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 63.4%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -8.5 \cdot 10^{+68} \lor \neg \left(z \cdot t \leq 2.35 \cdot 10^{+166}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 62.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \end{array} \]

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) -5e+68)
   (/ (/ x t) z)
   (if (<= (* z t) 5e+162) (/ x y) (/ x (* z t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -5e+68) {
		tmp = (x / t) / z;
	} else if ((z * t) <= 5e+162) {
		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) <= (-5d+68)) then
        tmp = (x / t) / z
    else if ((z * t) <= 5d+162) 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) <= -5e+68) {
		tmp = (x / t) / z;
	} else if ((z * t) <= 5e+162) {
		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) <= -5e+68:
		tmp = (x / t) / z
	elif (z * t) <= 5e+162:
		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) <= -5e+68)
		tmp = Float64(Float64(x / t) / z);
	elseif (Float64(z * t) <= 5e+162)
		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) <= -5e+68)
		tmp = (x / t) / z;
	elseif ((z * t) <= 5e+162)
		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], -5e+68], N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+162], N[(x / y), $MachinePrecision], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000004e68

   1. Initial program 97.2%

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

      1. clear-num 95.5%

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

      2. associate-/r/ 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in y around 0 92.0%

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

      1. expm1-log1p-u 92.0%

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

      2. expm1-udef 48.7%

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

      3. *-commutative 48.7%

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

   6. Applied egg-rr 48.7%

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

      1. expm1-def 92.0%

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

      2. expm1-log1p 92.0%

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

      3. *-commutative 92.0%

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

      4. associate-/r* 92.1%

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

   8. Simplified 92.1%

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

      1. associate-*l/ 89.4%

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

      2. associate-*l/ 89.5%

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

      3. neg-mul-1 89.5%

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

      4. add-sqr-sqrt 41.9%

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

      5. sqrt-unprod 53.4%

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

      6. sqr-neg 53.4%

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

      7. sqrt-unprod 24.3%

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

      8. add-sqr-sqrt 47.9%

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

   10. Applied egg-rr 47.9%

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

   if -5.0000000000000004e68 < (*.f64 z t) < 4.9999999999999997e162

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 63.4%

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

   if 4.9999999999999997e162 < (*.f64 z t)

   1. Initial program 82.4%

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

      1. clear-num 82.3%

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

      2. associate-/r/ 82.4%

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

   3. Applied egg-rr 82.4%

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

   4. Taylor expanded in y around 0 79.4%

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

      1. expm1-log1p-u 79.4%

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

      2. expm1-udef 55.9%

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

      3. *-commutative 55.9%

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

   6. Applied egg-rr 55.9%

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

      1. expm1-def 79.4%

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

      2. expm1-log1p 79.4%

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

      3. *-commutative 79.4%

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

      4. associate-/r* 79.3%

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

   8. Simplified 79.3%

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

      1. associate-/l/ 79.4%

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

      2. associate-*l/ 79.4%

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

      3. neg-mul-1 79.4%

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

      4. add-sqr-sqrt 31.5%

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

      5. sqrt-unprod 63.5%

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

      6. sqr-neg 63.5%

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

      7. sqrt-unprod 35.9%

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

      8. add-sqr-sqrt 55.7%

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

   10. Applied egg-rr 55.7%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 7: 62.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \end{array} \]

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) -5e+68)
   (/ (/ x z) t)
   (if (<= (* z t) 5e+162) (/ x y) (/ x (* z t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -5e+68) {
		tmp = (x / z) / t;
	} else if ((z * t) <= 5e+162) {
		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) <= (-5d+68)) then
        tmp = (x / z) / t
    else if ((z * t) <= 5d+162) 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) <= -5e+68) {
		tmp = (x / z) / t;
	} else if ((z * t) <= 5e+162) {
		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) <= -5e+68:
		tmp = (x / z) / t
	elif (z * t) <= 5e+162:
		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) <= -5e+68)
		tmp = Float64(Float64(x / z) / t);
	elseif (Float64(z * t) <= 5e+162)
		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) <= -5e+68)
		tmp = (x / z) / t;
	elseif ((z * t) <= 5e+162)
		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], -5e+68], N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+162], N[(x / y), $MachinePrecision], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -5.0000000000000004e68

   1. Initial program 97.2%

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

      1. clear-num 95.5%

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

      2. associate-/r/ 97.2%

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

   3. Applied egg-rr 97.2%

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

   4. Taylor expanded in y around 0 92.0%

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

      1. expm1-log1p-u 92.0%

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

      2. expm1-udef 48.7%

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

      3. *-commutative 48.7%

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

   6. Applied egg-rr 48.7%

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

      1. expm1-def 92.0%

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

      2. expm1-log1p 92.0%

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

      3. *-commutative 92.0%

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

      4. associate-/r* 92.1%

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

   8. Simplified 92.1%

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

      1. associate-/l/ 92.0%

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

      2. associate-*l/ 92.1%

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

      3. neg-mul-1 92.1%

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

      4. associate-/r* 90.8%

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

      5. add-sqr-sqrt 42.2%

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

      6. sqrt-unprod 56.0%

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

      7. sqr-neg 56.0%

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

      8. sqrt-unprod 24.4%

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

      9. add-sqr-sqrt 47.9%

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

   10. Applied egg-rr 47.9%

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

   if -5.0000000000000004e68 < (*.f64 z t) < 4.9999999999999997e162

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 63.4%

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

   if 4.9999999999999997e162 < (*.f64 z t)

   1. Initial program 82.4%

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

      1. clear-num 82.3%

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

      2. associate-/r/ 82.4%

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

   3. Applied egg-rr 82.4%

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

   4. Taylor expanded in y around 0 79.4%

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

      1. expm1-log1p-u 79.4%

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

      2. expm1-udef 55.9%

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

      3. *-commutative 55.9%

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

   6. Applied egg-rr 55.9%

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

      1. expm1-def 79.4%

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

      2. expm1-log1p 79.4%

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

      3. *-commutative 79.4%

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

      4. associate-/r* 79.3%

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

   8. Simplified 79.3%

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

      1. associate-/l/ 79.4%

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

      2. associate-*l/ 79.4%

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

      3. neg-mul-1 79.4%

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

      4. add-sqr-sqrt 31.5%

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

      5. sqrt-unprod 63.5%

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

      6. sqr-neg 63.5%

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

      7. sqrt-unprod 35.9%

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

      8. add-sqr-sqrt 55.7%

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

   10. Applied egg-rr 55.7%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 8: 97.1% accurate, 0.6× speedup

Math

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

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) 5e+182) (/ x (- y (* z t))) (/ (/ x t) (- z))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 5e+182) {
		tmp = x / (y - (z * t));
	} 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) <= 5d+182) then
        tmp = x / (y - (z * t))
    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) <= 5e+182) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / t) / -z;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= 5e+182:
		tmp = x / (y - (z * t))
	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) <= 5e+182)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / t) / Float64(-z));
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= 5e+182)
		tmp = x / (y - (z * t));
	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], 5e+182], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 4.99999999999999973e182

   1. Initial program 99.5%

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

   if 4.99999999999999973e182 < (*.f64 z t)

   1. Initial program 80.6%

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

      1. clear-num 80.6%

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

      2. associate-/r/ 80.6%

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

   3. Applied egg-rr 80.6%

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

   4. Taylor expanded in y around 0 80.6%

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

      1. expm1-log1p-u 80.6%

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

      2. expm1-udef 58.0%

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

      3. *-commutative 58.0%

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

   6. Applied egg-rr 58.0%

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

      1. expm1-def 80.6%

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

      2. expm1-log1p 80.6%

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

      3. *-commutative 80.6%

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

      4. associate-/r* 80.5%

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

   8. Simplified 80.5%

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

      1. associate-*l/ 96.6%

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

      2. frac-2neg 96.6%

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

      3. associate-*l/ 96.7%

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

      4. neg-mul-1 96.7%

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

      5. add-sqr-sqrt 41.2%

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

      6. sqrt-unprod 66.8%

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

      7. sqr-neg 66.8%

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

      8. sqrt-unprod 36.0%

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

      9. add-sqr-sqrt 57.4%

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

      10. distribute-frac-neg 57.4%

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

      11. add-sqr-sqrt 21.4%

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

      12. sqrt-unprod 66.6%

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

      13. sqr-neg 66.6%

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

      14. sqrt-unprod 55.3%

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

      15. add-sqr-sqrt 96.7%

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

   10. Applied egg-rr 96.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \end{array} \]

Alternative 9: 53.6% 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 97.3%

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

2. Taylor expanded in y around inf 51.0%

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

3. Final simplification 51.0%

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

Developer target: 96.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (- (/ y x) (* (/ z x) t)))))
   (if (< x -1.618195973607049e+50)
     t_1
     (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / ((y / x) - ((z / x) * t))
    if (x < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (x < 2.1378306434876444d+131) then
        tmp = x / (y - (z * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / ((y / x) - ((z / x) * t))
	tmp = 0
	if x < -1.618195973607049e+50:
		tmp = t_1
	elif x < 2.1378306434876444e+131:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(Float64(y / x) - Float64(Float64(z / x) * t)))
	tmp = 0.0
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / ((y / x) - ((z / x) * t));
	tmp = 0.0;
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[x, -1.618195973607049e+50], t$95$1, If[Less[x, 2.1378306434876444e+131], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023176 
(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))))