Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.3% → 96.6%

Time: 11.6s

Alternatives: 16

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{y \cdot z - t \cdot z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

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 * 2.0d0) / ((y * z) - (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

Python

def code(x, y, z, t):
	return (x * 2.0) / ((y * z) - (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * 2.0) / ((y * z) - (t * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2}{y \cdot z - t \cdot z} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ (* x 2.0) (- (* y z) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

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 * 2.0d0) / ((y * z) - (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x * 2.0) / ((y * z) - (t * z));
}

Python

def code(x, y, z, t):
	return (x * 2.0) / ((y * z) - (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x * 2.0) / ((y * z) - (t * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x\_m \cdot -2}{z \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{t - y}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 5e-88)
    (/ (* x_m -2.0) (* z (- t y)))
    (/ (/ (* x_m -2.0) (- t y)) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-88) {
		tmp = (x_m * -2.0) / (z * (t - y));
	} else {
		tmp = ((x_m * -2.0) / (t - y)) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-88) then
        tmp = (x_m * (-2.0d0)) / (z * (t - y))
    else
        tmp = ((x_m * (-2.0d0)) / (t - y)) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-88) {
		tmp = (x_m * -2.0) / (z * (t - y));
	} else {
		tmp = ((x_m * -2.0) / (t - y)) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-88:
		tmp = (x_m * -2.0) / (z * (t - y))
	else:
		tmp = ((x_m * -2.0) / (t - y)) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-88)
		tmp = Float64(Float64(x_m * -2.0) / Float64(z * Float64(t - y)));
	else
		tmp = Float64(Float64(Float64(x_m * -2.0) / Float64(t - y)) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-88)
		tmp = (x_m * -2.0) / (z * (t - y));
	else
		tmp = ((x_m * -2.0) / (t - y)) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-88], N[(N[(x$95$m * -2.0), $MachinePrecision] / N[(z * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / N[(t - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 5.00000000000000009e-88

   1. Initial program 88.6%

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

      1. remove-double-neg 88.6%

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

      2. distribute-rgt-out-- 89.8%

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

      3. distribute-lft-neg-in 89.8%

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

      4. distribute-rgt-out-- 88.6%

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

      5. remove-double-neg 88.6%

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

      6. distribute-lft-neg-out 88.6%

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

      7. distribute-rgt-neg-in 88.6%

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

      8. neg-mul-1 88.6%

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

      9. times-frac 88.5%

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

      10. associate-*r/ 88.6%

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

      11. distribute-frac-neg 88.6%

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

      12. distribute-neg-frac2 88.6%

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

      13. metadata-eval 88.6%

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

      14. /-rgt-identity 88.6%

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

      15. metadata-eval 88.6%

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

      16. distribute-rgt-neg-out 88.6%

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

      17. distribute-lft-neg-out 88.6%

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

      18. *-commutative 88.6%

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

      19. cancel-sign-sub 88.6%

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

      20. *-commutative 88.6%

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

   3. Simplified 89.8%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   if 5.00000000000000009e-88 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 90.1%

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

      1. remove-double-neg 90.1%

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

      2. distribute-rgt-out-- 91.5%

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

      3. distribute-lft-neg-in 91.5%

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

      4. distribute-rgt-out-- 87.1%

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

      5. remove-double-neg 87.1%

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

      6. distribute-lft-neg-out 87.1%

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

      7. distribute-rgt-neg-in 87.1%

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

      8. neg-mul-1 87.1%

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

      9. times-frac 86.5%

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

      10. associate-*r/ 87.1%

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

      11. distribute-frac-neg 87.1%

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

      12. distribute-neg-frac2 87.1%

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

      13. metadata-eval 87.1%

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

      14. /-rgt-identity 87.1%

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

      15. metadata-eval 87.1%

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

      16. distribute-rgt-neg-out 87.1%

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

      17. distribute-lft-neg-out 87.1%

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

      18. *-commutative 87.1%

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

      19. cancel-sign-sub 87.1%

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

      20. *-commutative 87.1%

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

   3. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 91.5%

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

      2. associate-/r* 98.6%

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

   6. Applied egg-rr 98.6%

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

Alternative 2: 71.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{\frac{x\_m \cdot -2}{z}}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x\_m}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{t}}{z}\\ \mathbf{elif}\;y \leq 4:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x\_m}}\\ \mathbf{elif}\;y \leq 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (/ (* x_m -2.0) z) t)))
   (*
    x_s
    (if (<= y -1.25e+101)
      (* 2.0 (/ (/ x_m z) y))
      (if (<= y -1.1e+52)
        t_1
        (if (<= y -1.05e-31)
          (/ 2.0 (* y (/ z x_m)))
          (if (<= y 4.2e-55)
            (/ (/ (* x_m -2.0) t) z)
            (if (<= y 4.0)
              (/ 2.0 (* z (/ y x_m)))
              (if (<= y 1e+79) t_1 (* (/ 2.0 z) (/ x_m y)))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = ((x_m * -2.0) / z) / t;
	double tmp;
	if (y <= -1.25e+101) {
		tmp = 2.0 * ((x_m / z) / y);
	} else if (y <= -1.1e+52) {
		tmp = t_1;
	} else if (y <= -1.05e-31) {
		tmp = 2.0 / (y * (z / x_m));
	} else if (y <= 4.2e-55) {
		tmp = ((x_m * -2.0) / t) / z;
	} else if (y <= 4.0) {
		tmp = 2.0 / (z * (y / x_m));
	} else if (y <= 1e+79) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m * (-2.0d0)) / z) / t
    if (y <= (-1.25d+101)) then
        tmp = 2.0d0 * ((x_m / z) / y)
    else if (y <= (-1.1d+52)) then
        tmp = t_1
    else if (y <= (-1.05d-31)) then
        tmp = 2.0d0 / (y * (z / x_m))
    else if (y <= 4.2d-55) then
        tmp = ((x_m * (-2.0d0)) / t) / z
    else if (y <= 4.0d0) then
        tmp = 2.0d0 / (z * (y / x_m))
    else if (y <= 1d+79) then
        tmp = t_1
    else
        tmp = (2.0d0 / z) * (x_m / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = ((x_m * -2.0) / z) / t;
	double tmp;
	if (y <= -1.25e+101) {
		tmp = 2.0 * ((x_m / z) / y);
	} else if (y <= -1.1e+52) {
		tmp = t_1;
	} else if (y <= -1.05e-31) {
		tmp = 2.0 / (y * (z / x_m));
	} else if (y <= 4.2e-55) {
		tmp = ((x_m * -2.0) / t) / z;
	} else if (y <= 4.0) {
		tmp = 2.0 / (z * (y / x_m));
	} else if (y <= 1e+79) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = ((x_m * -2.0) / z) / t
	tmp = 0
	if y <= -1.25e+101:
		tmp = 2.0 * ((x_m / z) / y)
	elif y <= -1.1e+52:
		tmp = t_1
	elif y <= -1.05e-31:
		tmp = 2.0 / (y * (z / x_m))
	elif y <= 4.2e-55:
		tmp = ((x_m * -2.0) / t) / z
	elif y <= 4.0:
		tmp = 2.0 / (z * (y / x_m))
	elif y <= 1e+79:
		tmp = t_1
	else:
		tmp = (2.0 / z) * (x_m / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(Float64(x_m * -2.0) / z) / t)
	tmp = 0.0
	if (y <= -1.25e+101)
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / y));
	elseif (y <= -1.1e+52)
		tmp = t_1;
	elseif (y <= -1.05e-31)
		tmp = Float64(2.0 / Float64(y * Float64(z / x_m)));
	elseif (y <= 4.2e-55)
		tmp = Float64(Float64(Float64(x_m * -2.0) / t) / z);
	elseif (y <= 4.0)
		tmp = Float64(2.0 / Float64(z * Float64(y / x_m)));
	elseif (y <= 1e+79)
		tmp = t_1;
	else
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = ((x_m * -2.0) / z) / t;
	tmp = 0.0;
	if (y <= -1.25e+101)
		tmp = 2.0 * ((x_m / z) / y);
	elseif (y <= -1.1e+52)
		tmp = t_1;
	elseif (y <= -1.05e-31)
		tmp = 2.0 / (y * (z / x_m));
	elseif (y <= 4.2e-55)
		tmp = ((x_m * -2.0) / t) / z;
	elseif (y <= 4.0)
		tmp = 2.0 / (z * (y / x_m));
	elseif (y <= 1e+79)
		tmp = t_1;
	else
		tmp = (2.0 / z) * (x_m / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / z), $MachinePrecision] / t), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.25e+101], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.1e+52], t$95$1, If[LessEqual[y, -1.05e-31], N[(2.0 / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e-55], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 4.0], N[(2.0 / N[(z * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+79], t$95$1, N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 6 regimes

2. if y < -1.24999999999999997e101

   1. Initial program 80.6%

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

      1. remove-double-neg 80.6%

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

      2. distribute-rgt-out-- 82.6%

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

      3. distribute-lft-neg-in 82.6%

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

      4. distribute-rgt-out-- 80.6%

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

      5. remove-double-neg 80.6%

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

      6. distribute-lft-neg-out 80.6%

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

      7. distribute-rgt-neg-in 80.6%

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

      8. neg-mul-1 80.6%

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

      9. times-frac 80.5%

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

      10. associate-*r/ 80.6%

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

      11. distribute-frac-neg 80.6%

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

      12. distribute-neg-frac2 80.6%

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

      13. metadata-eval 80.6%

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

      14. /-rgt-identity 80.6%

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

      15. metadata-eval 80.6%

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

      16. distribute-rgt-neg-out 80.6%

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

      17. distribute-lft-neg-out 80.6%

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

      18. *-commutative 80.6%

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

      19. cancel-sign-sub 80.6%

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

      20. *-commutative 80.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 95.8%

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

   6. Applied egg-rr 95.8%

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

   7. Taylor expanded in t around 0 76.0%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 76.0%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 87.6%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 87.6%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]

   if -1.24999999999999997e101 < y < -1.1e52 or 4 < y < 9.99999999999999967e78

   1. Initial program 84.9%

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

      1. remove-double-neg 84.9%

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

      2. distribute-rgt-out-- 88.8%

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

      3. distribute-lft-neg-in 88.8%

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

      4. distribute-rgt-out-- 84.9%

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

      5. remove-double-neg 84.9%

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

      6. distribute-lft-neg-out 84.9%

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

      7. distribute-rgt-neg-in 84.9%

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

      8. neg-mul-1 84.9%

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

      9. times-frac 84.7%

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

      10. associate-*r/ 84.9%

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

      11. distribute-frac-neg 84.9%

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

      12. distribute-neg-frac2 84.9%

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

      13. metadata-eval 84.9%

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

      14. /-rgt-identity 84.9%

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

      15. metadata-eval 84.9%

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

      16. distribute-rgt-neg-out 84.9%

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

      17. distribute-lft-neg-out 84.9%

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

      18. *-commutative 84.9%

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

      19. cancel-sign-sub 84.9%

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

      20. *-commutative 84.9%

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

   3. Simplified 88.8%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 95.9%

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

   6. Applied egg-rr 95.9%

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

   7. Taylor expanded in t around inf 70.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   8. Step-by-step derivation

      1. associate-*r/ 70.4%

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

      2. *-commutative 70.4%

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

      3. *-commutative 70.4%

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

      4. times-frac 80.9%

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

      5. associate-*r/ 81.2%

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

      6. *-commutative 81.2%

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

      7. associate-*r/ 81.2%

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

   9. Simplified 81.2%

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

   if -1.1e52 < y < -1.04999999999999996e-31

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-rgt-out-- 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-out-- 99.8%

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

      5. remove-double-neg 99.8%

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

      6. distribute-lft-neg-out 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

      8. neg-mul-1 99.8%

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

      9. times-frac 99.7%

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

      10. associate-*r/ 99.8%

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

      11. distribute-frac-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. metadata-eval 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. metadata-eval 99.8%

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

      16. distribute-rgt-neg-out 99.8%

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

      17. distribute-lft-neg-out 99.8%

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

      18. *-commutative 99.8%

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

      19. cancel-sign-sub 99.8%

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

      20. *-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t around 0 87.7%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 87.7%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 87.8%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 87.8%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]
   10. Step-by-step derivation

       1. *-commutative 87.8%

          \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y} \cdot 2} \]

       2. associate-*l/ 87.8%

          \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot 2}{y}} \]

       3. associate-*r/ 87.7%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y}} \]

       4. clear-num 87.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{2}{y} \]

       5. frac-times 87.8%

          \[\leadsto \color{blue}{\frac{1 \cdot 2}{\frac{z}{x} \cdot y}} \]

       6. metadata-eval 87.8%

          \[\leadsto \frac{\color{blue}{2}}{\frac{z}{x} \cdot y} \]

   11. Applied egg-rr 87.8%

       \[\leadsto \color{blue}{\frac{2}{\frac{z}{x} \cdot y}} \]

   if -1.04999999999999996e-31 < y < 4.2000000000000003e-55

   1. Initial program 94.1%

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

      1. remove-double-neg 94.1%

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

      2. distribute-rgt-out-- 95.1%

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

      3. distribute-lft-neg-in 95.1%

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

      4. distribute-rgt-out-- 92.4%

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

      5. remove-double-neg 92.4%

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

      6. distribute-lft-neg-out 92.4%

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

      7. distribute-rgt-neg-in 92.4%

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

      8. neg-mul-1 92.4%

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

      9. times-frac 91.9%

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

      10. associate-*r/ 92.4%

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

      11. distribute-frac-neg 92.4%

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

      12. distribute-neg-frac2 92.4%

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

      13. metadata-eval 92.4%

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

      14. /-rgt-identity 92.4%

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

      15. metadata-eval 92.4%

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

      16. distribute-rgt-neg-out 92.4%

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

      17. distribute-lft-neg-out 92.4%

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

      18. *-commutative 92.4%

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

      19. cancel-sign-sub 92.4%

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

      20. *-commutative 92.4%

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

   3. Simplified 95.1%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 80.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 80.8%

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

      2. *-commutative 80.8%

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

      3. *-commutative 80.8%

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

      4. associate-/l* 80.4%

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

   7. Simplified 80.4%

      \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]
   8. Step-by-step derivation

      1. associate-*r/ 80.8%

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

      2. *-commutative 80.8%

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

      3. associate-/r* 81.4%

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

   9. Applied egg-rr 81.4%

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

   if 4.2000000000000003e-55 < y < 4

   1. Initial program 89.7%

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

      1. remove-double-neg 89.7%

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

      2. distribute-rgt-out-- 89.7%

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

      3. distribute-lft-neg-in 89.7%

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

      4. distribute-rgt-out-- 78.6%

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

      5. remove-double-neg 78.6%

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

      6. distribute-lft-neg-out 78.6%

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

      7. distribute-rgt-neg-in 78.6%

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

      8. neg-mul-1 78.6%

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

      9. times-frac 78.4%

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

      10. associate-*r/ 78.6%

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

      11. distribute-frac-neg 78.6%

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

      12. distribute-neg-frac2 78.6%

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

      13. metadata-eval 78.6%

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

      14. /-rgt-identity 78.6%

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

      15. metadata-eval 78.6%

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

      16. distribute-rgt-neg-out 78.6%

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

      17. distribute-lft-neg-out 78.6%

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

      18. *-commutative 78.6%

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

      19. cancel-sign-sub 78.6%

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

      20. *-commutative 78.6%

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

   3. Simplified 89.7%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around 0 89.2%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 89.2%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 89.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 89.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]
   10. Step-by-step derivation

       1. clear-num 89.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{y}{\frac{x}{z}}}} \]

       2. un-div-inv 89.2%

          \[\leadsto \color{blue}{\frac{2}{\frac{y}{\frac{x}{z}}}} \]

       3. associate-/r/ 89.2%

          \[\leadsto \frac{2}{\color{blue}{\frac{y}{x} \cdot z}} \]

   11. Applied egg-rr 89.2%

       \[\leadsto \color{blue}{\frac{2}{\frac{y}{x} \cdot z}} \]

   if 9.99999999999999967e78 < y

   1. Initial program 85.0%

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

      1. remove-double-neg 85.0%

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

      2. distribute-rgt-out-- 85.3%

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

      3. distribute-lft-neg-in 85.3%

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

      4. distribute-rgt-out-- 85.0%

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

      5. remove-double-neg 85.0%

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

      6. distribute-lft-neg-out 85.0%

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

      7. distribute-rgt-neg-in 85.0%

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

      8. neg-mul-1 85.0%

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

      9. times-frac 84.8%

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

      10. associate-*r/ 85.0%

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

      11. distribute-frac-neg 85.0%

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

      12. distribute-neg-frac2 85.0%

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

      13. metadata-eval 85.0%

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

      14. /-rgt-identity 85.0%

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

      15. metadata-eval 85.0%

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

      16. distribute-rgt-neg-out 85.0%

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

      17. distribute-lft-neg-out 85.0%

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

      18. *-commutative 85.0%

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

      19. cancel-sign-sub 85.0%

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

      20. *-commutative 85.0%

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

   3. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.4%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 73.4%

         \[\leadsto \color{blue}{\frac{2 \cdot x}{y \cdot z}} \]

      2. *-commutative 73.4%

         \[\leadsto \frac{2 \cdot x}{\color{blue}{z \cdot y}} \]

      3. times-frac 77.6%

         \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]

   7. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{t}}{z}\\ \mathbf{elif}\;y \leq 4:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 10^{+79}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{\frac{x\_m \cdot -2}{z}}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+100}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x\_m}}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-55}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x\_m}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (/ (* x_m -2.0) z) t)))
   (*
    x_s
    (if (<= y -1.16e+100)
      (* 2.0 (/ (/ x_m z) y))
      (if (<= y -2.8e+51)
        t_1
        (if (<= y -1.35e-33)
          (/ 2.0 (* y (/ z x_m)))
          (if (<= y 3.7e-55)
            (* (/ -2.0 z) (/ x_m t))
            (if (<= y 1.75e-6)
              (/ 2.0 (* z (/ y x_m)))
              (if (<= y 1.1e+81) t_1 (* (/ 2.0 z) (/ x_m y)))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = ((x_m * -2.0) / z) / t;
	double tmp;
	if (y <= -1.16e+100) {
		tmp = 2.0 * ((x_m / z) / y);
	} else if (y <= -2.8e+51) {
		tmp = t_1;
	} else if (y <= -1.35e-33) {
		tmp = 2.0 / (y * (z / x_m));
	} else if (y <= 3.7e-55) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (y <= 1.75e-6) {
		tmp = 2.0 / (z * (y / x_m));
	} else if (y <= 1.1e+81) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m * (-2.0d0)) / z) / t
    if (y <= (-1.16d+100)) then
        tmp = 2.0d0 * ((x_m / z) / y)
    else if (y <= (-2.8d+51)) then
        tmp = t_1
    else if (y <= (-1.35d-33)) then
        tmp = 2.0d0 / (y * (z / x_m))
    else if (y <= 3.7d-55) then
        tmp = ((-2.0d0) / z) * (x_m / t)
    else if (y <= 1.75d-6) then
        tmp = 2.0d0 / (z * (y / x_m))
    else if (y <= 1.1d+81) then
        tmp = t_1
    else
        tmp = (2.0d0 / z) * (x_m / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = ((x_m * -2.0) / z) / t;
	double tmp;
	if (y <= -1.16e+100) {
		tmp = 2.0 * ((x_m / z) / y);
	} else if (y <= -2.8e+51) {
		tmp = t_1;
	} else if (y <= -1.35e-33) {
		tmp = 2.0 / (y * (z / x_m));
	} else if (y <= 3.7e-55) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (y <= 1.75e-6) {
		tmp = 2.0 / (z * (y / x_m));
	} else if (y <= 1.1e+81) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = ((x_m * -2.0) / z) / t
	tmp = 0
	if y <= -1.16e+100:
		tmp = 2.0 * ((x_m / z) / y)
	elif y <= -2.8e+51:
		tmp = t_1
	elif y <= -1.35e-33:
		tmp = 2.0 / (y * (z / x_m))
	elif y <= 3.7e-55:
		tmp = (-2.0 / z) * (x_m / t)
	elif y <= 1.75e-6:
		tmp = 2.0 / (z * (y / x_m))
	elif y <= 1.1e+81:
		tmp = t_1
	else:
		tmp = (2.0 / z) * (x_m / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(Float64(x_m * -2.0) / z) / t)
	tmp = 0.0
	if (y <= -1.16e+100)
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / y));
	elseif (y <= -2.8e+51)
		tmp = t_1;
	elseif (y <= -1.35e-33)
		tmp = Float64(2.0 / Float64(y * Float64(z / x_m)));
	elseif (y <= 3.7e-55)
		tmp = Float64(Float64(-2.0 / z) * Float64(x_m / t));
	elseif (y <= 1.75e-6)
		tmp = Float64(2.0 / Float64(z * Float64(y / x_m)));
	elseif (y <= 1.1e+81)
		tmp = t_1;
	else
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = ((x_m * -2.0) / z) / t;
	tmp = 0.0;
	if (y <= -1.16e+100)
		tmp = 2.0 * ((x_m / z) / y);
	elseif (y <= -2.8e+51)
		tmp = t_1;
	elseif (y <= -1.35e-33)
		tmp = 2.0 / (y * (z / x_m));
	elseif (y <= 3.7e-55)
		tmp = (-2.0 / z) * (x_m / t);
	elseif (y <= 1.75e-6)
		tmp = 2.0 / (z * (y / x_m));
	elseif (y <= 1.1e+81)
		tmp = t_1;
	else
		tmp = (2.0 / z) * (x_m / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / z), $MachinePrecision] / t), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.16e+100], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.8e+51], t$95$1, If[LessEqual[y, -1.35e-33], N[(2.0 / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e-55], N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-6], N[(2.0 / N[(z * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+81], t$95$1, N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 6 regimes

2. if y < -1.1600000000000001e100

   1. Initial program 80.6%

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

      1. remove-double-neg 80.6%

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

      2. distribute-rgt-out-- 82.6%

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

      3. distribute-lft-neg-in 82.6%

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

      4. distribute-rgt-out-- 80.6%

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

      5. remove-double-neg 80.6%

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

      6. distribute-lft-neg-out 80.6%

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

      7. distribute-rgt-neg-in 80.6%

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

      8. neg-mul-1 80.6%

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

      9. times-frac 80.5%

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

      10. associate-*r/ 80.6%

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

      11. distribute-frac-neg 80.6%

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

      12. distribute-neg-frac2 80.6%

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

      13. metadata-eval 80.6%

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

      14. /-rgt-identity 80.6%

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

      15. metadata-eval 80.6%

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

      16. distribute-rgt-neg-out 80.6%

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

      17. distribute-lft-neg-out 80.6%

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

      18. *-commutative 80.6%

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

      19. cancel-sign-sub 80.6%

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

      20. *-commutative 80.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 95.8%

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

   6. Applied egg-rr 95.8%

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

   7. Taylor expanded in t around 0 76.0%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 76.0%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 87.6%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 87.6%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]

   if -1.1600000000000001e100 < y < -2.80000000000000005e51 or 1.74999999999999997e-6 < y < 1.09999999999999993e81

   1. Initial program 84.9%

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

      1. remove-double-neg 84.9%

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

      2. distribute-rgt-out-- 88.8%

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

      3. distribute-lft-neg-in 88.8%

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

      4. distribute-rgt-out-- 84.9%

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

      5. remove-double-neg 84.9%

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

      6. distribute-lft-neg-out 84.9%

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

      7. distribute-rgt-neg-in 84.9%

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

      8. neg-mul-1 84.9%

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

      9. times-frac 84.7%

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

      10. associate-*r/ 84.9%

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

      11. distribute-frac-neg 84.9%

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

      12. distribute-neg-frac2 84.9%

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

      13. metadata-eval 84.9%

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

      14. /-rgt-identity 84.9%

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

      15. metadata-eval 84.9%

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

      16. distribute-rgt-neg-out 84.9%

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

      17. distribute-lft-neg-out 84.9%

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

      18. *-commutative 84.9%

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

      19. cancel-sign-sub 84.9%

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

      20. *-commutative 84.9%

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

   3. Simplified 88.8%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 95.9%

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

   6. Applied egg-rr 95.9%

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

   7. Taylor expanded in t around inf 70.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   8. Step-by-step derivation

      1. associate-*r/ 70.4%

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

      2. *-commutative 70.4%

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

      3. *-commutative 70.4%

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

      4. times-frac 80.9%

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

      5. associate-*r/ 81.2%

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

      6. *-commutative 81.2%

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

      7. associate-*r/ 81.2%

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

   9. Simplified 81.2%

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

   if -2.80000000000000005e51 < y < -1.35e-33

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-rgt-out-- 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. distribute-rgt-out-- 99.8%

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

      5. remove-double-neg 99.8%

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

      6. distribute-lft-neg-out 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

      8. neg-mul-1 99.8%

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

      9. times-frac 99.7%

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

      10. associate-*r/ 99.8%

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

      11. distribute-frac-neg 99.8%

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

      12. distribute-neg-frac2 99.8%

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

      13. metadata-eval 99.8%

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

      14. /-rgt-identity 99.8%

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

      15. metadata-eval 99.8%

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

      16. distribute-rgt-neg-out 99.8%

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

      17. distribute-lft-neg-out 99.8%

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

      18. *-commutative 99.8%

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

      19. cancel-sign-sub 99.8%

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

      20. *-commutative 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in t around 0 87.7%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 87.7%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 87.8%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 87.8%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]
   10. Step-by-step derivation

       1. *-commutative 87.8%

          \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y} \cdot 2} \]

       2. associate-*l/ 87.8%

          \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot 2}{y}} \]

       3. associate-*r/ 87.7%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y}} \]

       4. clear-num 87.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{2}{y} \]

       5. frac-times 87.8%

          \[\leadsto \color{blue}{\frac{1 \cdot 2}{\frac{z}{x} \cdot y}} \]

       6. metadata-eval 87.8%

          \[\leadsto \frac{\color{blue}{2}}{\frac{z}{x} \cdot y} \]

   11. Applied egg-rr 87.8%

       \[\leadsto \color{blue}{\frac{2}{\frac{z}{x} \cdot y}} \]

   if -1.35e-33 < y < 3.69999999999999985e-55

   1. Initial program 94.1%

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

      1. remove-double-neg 94.1%

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

      2. distribute-rgt-out-- 95.1%

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

      3. distribute-lft-neg-in 95.1%

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

      4. distribute-rgt-out-- 92.4%

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

      5. remove-double-neg 92.4%

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

      6. distribute-lft-neg-out 92.4%

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

      7. distribute-rgt-neg-in 92.4%

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

      8. neg-mul-1 92.4%

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

      9. times-frac 91.9%

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

      10. associate-*r/ 92.4%

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

      11. distribute-frac-neg 92.4%

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

      12. distribute-neg-frac2 92.4%

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

      13. metadata-eval 92.4%

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

      14. /-rgt-identity 92.4%

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

      15. metadata-eval 92.4%

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

      16. distribute-rgt-neg-out 92.4%

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

      17. distribute-lft-neg-out 92.4%

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

      18. *-commutative 92.4%

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

      19. cancel-sign-sub 92.4%

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

      20. *-commutative 92.4%

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

   3. Simplified 95.1%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 92.3%

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

   6. Applied egg-rr 92.3%

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

   7. Taylor expanded in t around inf 80.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   8. Step-by-step derivation

      1. associate-*r/ 80.8%

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

      2. *-commutative 80.8%

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

      3. *-commutative 80.8%

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

      4. times-frac 81.0%

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

      5. associate-*r/ 81.0%

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

      6. *-commutative 81.0%

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

      7. associate-*r/ 81.0%

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

   9. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\frac{-2 \cdot x}{z}}{t}} \]
   10. Step-by-step derivation

       1. associate-/l/ 80.8%

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

       2. *-commutative 80.8%

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

       3. times-frac 81.4%

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

   11. Applied egg-rr 81.4%

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

   if 3.69999999999999985e-55 < y < 1.74999999999999997e-6

   1. Initial program 89.7%

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

      1. remove-double-neg 89.7%

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

      2. distribute-rgt-out-- 89.7%

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

      3. distribute-lft-neg-in 89.7%

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

      4. distribute-rgt-out-- 78.6%

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

      5. remove-double-neg 78.6%

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

      6. distribute-lft-neg-out 78.6%

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

      7. distribute-rgt-neg-in 78.6%

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

      8. neg-mul-1 78.6%

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

      9. times-frac 78.4%

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

      10. associate-*r/ 78.6%

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

      11. distribute-frac-neg 78.6%

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

      12. distribute-neg-frac2 78.6%

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

      13. metadata-eval 78.6%

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

      14. /-rgt-identity 78.6%

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

      15. metadata-eval 78.6%

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

      16. distribute-rgt-neg-out 78.6%

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

      17. distribute-lft-neg-out 78.6%

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

      18. *-commutative 78.6%

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

      19. cancel-sign-sub 78.6%

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

      20. *-commutative 78.6%

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

   3. Simplified 89.7%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around 0 89.2%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 89.2%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 89.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 89.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]
   10. Step-by-step derivation

       1. clear-num 89.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{y}{\frac{x}{z}}}} \]

       2. un-div-inv 89.2%

          \[\leadsto \color{blue}{\frac{2}{\frac{y}{\frac{x}{z}}}} \]

       3. associate-/r/ 89.2%

          \[\leadsto \frac{2}{\color{blue}{\frac{y}{x} \cdot z}} \]

   11. Applied egg-rr 89.2%

       \[\leadsto \color{blue}{\frac{2}{\frac{y}{x} \cdot z}} \]

   if 1.09999999999999993e81 < y

   1. Initial program 85.0%

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

      1. remove-double-neg 85.0%

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

      2. distribute-rgt-out-- 85.3%

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

      3. distribute-lft-neg-in 85.3%

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

      4. distribute-rgt-out-- 85.0%

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

      5. remove-double-neg 85.0%

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

      6. distribute-lft-neg-out 85.0%

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

      7. distribute-rgt-neg-in 85.0%

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

      8. neg-mul-1 85.0%

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

      9. times-frac 84.8%

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

      10. associate-*r/ 85.0%

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

      11. distribute-frac-neg 85.0%

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

      12. distribute-neg-frac2 85.0%

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

      13. metadata-eval 85.0%

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

      14. /-rgt-identity 85.0%

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

      15. metadata-eval 85.0%

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

      16. distribute-rgt-neg-out 85.0%

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

      17. distribute-lft-neg-out 85.0%

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

      18. *-commutative 85.0%

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

      19. cancel-sign-sub 85.0%

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

      20. *-commutative 85.0%

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

   3. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.4%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 73.4%

         \[\leadsto \color{blue}{\frac{2 \cdot x}{y \cdot z}} \]

      2. *-commutative 73.4%

         \[\leadsto \frac{2 \cdot x}{\color{blue}{z \cdot y}} \]

      3. times-frac 77.6%

         \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]

   7. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+100}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-55}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := 2 \cdot \frac{\frac{x\_m}{z}}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{+51}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;y \leq 0.00016:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x\_m}}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{-2}{\frac{z \cdot t}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (/ x_m z) y))))
   (*
    x_s
    (if (<= y -7.4e+99)
      t_1
      (if (<= y -9e+51)
        (* x_m (/ (/ -2.0 z) t))
        (if (<= y -7.6e-33)
          t_1
          (if (<= y 2.8e-55)
            (* (/ -2.0 z) (/ x_m t))
            (if (<= y 0.00016)
              (/ 2.0 (* z (/ y x_m)))
              (if (<= y 7.8e+78)
                (/ -2.0 (/ (* z t) x_m))
                (* (/ 2.0 z) (/ x_m y)))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = 2.0 * ((x_m / z) / y);
	double tmp;
	if (y <= -7.4e+99) {
		tmp = t_1;
	} else if (y <= -9e+51) {
		tmp = x_m * ((-2.0 / z) / t);
	} else if (y <= -7.6e-33) {
		tmp = t_1;
	} else if (y <= 2.8e-55) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (y <= 0.00016) {
		tmp = 2.0 / (z * (y / x_m));
	} else if (y <= 7.8e+78) {
		tmp = -2.0 / ((z * t) / x_m);
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * ((x_m / z) / y)
    if (y <= (-7.4d+99)) then
        tmp = t_1
    else if (y <= (-9d+51)) then
        tmp = x_m * (((-2.0d0) / z) / t)
    else if (y <= (-7.6d-33)) then
        tmp = t_1
    else if (y <= 2.8d-55) then
        tmp = ((-2.0d0) / z) * (x_m / t)
    else if (y <= 0.00016d0) then
        tmp = 2.0d0 / (z * (y / x_m))
    else if (y <= 7.8d+78) then
        tmp = (-2.0d0) / ((z * t) / x_m)
    else
        tmp = (2.0d0 / z) * (x_m / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = 2.0 * ((x_m / z) / y);
	double tmp;
	if (y <= -7.4e+99) {
		tmp = t_1;
	} else if (y <= -9e+51) {
		tmp = x_m * ((-2.0 / z) / t);
	} else if (y <= -7.6e-33) {
		tmp = t_1;
	} else if (y <= 2.8e-55) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (y <= 0.00016) {
		tmp = 2.0 / (z * (y / x_m));
	} else if (y <= 7.8e+78) {
		tmp = -2.0 / ((z * t) / x_m);
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = 2.0 * ((x_m / z) / y)
	tmp = 0
	if y <= -7.4e+99:
		tmp = t_1
	elif y <= -9e+51:
		tmp = x_m * ((-2.0 / z) / t)
	elif y <= -7.6e-33:
		tmp = t_1
	elif y <= 2.8e-55:
		tmp = (-2.0 / z) * (x_m / t)
	elif y <= 0.00016:
		tmp = 2.0 / (z * (y / x_m))
	elif y <= 7.8e+78:
		tmp = -2.0 / ((z * t) / x_m)
	else:
		tmp = (2.0 / z) * (x_m / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(2.0 * Float64(Float64(x_m / z) / y))
	tmp = 0.0
	if (y <= -7.4e+99)
		tmp = t_1;
	elseif (y <= -9e+51)
		tmp = Float64(x_m * Float64(Float64(-2.0 / z) / t));
	elseif (y <= -7.6e-33)
		tmp = t_1;
	elseif (y <= 2.8e-55)
		tmp = Float64(Float64(-2.0 / z) * Float64(x_m / t));
	elseif (y <= 0.00016)
		tmp = Float64(2.0 / Float64(z * Float64(y / x_m)));
	elseif (y <= 7.8e+78)
		tmp = Float64(-2.0 / Float64(Float64(z * t) / x_m));
	else
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = 2.0 * ((x_m / z) / y);
	tmp = 0.0;
	if (y <= -7.4e+99)
		tmp = t_1;
	elseif (y <= -9e+51)
		tmp = x_m * ((-2.0 / z) / t);
	elseif (y <= -7.6e-33)
		tmp = t_1;
	elseif (y <= 2.8e-55)
		tmp = (-2.0 / z) * (x_m / t);
	elseif (y <= 0.00016)
		tmp = 2.0 / (z * (y / x_m));
	elseif (y <= 7.8e+78)
		tmp = -2.0 / ((z * t) / x_m);
	else
		tmp = (2.0 / z) * (x_m / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -7.4e+99], t$95$1, If[LessEqual[y, -9e+51], N[(x$95$m * N[(N[(-2.0 / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -7.6e-33], t$95$1, If[LessEqual[y, 2.8e-55], N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00016], N[(2.0 / N[(z * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.8e+78], N[(-2.0 / N[(N[(z * t), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 6 regimes

2. if y < -7.4000000000000002e99 or -8.9999999999999999e51 < y < -7.59999999999999988e-33

   1. Initial program 85.3%

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

      1. remove-double-neg 85.3%

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

      2. distribute-rgt-out-- 86.9%

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

      3. distribute-lft-neg-in 86.9%

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

      4. distribute-rgt-out-- 85.3%

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

      5. remove-double-neg 85.3%

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

      6. distribute-lft-neg-out 85.3%

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

      7. distribute-rgt-neg-in 85.3%

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

      8. neg-mul-1 85.3%

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

      9. times-frac 85.2%

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

      10. associate-*r/ 85.3%

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

      11. distribute-frac-neg 85.3%

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

      12. distribute-neg-frac2 85.3%

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

      13. metadata-eval 85.3%

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

      14. /-rgt-identity 85.3%

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

      15. metadata-eval 85.3%

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

      16. distribute-rgt-neg-out 85.3%

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

      17. distribute-lft-neg-out 85.3%

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

      18. *-commutative 85.3%

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

      19. cancel-sign-sub 85.3%

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

      20. *-commutative 85.3%

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

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 96.8%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in t around 0 78.9%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 78.9%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 87.7%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 87.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]

   if -7.4000000000000002e99 < y < -8.9999999999999999e51

   1. Initial program 91.2%

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

      1. remove-double-neg 91.2%

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

      2. distribute-rgt-out-- 91.2%

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

      3. distribute-lft-neg-in 91.2%

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

      4. distribute-rgt-out-- 91.2%

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

      5. remove-double-neg 91.2%

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

      6. distribute-lft-neg-out 91.2%

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

      7. distribute-rgt-neg-in 91.2%

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

      8. neg-mul-1 91.2%

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

      9. times-frac 91.1%

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

      10. associate-*r/ 91.2%

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

      11. distribute-frac-neg 91.2%

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

      12. distribute-neg-frac2 91.2%

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

      13. metadata-eval 91.2%

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

      14. /-rgt-identity 91.2%

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

      15. metadata-eval 91.2%

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

      16. distribute-rgt-neg-out 91.2%

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

      17. distribute-lft-neg-out 91.2%

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

      18. *-commutative 91.2%

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

      19. cancel-sign-sub 91.2%

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

      20. *-commutative 91.2%

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

   3. Simplified 91.2%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 74.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 74.1%

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

      2. *-commutative 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-/l* 73.9%

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

   7. Simplified 73.9%

      \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]
   8. Step-by-step derivation

      1. associate-/r* 74.1%

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

   9. Applied egg-rr 74.1%

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

   if -7.59999999999999988e-33 < y < 2.79999999999999984e-55

   1. Initial program 94.1%

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

      1. remove-double-neg 94.1%

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

      2. distribute-rgt-out-- 95.1%

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

      3. distribute-lft-neg-in 95.1%

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

      4. distribute-rgt-out-- 92.4%

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

      5. remove-double-neg 92.4%

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

      6. distribute-lft-neg-out 92.4%

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

      7. distribute-rgt-neg-in 92.4%

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

      8. neg-mul-1 92.4%

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

      9. times-frac 91.9%

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

      10. associate-*r/ 92.4%

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

      11. distribute-frac-neg 92.4%

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

      12. distribute-neg-frac2 92.4%

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

      13. metadata-eval 92.4%

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

      14. /-rgt-identity 92.4%

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

      15. metadata-eval 92.4%

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

      16. distribute-rgt-neg-out 92.4%

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

      17. distribute-lft-neg-out 92.4%

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

      18. *-commutative 92.4%

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

      19. cancel-sign-sub 92.4%

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

      20. *-commutative 92.4%

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

   3. Simplified 95.1%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 92.3%

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

   6. Applied egg-rr 92.3%

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

   7. Taylor expanded in t around inf 80.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   8. Step-by-step derivation

      1. associate-*r/ 80.8%

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

      2. *-commutative 80.8%

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

      3. *-commutative 80.8%

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

      4. times-frac 81.0%

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

      5. associate-*r/ 81.0%

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

      6. *-commutative 81.0%

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

      7. associate-*r/ 81.0%

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

   9. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\frac{-2 \cdot x}{z}}{t}} \]
   10. Step-by-step derivation

       1. associate-/l/ 80.8%

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

       2. *-commutative 80.8%

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

       3. times-frac 81.4%

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

   11. Applied egg-rr 81.4%

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

   if 2.79999999999999984e-55 < y < 1.60000000000000013e-4

   1. Initial program 89.7%

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

      1. remove-double-neg 89.7%

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

      2. distribute-rgt-out-- 89.7%

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

      3. distribute-lft-neg-in 89.7%

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

      4. distribute-rgt-out-- 78.6%

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

      5. remove-double-neg 78.6%

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

      6. distribute-lft-neg-out 78.6%

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

      7. distribute-rgt-neg-in 78.6%

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

      8. neg-mul-1 78.6%

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

      9. times-frac 78.4%

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

      10. associate-*r/ 78.6%

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

      11. distribute-frac-neg 78.6%

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

      12. distribute-neg-frac2 78.6%

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

      13. metadata-eval 78.6%

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

      14. /-rgt-identity 78.6%

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

      15. metadata-eval 78.6%

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

      16. distribute-rgt-neg-out 78.6%

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

      17. distribute-lft-neg-out 78.6%

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

      18. *-commutative 78.6%

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

      19. cancel-sign-sub 78.6%

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

      20. *-commutative 78.6%

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

   3. Simplified 89.7%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around 0 89.2%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 89.2%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 89.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 89.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]
   10. Step-by-step derivation

       1. clear-num 89.2%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{y}{\frac{x}{z}}}} \]

       2. un-div-inv 89.2%

          \[\leadsto \color{blue}{\frac{2}{\frac{y}{\frac{x}{z}}}} \]

       3. associate-/r/ 89.2%

          \[\leadsto \frac{2}{\color{blue}{\frac{y}{x} \cdot z}} \]

   11. Applied egg-rr 89.2%

       \[\leadsto \color{blue}{\frac{2}{\frac{y}{x} \cdot z}} \]

   if 1.60000000000000013e-4 < y < 7.8000000000000008e78

   1. Initial program 80.3%

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

      1. remove-double-neg 80.3%

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

      2. distribute-rgt-out-- 87.1%

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

      3. distribute-lft-neg-in 87.1%

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

      4. distribute-rgt-out-- 80.3%

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

      5. remove-double-neg 80.3%

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

      6. distribute-lft-neg-out 80.3%

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

      7. distribute-rgt-neg-in 80.3%

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

      8. neg-mul-1 80.3%

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

      9. times-frac 80.0%

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

      10. associate-*r/ 80.3%

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

      11. distribute-frac-neg 80.3%

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

      12. distribute-neg-frac2 80.3%

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

      13. metadata-eval 80.3%

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

      14. /-rgt-identity 80.3%

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

      15. metadata-eval 80.3%

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

      16. distribute-rgt-neg-out 80.3%

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

      17. distribute-lft-neg-out 80.3%

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

      18. *-commutative 80.3%

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

      19. cancel-sign-sub 80.3%

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

      20. *-commutative 80.3%

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

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. associate-/l/ 87.1%

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

      2. frac-times 80.7%

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

      3. clear-num 80.5%

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

      4. frac-times 80.7%

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

      5. metadata-eval 80.7%

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

   8. Applied egg-rr 80.7%

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

   9. Taylor expanded in t around inf 67.9%

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

   if 7.8000000000000008e78 < y

   1. Initial program 85.0%

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

      1. remove-double-neg 85.0%

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

      2. distribute-rgt-out-- 85.3%

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

      3. distribute-lft-neg-in 85.3%

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

      4. distribute-rgt-out-- 85.0%

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

      5. remove-double-neg 85.0%

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

      6. distribute-lft-neg-out 85.0%

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

      7. distribute-rgt-neg-in 85.0%

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

      8. neg-mul-1 85.0%

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

      9. times-frac 84.8%

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

      10. associate-*r/ 85.0%

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

      11. distribute-frac-neg 85.0%

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

      12. distribute-neg-frac2 85.0%

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

      13. metadata-eval 85.0%

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

      14. /-rgt-identity 85.0%

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

      15. metadata-eval 85.0%

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

      16. distribute-rgt-neg-out 85.0%

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

      17. distribute-lft-neg-out 85.0%

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

      18. *-commutative 85.0%

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

      19. cancel-sign-sub 85.0%

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

      20. *-commutative 85.0%

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

   3. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.4%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 73.4%

         \[\leadsto \color{blue}{\frac{2 \cdot x}{y \cdot z}} \]

      2. *-commutative 73.4%

         \[\leadsto \frac{2 \cdot x}{\color{blue}{z \cdot y}} \]

      3. times-frac 77.6%

         \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]

   7. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+99}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq -9 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-55}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 0.00016:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{-2}{\frac{z \cdot t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := 2 \cdot \frac{\frac{x\_m}{z}}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{+51}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-56}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;y \leq 0.0045:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{-2}{\frac{z \cdot t}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (/ x_m z) y))))
   (*
    x_s
    (if (<= y -4.8e+99)
      t_1
      (if (<= y -9e+51)
        (* x_m (/ (/ -2.0 z) t))
        (if (<= y -7.5e-33)
          t_1
          (if (<= y 7e-56)
            (* (/ -2.0 z) (/ x_m t))
            (if (<= y 0.0045)
              t_1
              (if (<= y 7.8e+78)
                (/ -2.0 (/ (* z t) x_m))
                (* (/ 2.0 z) (/ x_m y)))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = 2.0 * ((x_m / z) / y);
	double tmp;
	if (y <= -4.8e+99) {
		tmp = t_1;
	} else if (y <= -9e+51) {
		tmp = x_m * ((-2.0 / z) / t);
	} else if (y <= -7.5e-33) {
		tmp = t_1;
	} else if (y <= 7e-56) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (y <= 0.0045) {
		tmp = t_1;
	} else if (y <= 7.8e+78) {
		tmp = -2.0 / ((z * t) / x_m);
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * ((x_m / z) / y)
    if (y <= (-4.8d+99)) then
        tmp = t_1
    else if (y <= (-9d+51)) then
        tmp = x_m * (((-2.0d0) / z) / t)
    else if (y <= (-7.5d-33)) then
        tmp = t_1
    else if (y <= 7d-56) then
        tmp = ((-2.0d0) / z) * (x_m / t)
    else if (y <= 0.0045d0) then
        tmp = t_1
    else if (y <= 7.8d+78) then
        tmp = (-2.0d0) / ((z * t) / x_m)
    else
        tmp = (2.0d0 / z) * (x_m / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = 2.0 * ((x_m / z) / y);
	double tmp;
	if (y <= -4.8e+99) {
		tmp = t_1;
	} else if (y <= -9e+51) {
		tmp = x_m * ((-2.0 / z) / t);
	} else if (y <= -7.5e-33) {
		tmp = t_1;
	} else if (y <= 7e-56) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (y <= 0.0045) {
		tmp = t_1;
	} else if (y <= 7.8e+78) {
		tmp = -2.0 / ((z * t) / x_m);
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = 2.0 * ((x_m / z) / y)
	tmp = 0
	if y <= -4.8e+99:
		tmp = t_1
	elif y <= -9e+51:
		tmp = x_m * ((-2.0 / z) / t)
	elif y <= -7.5e-33:
		tmp = t_1
	elif y <= 7e-56:
		tmp = (-2.0 / z) * (x_m / t)
	elif y <= 0.0045:
		tmp = t_1
	elif y <= 7.8e+78:
		tmp = -2.0 / ((z * t) / x_m)
	else:
		tmp = (2.0 / z) * (x_m / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(2.0 * Float64(Float64(x_m / z) / y))
	tmp = 0.0
	if (y <= -4.8e+99)
		tmp = t_1;
	elseif (y <= -9e+51)
		tmp = Float64(x_m * Float64(Float64(-2.0 / z) / t));
	elseif (y <= -7.5e-33)
		tmp = t_1;
	elseif (y <= 7e-56)
		tmp = Float64(Float64(-2.0 / z) * Float64(x_m / t));
	elseif (y <= 0.0045)
		tmp = t_1;
	elseif (y <= 7.8e+78)
		tmp = Float64(-2.0 / Float64(Float64(z * t) / x_m));
	else
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = 2.0 * ((x_m / z) / y);
	tmp = 0.0;
	if (y <= -4.8e+99)
		tmp = t_1;
	elseif (y <= -9e+51)
		tmp = x_m * ((-2.0 / z) / t);
	elseif (y <= -7.5e-33)
		tmp = t_1;
	elseif (y <= 7e-56)
		tmp = (-2.0 / z) * (x_m / t);
	elseif (y <= 0.0045)
		tmp = t_1;
	elseif (y <= 7.8e+78)
		tmp = -2.0 / ((z * t) / x_m);
	else
		tmp = (2.0 / z) * (x_m / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -4.8e+99], t$95$1, If[LessEqual[y, -9e+51], N[(x$95$m * N[(N[(-2.0 / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -7.5e-33], t$95$1, If[LessEqual[y, 7e-56], N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.0045], t$95$1, If[LessEqual[y, 7.8e+78], N[(-2.0 / N[(N[(z * t), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if y < -4.8000000000000002e99 or -8.9999999999999999e51 < y < -7.5000000000000001e-33 or 6.9999999999999996e-56 < y < 0.00449999999999999966

   1. Initial program 85.9%

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

      1. remove-double-neg 85.9%

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

      2. distribute-rgt-out-- 87.2%

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

      3. distribute-lft-neg-in 87.2%

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

      4. distribute-rgt-out-- 84.5%

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

      5. remove-double-neg 84.5%

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

      6. distribute-lft-neg-out 84.5%

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

      7. distribute-rgt-neg-in 84.5%

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

      8. neg-mul-1 84.5%

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

      9. times-frac 84.4%

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

      10. associate-*r/ 84.5%

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

      11. distribute-frac-neg 84.5%

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

      12. distribute-neg-frac2 84.5%

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

      13. metadata-eval 84.5%

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

      14. /-rgt-identity 84.5%

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

      15. metadata-eval 84.5%

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

      16. distribute-rgt-neg-out 84.5%

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

      17. distribute-lft-neg-out 84.5%

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

      18. *-commutative 84.5%

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

      19. cancel-sign-sub 84.5%

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

      20. *-commutative 84.5%

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

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 97.2%

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

   6. Applied egg-rr 97.2%

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

   7. Taylor expanded in t around 0 80.2%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 80.2%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 87.8%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 87.8%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]

   if -4.8000000000000002e99 < y < -8.9999999999999999e51

   1. Initial program 91.2%

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

      1. remove-double-neg 91.2%

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

      2. distribute-rgt-out-- 91.2%

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

      3. distribute-lft-neg-in 91.2%

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

      4. distribute-rgt-out-- 91.2%

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

      5. remove-double-neg 91.2%

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

      6. distribute-lft-neg-out 91.2%

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

      7. distribute-rgt-neg-in 91.2%

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

      8. neg-mul-1 91.2%

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

      9. times-frac 91.1%

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

      10. associate-*r/ 91.2%

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

      11. distribute-frac-neg 91.2%

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

      12. distribute-neg-frac2 91.2%

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

      13. metadata-eval 91.2%

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

      14. /-rgt-identity 91.2%

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

      15. metadata-eval 91.2%

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

      16. distribute-rgt-neg-out 91.2%

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

      17. distribute-lft-neg-out 91.2%

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

      18. *-commutative 91.2%

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

      19. cancel-sign-sub 91.2%

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

      20. *-commutative 91.2%

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

   3. Simplified 91.2%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 74.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 74.1%

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

      2. *-commutative 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-/l* 73.9%

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

   7. Simplified 73.9%

      \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]
   8. Step-by-step derivation

      1. associate-/r* 74.1%

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

   9. Applied egg-rr 74.1%

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

   if -7.5000000000000001e-33 < y < 6.9999999999999996e-56

   1. Initial program 94.1%

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

      1. remove-double-neg 94.1%

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

      2. distribute-rgt-out-- 95.1%

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

      3. distribute-lft-neg-in 95.1%

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

      4. distribute-rgt-out-- 92.4%

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

      5. remove-double-neg 92.4%

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

      6. distribute-lft-neg-out 92.4%

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

      7. distribute-rgt-neg-in 92.4%

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

      8. neg-mul-1 92.4%

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

      9. times-frac 91.9%

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

      10. associate-*r/ 92.4%

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

      11. distribute-frac-neg 92.4%

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

      12. distribute-neg-frac2 92.4%

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

      13. metadata-eval 92.4%

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

      14. /-rgt-identity 92.4%

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

      15. metadata-eval 92.4%

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

      16. distribute-rgt-neg-out 92.4%

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

      17. distribute-lft-neg-out 92.4%

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

      18. *-commutative 92.4%

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

      19. cancel-sign-sub 92.4%

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

      20. *-commutative 92.4%

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

   3. Simplified 95.1%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 92.3%

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

   6. Applied egg-rr 92.3%

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

   7. Taylor expanded in t around inf 80.8%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   8. Step-by-step derivation

      1. associate-*r/ 80.8%

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

      2. *-commutative 80.8%

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

      3. *-commutative 80.8%

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

      4. times-frac 81.0%

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

      5. associate-*r/ 81.0%

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

      6. *-commutative 81.0%

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

      7. associate-*r/ 81.0%

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

   9. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\frac{-2 \cdot x}{z}}{t}} \]
   10. Step-by-step derivation

       1. associate-/l/ 80.8%

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

       2. *-commutative 80.8%

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

       3. times-frac 81.4%

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

   11. Applied egg-rr 81.4%

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

   if 0.00449999999999999966 < y < 7.8000000000000008e78

   1. Initial program 80.3%

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

      1. remove-double-neg 80.3%

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

      2. distribute-rgt-out-- 87.1%

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

      3. distribute-lft-neg-in 87.1%

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

      4. distribute-rgt-out-- 80.3%

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

      5. remove-double-neg 80.3%

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

      6. distribute-lft-neg-out 80.3%

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

      7. distribute-rgt-neg-in 80.3%

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

      8. neg-mul-1 80.3%

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

      9. times-frac 80.0%

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

      10. associate-*r/ 80.3%

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

      11. distribute-frac-neg 80.3%

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

      12. distribute-neg-frac2 80.3%

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

      13. metadata-eval 80.3%

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

      14. /-rgt-identity 80.3%

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

      15. metadata-eval 80.3%

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

      16. distribute-rgt-neg-out 80.3%

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

      17. distribute-lft-neg-out 80.3%

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

      18. *-commutative 80.3%

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

      19. cancel-sign-sub 80.3%

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

      20. *-commutative 80.3%

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

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. associate-/l/ 87.1%

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

      2. frac-times 80.7%

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

      3. clear-num 80.5%

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

      4. frac-times 80.7%

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

      5. metadata-eval 80.7%

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

   8. Applied egg-rr 80.7%

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

   9. Taylor expanded in t around inf 67.9%

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

   if 7.8000000000000008e78 < y

   1. Initial program 85.0%

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

      1. remove-double-neg 85.0%

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

      2. distribute-rgt-out-- 85.3%

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

      3. distribute-lft-neg-in 85.3%

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

      4. distribute-rgt-out-- 85.0%

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

      5. remove-double-neg 85.0%

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

      6. distribute-lft-neg-out 85.0%

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

      7. distribute-rgt-neg-in 85.0%

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

      8. neg-mul-1 85.0%

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

      9. times-frac 84.8%

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

      10. associate-*r/ 85.0%

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

      11. distribute-frac-neg 85.0%

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

      12. distribute-neg-frac2 85.0%

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

      13. metadata-eval 85.0%

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

      14. /-rgt-identity 85.0%

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

      15. metadata-eval 85.0%

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

      16. distribute-rgt-neg-out 85.0%

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

      17. distribute-lft-neg-out 85.0%

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

      18. *-commutative 85.0%

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

      19. cancel-sign-sub 85.0%

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

      20. *-commutative 85.0%

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

   3. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.4%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 73.4%

         \[\leadsto \color{blue}{\frac{2 \cdot x}{y \cdot z}} \]

      2. *-commutative 73.4%

         \[\leadsto \frac{2 \cdot x}{\color{blue}{z \cdot y}} \]

      3. times-frac 77.6%

         \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]

   7. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+99}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq -9 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-56}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 0.0045:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{-2}{\frac{z \cdot t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := 2 \cdot \frac{\frac{x\_m}{z}}{y}\\ t_2 := \frac{-2}{z} \cdot \frac{x\_m}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+51}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 0.08:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+79}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (/ x_m z) y))) (t_2 (* (/ -2.0 z) (/ x_m t))))
   (*
    x_s
    (if (<= y -4.8e+99)
      t_1
      (if (<= y -2.1e+51)
        (* x_m (/ (/ -2.0 z) t))
        (if (<= y -1.4e-33)
          t_1
          (if (<= y 3.25e-55)
            t_2
            (if (<= y 0.08)
              t_1
              (if (<= y 1.6e+79) t_2 (* (/ 2.0 z) (/ x_m y)))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = 2.0 * ((x_m / z) / y);
	double t_2 = (-2.0 / z) * (x_m / t);
	double tmp;
	if (y <= -4.8e+99) {
		tmp = t_1;
	} else if (y <= -2.1e+51) {
		tmp = x_m * ((-2.0 / z) / t);
	} else if (y <= -1.4e-33) {
		tmp = t_1;
	} else if (y <= 3.25e-55) {
		tmp = t_2;
	} else if (y <= 0.08) {
		tmp = t_1;
	} else if (y <= 1.6e+79) {
		tmp = t_2;
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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 = 2.0d0 * ((x_m / z) / y)
    t_2 = ((-2.0d0) / z) * (x_m / t)
    if (y <= (-4.8d+99)) then
        tmp = t_1
    else if (y <= (-2.1d+51)) then
        tmp = x_m * (((-2.0d0) / z) / t)
    else if (y <= (-1.4d-33)) then
        tmp = t_1
    else if (y <= 3.25d-55) then
        tmp = t_2
    else if (y <= 0.08d0) then
        tmp = t_1
    else if (y <= 1.6d+79) then
        tmp = t_2
    else
        tmp = (2.0d0 / z) * (x_m / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = 2.0 * ((x_m / z) / y);
	double t_2 = (-2.0 / z) * (x_m / t);
	double tmp;
	if (y <= -4.8e+99) {
		tmp = t_1;
	} else if (y <= -2.1e+51) {
		tmp = x_m * ((-2.0 / z) / t);
	} else if (y <= -1.4e-33) {
		tmp = t_1;
	} else if (y <= 3.25e-55) {
		tmp = t_2;
	} else if (y <= 0.08) {
		tmp = t_1;
	} else if (y <= 1.6e+79) {
		tmp = t_2;
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = 2.0 * ((x_m / z) / y)
	t_2 = (-2.0 / z) * (x_m / t)
	tmp = 0
	if y <= -4.8e+99:
		tmp = t_1
	elif y <= -2.1e+51:
		tmp = x_m * ((-2.0 / z) / t)
	elif y <= -1.4e-33:
		tmp = t_1
	elif y <= 3.25e-55:
		tmp = t_2
	elif y <= 0.08:
		tmp = t_1
	elif y <= 1.6e+79:
		tmp = t_2
	else:
		tmp = (2.0 / z) * (x_m / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(2.0 * Float64(Float64(x_m / z) / y))
	t_2 = Float64(Float64(-2.0 / z) * Float64(x_m / t))
	tmp = 0.0
	if (y <= -4.8e+99)
		tmp = t_1;
	elseif (y <= -2.1e+51)
		tmp = Float64(x_m * Float64(Float64(-2.0 / z) / t));
	elseif (y <= -1.4e-33)
		tmp = t_1;
	elseif (y <= 3.25e-55)
		tmp = t_2;
	elseif (y <= 0.08)
		tmp = t_1;
	elseif (y <= 1.6e+79)
		tmp = t_2;
	else
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = 2.0 * ((x_m / z) / y);
	t_2 = (-2.0 / z) * (x_m / t);
	tmp = 0.0;
	if (y <= -4.8e+99)
		tmp = t_1;
	elseif (y <= -2.1e+51)
		tmp = x_m * ((-2.0 / z) / t);
	elseif (y <= -1.4e-33)
		tmp = t_1;
	elseif (y <= 3.25e-55)
		tmp = t_2;
	elseif (y <= 0.08)
		tmp = t_1;
	elseif (y <= 1.6e+79)
		tmp = t_2;
	else
		tmp = (2.0 / z) * (x_m / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -4.8e+99], t$95$1, If[LessEqual[y, -2.1e+51], N[(x$95$m * N[(N[(-2.0 / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.4e-33], t$95$1, If[LessEqual[y, 3.25e-55], t$95$2, If[LessEqual[y, 0.08], t$95$1, If[LessEqual[y, 1.6e+79], t$95$2, N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.8000000000000002e99 or -2.1000000000000001e51 < y < -1.4e-33 or 3.25000000000000003e-55 < y < 0.0800000000000000017

   1. Initial program 85.9%

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

      1. remove-double-neg 85.9%

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

      2. distribute-rgt-out-- 87.2%

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

      3. distribute-lft-neg-in 87.2%

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

      4. distribute-rgt-out-- 84.5%

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

      5. remove-double-neg 84.5%

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

      6. distribute-lft-neg-out 84.5%

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

      7. distribute-rgt-neg-in 84.5%

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

      8. neg-mul-1 84.5%

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

      9. times-frac 84.4%

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

      10. associate-*r/ 84.5%

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

      11. distribute-frac-neg 84.5%

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

      12. distribute-neg-frac2 84.5%

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

      13. metadata-eval 84.5%

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

      14. /-rgt-identity 84.5%

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

      15. metadata-eval 84.5%

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

      16. distribute-rgt-neg-out 84.5%

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

      17. distribute-lft-neg-out 84.5%

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

      18. *-commutative 84.5%

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

      19. cancel-sign-sub 84.5%

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

      20. *-commutative 84.5%

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

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 97.2%

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

   6. Applied egg-rr 97.2%

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

   7. Taylor expanded in t around 0 80.2%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 80.2%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 87.8%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 87.8%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]

   if -4.8000000000000002e99 < y < -2.1000000000000001e51

   1. Initial program 91.2%

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

      1. remove-double-neg 91.2%

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

      2. distribute-rgt-out-- 91.2%

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

      3. distribute-lft-neg-in 91.2%

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

      4. distribute-rgt-out-- 91.2%

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

      5. remove-double-neg 91.2%

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

      6. distribute-lft-neg-out 91.2%

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

      7. distribute-rgt-neg-in 91.2%

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

      8. neg-mul-1 91.2%

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

      9. times-frac 91.1%

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

      10. associate-*r/ 91.2%

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

      11. distribute-frac-neg 91.2%

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

      12. distribute-neg-frac2 91.2%

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

      13. metadata-eval 91.2%

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

      14. /-rgt-identity 91.2%

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

      15. metadata-eval 91.2%

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

      16. distribute-rgt-neg-out 91.2%

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

      17. distribute-lft-neg-out 91.2%

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

      18. *-commutative 91.2%

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

      19. cancel-sign-sub 91.2%

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

      20. *-commutative 91.2%

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

   3. Simplified 91.2%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 74.1%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 74.1%

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

      2. *-commutative 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-/l* 73.9%

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

   7. Simplified 73.9%

      \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]
   8. Step-by-step derivation

      1. associate-/r* 74.1%

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

   9. Applied egg-rr 74.1%

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

   if -1.4e-33 < y < 3.25000000000000003e-55 or 0.0800000000000000017 < y < 1.60000000000000001e79

   1. Initial program 92.4%

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

      1. remove-double-neg 92.4%

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

      2. distribute-rgt-out-- 94.1%

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

      3. distribute-lft-neg-in 94.1%

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

      4. distribute-rgt-out-- 91.0%

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

      5. remove-double-neg 91.0%

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

      6. distribute-lft-neg-out 91.0%

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

      7. distribute-rgt-neg-in 91.0%

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

      8. neg-mul-1 91.0%

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

      9. times-frac 90.5%

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

      10. associate-*r/ 91.0%

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

      11. distribute-frac-neg 91.0%

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

      12. distribute-neg-frac2 91.0%

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

      13. metadata-eval 91.0%

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

      14. /-rgt-identity 91.0%

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

      15. metadata-eval 91.0%

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

      16. distribute-rgt-neg-out 91.0%

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

      17. distribute-lft-neg-out 91.0%

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

      18. *-commutative 91.0%

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

      19. cancel-sign-sub 91.0%

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

      20. *-commutative 91.0%

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

   3. Simplified 94.1%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 93.2%

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

   6. Applied egg-rr 93.2%

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

   7. Taylor expanded in t around inf 79.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   8. Step-by-step derivation

      1. associate-*r/ 79.2%

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

      2. *-commutative 79.2%

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

      3. *-commutative 79.2%

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

      4. times-frac 80.8%

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

      5. associate-*r/ 80.9%

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

      6. *-commutative 80.9%

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

      7. associate-*r/ 80.9%

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

   9. Simplified 80.9%

      \[\leadsto \color{blue}{\frac{\frac{-2 \cdot x}{z}}{t}} \]
   10. Step-by-step derivation

       1. associate-/l/ 79.2%

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

       2. *-commutative 79.2%

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

       3. times-frac 79.8%

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

   11. Applied egg-rr 79.8%

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

   if 1.60000000000000001e79 < y

   1. Initial program 85.0%

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

      1. remove-double-neg 85.0%

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

      2. distribute-rgt-out-- 85.3%

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

      3. distribute-lft-neg-in 85.3%

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

      4. distribute-rgt-out-- 85.0%

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

      5. remove-double-neg 85.0%

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

      6. distribute-lft-neg-out 85.0%

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

      7. distribute-rgt-neg-in 85.0%

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

      8. neg-mul-1 85.0%

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

      9. times-frac 84.8%

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

      10. associate-*r/ 85.0%

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

      11. distribute-frac-neg 85.0%

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

      12. distribute-neg-frac2 85.0%

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

      13. metadata-eval 85.0%

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

      14. /-rgt-identity 85.0%

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

      15. metadata-eval 85.0%

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

      16. distribute-rgt-neg-out 85.0%

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

      17. distribute-lft-neg-out 85.0%

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

      18. *-commutative 85.0%

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

      19. cancel-sign-sub 85.0%

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

      20. *-commutative 85.0%

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

   3. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 73.4%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 73.4%

         \[\leadsto \color{blue}{\frac{2 \cdot x}{y \cdot z}} \]

      2. *-commutative 73.4%

         \[\leadsto \frac{2 \cdot x}{\color{blue}{z \cdot y}} \]

      3. times-frac 77.6%

         \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]

   7. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+99}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+51}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-55}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;y \leq 0.08:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+99} \lor \neg \left(y \leq -2.7 \cdot 10^{+51} \lor \neg \left(y \leq -1.1 \cdot 10^{-31}\right) \land y \leq 4.6 \cdot 10^{-56}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -4.8e+99)
          (not
           (or (<= y -2.7e+51) (and (not (<= y -1.1e-31)) (<= y 4.6e-56)))))
    (* 2.0 (/ (/ x_m z) y))
    (* x_m (/ (/ -2.0 z) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -4.8e+99) || !((y <= -2.7e+51) || (!(y <= -1.1e-31) && (y <= 4.6e-56)))) {
		tmp = 2.0 * ((x_m / z) / y);
	} else {
		tmp = x_m * ((-2.0 / z) / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-4.8d+99)) .or. (.not. (y <= (-2.7d+51)) .or. (.not. (y <= (-1.1d-31))) .and. (y <= 4.6d-56))) then
        tmp = 2.0d0 * ((x_m / z) / y)
    else
        tmp = x_m * (((-2.0d0) / z) / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -4.8e+99) || !((y <= -2.7e+51) || (!(y <= -1.1e-31) && (y <= 4.6e-56)))) {
		tmp = 2.0 * ((x_m / z) / y);
	} else {
		tmp = x_m * ((-2.0 / z) / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -4.8e+99) or not ((y <= -2.7e+51) or (not (y <= -1.1e-31) and (y <= 4.6e-56))):
		tmp = 2.0 * ((x_m / z) / y)
	else:
		tmp = x_m * ((-2.0 / z) / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -4.8e+99) || !((y <= -2.7e+51) || (!(y <= -1.1e-31) && (y <= 4.6e-56))))
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / y));
	else
		tmp = Float64(x_m * Float64(Float64(-2.0 / z) / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -4.8e+99) || ~(((y <= -2.7e+51) || (~((y <= -1.1e-31)) && (y <= 4.6e-56)))))
		tmp = 2.0 * ((x_m / z) / y);
	else
		tmp = x_m * ((-2.0 / z) / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -4.8e+99], N[Not[Or[LessEqual[y, -2.7e+51], And[N[Not[LessEqual[y, -1.1e-31]], $MachinePrecision], LessEqual[y, 4.6e-56]]]], $MachinePrecision]], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(-2.0 / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000002e99 or -2.69999999999999992e51 < y < -1.10000000000000005e-31 or 4.60000000000000005e-56 < y

   1. Initial program 84.9%

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

      1. remove-double-neg 84.9%

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

      2. distribute-rgt-out-- 86.6%

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

      3. distribute-lft-neg-in 86.6%

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

      4. distribute-rgt-out-- 84.2%

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

      5. remove-double-neg 84.2%

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

      6. distribute-lft-neg-out 84.2%

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

      7. distribute-rgt-neg-in 84.2%

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

      8. neg-mul-1 84.2%

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

      9. times-frac 84.1%

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

      10. associate-*r/ 84.2%

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

      11. distribute-frac-neg 84.2%

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

      12. distribute-neg-frac2 84.2%

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

      13. metadata-eval 84.2%

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

      14. /-rgt-identity 84.2%

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

      15. metadata-eval 84.2%

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

      16. distribute-rgt-neg-out 84.2%

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

      17. distribute-lft-neg-out 84.2%

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

      18. *-commutative 84.2%

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

      19. cancel-sign-sub 84.2%

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

      20. *-commutative 84.2%

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

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 95.4%

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

   6. Applied egg-rr 95.4%

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

   7. Taylor expanded in t around 0 72.3%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 72.3%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 77.2%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 77.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]

   if -4.8000000000000002e99 < y < -2.69999999999999992e51 or -1.10000000000000005e-31 < y < 4.60000000000000005e-56

   1. Initial program 93.8%

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

      1. remove-double-neg 93.8%

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

      2. distribute-rgt-out-- 94.7%

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

      3. distribute-lft-neg-in 94.7%

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

      4. distribute-rgt-out-- 92.3%

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

      5. remove-double-neg 92.3%

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

      6. distribute-lft-neg-out 92.3%

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

      7. distribute-rgt-neg-in 92.3%

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

      8. neg-mul-1 92.3%

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

      9. times-frac 91.8%

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

      10. associate-*r/ 92.3%

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

      11. distribute-frac-neg 92.3%

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

      12. distribute-neg-frac2 92.3%

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

      13. metadata-eval 92.3%

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

      14. /-rgt-identity 92.3%

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

      15. metadata-eval 92.3%

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

      16. distribute-rgt-neg-out 92.3%

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

      17. distribute-lft-neg-out 92.3%

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

      18. *-commutative 92.3%

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

      19. cancel-sign-sub 92.3%

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

      20. *-commutative 92.3%

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

   3. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 80.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 80.2%

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

      2. *-commutative 80.2%

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

      3. *-commutative 80.2%

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

      4. associate-/l* 79.8%

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

   7. Simplified 79.8%

      \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]
   8. Step-by-step derivation

      1. associate-/r* 80.1%

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

   9. Applied egg-rr 80.1%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+99} \lor \neg \left(y \leq -2.7 \cdot 10^{+51} \lor \neg \left(y \leq -1.1 \cdot 10^{-31}\right) \land y \leq 4.6 \cdot 10^{-56}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+101} \lor \neg \left(y \leq -1 \cdot 10^{+52} \lor \neg \left(y \leq -5.2 \cdot 10^{-32}\right) \land y \leq 1.05 \cdot 10^{-55}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{-2}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -1.25e+101)
          (not (or (<= y -1e+52) (and (not (<= y -5.2e-32)) (<= y 1.05e-55)))))
    (* 2.0 (/ (/ x_m z) y))
    (* x_m (/ -2.0 (* z t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -1.25e+101) || !((y <= -1e+52) || (!(y <= -5.2e-32) && (y <= 1.05e-55)))) {
		tmp = 2.0 * ((x_m / z) / y);
	} else {
		tmp = x_m * (-2.0 / (z * t));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-1.25d+101)) .or. (.not. (y <= (-1d+52)) .or. (.not. (y <= (-5.2d-32))) .and. (y <= 1.05d-55))) then
        tmp = 2.0d0 * ((x_m / z) / y)
    else
        tmp = x_m * ((-2.0d0) / (z * t))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -1.25e+101) || !((y <= -1e+52) || (!(y <= -5.2e-32) && (y <= 1.05e-55)))) {
		tmp = 2.0 * ((x_m / z) / y);
	} else {
		tmp = x_m * (-2.0 / (z * t));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -1.25e+101) or not ((y <= -1e+52) or (not (y <= -5.2e-32) and (y <= 1.05e-55))):
		tmp = 2.0 * ((x_m / z) / y)
	else:
		tmp = x_m * (-2.0 / (z * t))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -1.25e+101) || !((y <= -1e+52) || (!(y <= -5.2e-32) && (y <= 1.05e-55))))
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / y));
	else
		tmp = Float64(x_m * Float64(-2.0 / Float64(z * t)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -1.25e+101) || ~(((y <= -1e+52) || (~((y <= -5.2e-32)) && (y <= 1.05e-55)))))
		tmp = 2.0 * ((x_m / z) / y);
	else
		tmp = x_m * (-2.0 / (z * t));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -1.25e+101], N[Not[Or[LessEqual[y, -1e+52], And[N[Not[LessEqual[y, -5.2e-32]], $MachinePrecision], LessEqual[y, 1.05e-55]]]], $MachinePrecision]], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(-2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.24999999999999997e101 or -9.9999999999999999e51 < y < -5.1999999999999995e-32 or 1.0500000000000001e-55 < y

   1. Initial program 84.9%

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

      1. remove-double-neg 84.9%

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

      2. distribute-rgt-out-- 86.6%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 86.6%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 84.2%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 84.2%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 84.2%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 84.2%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 84.2%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 84.1%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 84.2%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 84.2%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 84.2%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 84.2%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 84.2%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 84.2%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 84.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 84.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 84.2%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 84.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 84.2%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 95.4%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   6. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   7. Taylor expanded in t around 0 72.3%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 72.3%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 77.2%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 77.2%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]

   if -1.24999999999999997e101 < y < -9.9999999999999999e51 or -5.1999999999999995e-32 < y < 1.0500000000000001e-55

   1. Initial program 93.8%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 93.8%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 94.7%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 94.7%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 92.3%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 92.3%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 92.3%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 92.3%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 92.3%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 91.8%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 92.3%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 92.3%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 92.3%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 92.3%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 92.3%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 92.3%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 92.3%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 92.3%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 92.3%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 92.3%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 92.3%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 80.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 80.2%

         \[\leadsto \color{blue}{\frac{-2 \cdot x}{t \cdot z}} \]

      2. *-commutative 80.2%

         \[\leadsto \frac{\color{blue}{x \cdot -2}}{t \cdot z} \]

      3. *-commutative 80.2%

         \[\leadsto \frac{x \cdot -2}{\color{blue}{z \cdot t}} \]

      4. associate-/l* 79.8%

         \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]

   7. Simplified 79.8%

      \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+101} \lor \neg \left(y \leq -1 \cdot 10^{+52} \lor \neg \left(y \leq -5.2 \cdot 10^{-32}\right) \land y \leq 1.05 \cdot 10^{-55}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{-2}{z} \cdot \frac{x\_m}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.35 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-133} \lor \neg \left(t \leq 1.4 \cdot 10^{+75}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (/ -2.0 z) (/ x_m t))))
   (*
    x_s
    (if (<= t -3.35e+84)
      t_1
      (if (<= t -3.8e-94)
        (* 2.0 (/ (/ x_m z) y))
        (if (or (<= t -3.5e-133) (not (<= t 1.4e+75)))
          t_1
          (/ 2.0 (* y (/ z x_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (-2.0 / z) * (x_m / t);
	double tmp;
	if (t <= -3.35e+84) {
		tmp = t_1;
	} else if (t <= -3.8e-94) {
		tmp = 2.0 * ((x_m / z) / y);
	} else if ((t <= -3.5e-133) || !(t <= 1.4e+75)) {
		tmp = t_1;
	} else {
		tmp = 2.0 / (y * (z / x_m));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-2.0d0) / z) * (x_m / t)
    if (t <= (-3.35d+84)) then
        tmp = t_1
    else if (t <= (-3.8d-94)) then
        tmp = 2.0d0 * ((x_m / z) / y)
    else if ((t <= (-3.5d-133)) .or. (.not. (t <= 1.4d+75))) then
        tmp = t_1
    else
        tmp = 2.0d0 / (y * (z / x_m))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (-2.0 / z) * (x_m / t);
	double tmp;
	if (t <= -3.35e+84) {
		tmp = t_1;
	} else if (t <= -3.8e-94) {
		tmp = 2.0 * ((x_m / z) / y);
	} else if ((t <= -3.5e-133) || !(t <= 1.4e+75)) {
		tmp = t_1;
	} else {
		tmp = 2.0 / (y * (z / x_m));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (-2.0 / z) * (x_m / t)
	tmp = 0
	if t <= -3.35e+84:
		tmp = t_1
	elif t <= -3.8e-94:
		tmp = 2.0 * ((x_m / z) / y)
	elif (t <= -3.5e-133) or not (t <= 1.4e+75):
		tmp = t_1
	else:
		tmp = 2.0 / (y * (z / x_m))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(-2.0 / z) * Float64(x_m / t))
	tmp = 0.0
	if (t <= -3.35e+84)
		tmp = t_1;
	elseif (t <= -3.8e-94)
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / y));
	elseif ((t <= -3.5e-133) || !(t <= 1.4e+75))
		tmp = t_1;
	else
		tmp = Float64(2.0 / Float64(y * Float64(z / x_m)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (-2.0 / z) * (x_m / t);
	tmp = 0.0;
	if (t <= -3.35e+84)
		tmp = t_1;
	elseif (t <= -3.8e-94)
		tmp = 2.0 * ((x_m / z) / y);
	elseif ((t <= -3.5e-133) || ~((t <= 1.4e+75)))
		tmp = t_1;
	else
		tmp = 2.0 / (y * (z / x_m));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -3.35e+84], t$95$1, If[LessEqual[t, -3.8e-94], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -3.5e-133], N[Not[LessEqual[t, 1.4e+75]], $MachinePrecision]], t$95$1, N[(2.0 / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < -3.3500000000000002e84 or -3.79999999999999999e-94 < t < -3.50000000000000003e-133 or 1.40000000000000006e75 < t

   1. Initial program 85.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 85.3%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 88.4%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 88.4%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 85.4%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 85.4%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 85.4%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 85.4%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 85.4%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 85.4%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 85.4%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 85.4%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 85.4%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 85.4%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 85.4%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 85.4%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 85.4%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 85.4%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 85.4%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 85.4%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 85.4%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 88.4%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 95.2%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   6. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   7. Taylor expanded in t around inf 79.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   8. Step-by-step derivation

      1. associate-*r/ 79.9%

         \[\leadsto \color{blue}{\frac{-2 \cdot x}{t \cdot z}} \]

      2. *-commutative 79.9%

         \[\leadsto \frac{\color{blue}{x \cdot -2}}{t \cdot z} \]

      3. *-commutative 79.9%

         \[\leadsto \frac{x \cdot -2}{\color{blue}{z \cdot t}} \]

      4. times-frac 85.2%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{-2}{t}} \]

      5. associate-*r/ 85.3%

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot -2}{t}} \]

      6. *-commutative 85.3%

         \[\leadsto \frac{\color{blue}{-2 \cdot \frac{x}{z}}}{t} \]

      7. associate-*r/ 85.3%

         \[\leadsto \frac{\color{blue}{\frac{-2 \cdot x}{z}}}{t} \]

   9. Simplified 85.3%

      \[\leadsto \color{blue}{\frac{\frac{-2 \cdot x}{z}}{t}} \]
   10. Step-by-step derivation

       1. associate-/l/ 79.9%

          \[\leadsto \color{blue}{\frac{-2 \cdot x}{t \cdot z}} \]

       2. *-commutative 79.9%

          \[\leadsto \frac{\color{blue}{x \cdot -2}}{t \cdot z} \]

       3. times-frac 83.5%

          \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{-2}{z}} \]

   11. Applied egg-rr 83.5%

       \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{-2}{z}} \]

   if -3.3500000000000002e84 < t < -3.79999999999999999e-94

   1. Initial program 94.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 94.2%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 94.2%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 94.2%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 94.2%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 94.2%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 94.2%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 94.2%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 94.2%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 94.1%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 94.2%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 94.2%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 94.2%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 94.2%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 94.2%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 94.2%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 94.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 94.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 94.2%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 94.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 94.2%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 95.8%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   6. Applied egg-rr 95.8%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   7. Taylor expanded in t around 0 65.8%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 65.8%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 70.3%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 70.3%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]

   if -3.50000000000000003e-133 < t < 1.40000000000000006e75

   1. Initial program 91.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 91.2%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 91.3%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 91.3%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 88.7%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 88.7%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 88.7%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 88.7%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 88.7%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 88.1%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 88.7%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 88.7%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 88.7%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 88.7%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 88.7%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 88.7%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 88.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 88.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 88.7%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 88.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 88.7%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 91.3%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 92.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   6. Applied egg-rr 92.1%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   7. Taylor expanded in t around 0 72.1%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 72.1%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 75.6%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 75.6%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]
   10. Step-by-step derivation

       1. *-commutative 75.6%

          \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y} \cdot 2} \]

       2. associate-*l/ 75.6%

          \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot 2}{y}} \]

       3. associate-*r/ 75.5%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y}} \]

       4. clear-num 75.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{2}{y} \]

       5. frac-times 76.2%

          \[\leadsto \color{blue}{\frac{1 \cdot 2}{\frac{z}{x} \cdot y}} \]

       6. metadata-eval 76.2%

          \[\leadsto \frac{\color{blue}{2}}{\frac{z}{x} \cdot y} \]

   11. Applied egg-rr 76.2%

       \[\leadsto \color{blue}{\frac{2}{\frac{z}{x} \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.35 \cdot 10^{+84}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-133} \lor \neg \left(t \leq 1.4 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{-2}{z} \cdot \frac{x\_m}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{2}{y}}{\frac{z}{x\_m}}\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-126}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+75}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{z}}{\frac{t}{x\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (/ -2.0 z) (/ x_m t))))
   (*
    x_s
    (if (<= t -1.1e+92)
      t_1
      (if (<= t -1.8e-95)
        (/ (/ 2.0 y) (/ z x_m))
        (if (<= t -1.75e-126)
          t_1
          (if (<= t 1.3e+75)
            (/ 2.0 (* y (/ z x_m)))
            (/ (/ -2.0 z) (/ t x_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (-2.0 / z) * (x_m / t);
	double tmp;
	if (t <= -1.1e+92) {
		tmp = t_1;
	} else if (t <= -1.8e-95) {
		tmp = (2.0 / y) / (z / x_m);
	} else if (t <= -1.75e-126) {
		tmp = t_1;
	} else if (t <= 1.3e+75) {
		tmp = 2.0 / (y * (z / x_m));
	} else {
		tmp = (-2.0 / z) / (t / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-2.0d0) / z) * (x_m / t)
    if (t <= (-1.1d+92)) then
        tmp = t_1
    else if (t <= (-1.8d-95)) then
        tmp = (2.0d0 / y) / (z / x_m)
    else if (t <= (-1.75d-126)) then
        tmp = t_1
    else if (t <= 1.3d+75) then
        tmp = 2.0d0 / (y * (z / x_m))
    else
        tmp = ((-2.0d0) / z) / (t / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (-2.0 / z) * (x_m / t);
	double tmp;
	if (t <= -1.1e+92) {
		tmp = t_1;
	} else if (t <= -1.8e-95) {
		tmp = (2.0 / y) / (z / x_m);
	} else if (t <= -1.75e-126) {
		tmp = t_1;
	} else if (t <= 1.3e+75) {
		tmp = 2.0 / (y * (z / x_m));
	} else {
		tmp = (-2.0 / z) / (t / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (-2.0 / z) * (x_m / t)
	tmp = 0
	if t <= -1.1e+92:
		tmp = t_1
	elif t <= -1.8e-95:
		tmp = (2.0 / y) / (z / x_m)
	elif t <= -1.75e-126:
		tmp = t_1
	elif t <= 1.3e+75:
		tmp = 2.0 / (y * (z / x_m))
	else:
		tmp = (-2.0 / z) / (t / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(-2.0 / z) * Float64(x_m / t))
	tmp = 0.0
	if (t <= -1.1e+92)
		tmp = t_1;
	elseif (t <= -1.8e-95)
		tmp = Float64(Float64(2.0 / y) / Float64(z / x_m));
	elseif (t <= -1.75e-126)
		tmp = t_1;
	elseif (t <= 1.3e+75)
		tmp = Float64(2.0 / Float64(y * Float64(z / x_m)));
	else
		tmp = Float64(Float64(-2.0 / z) / Float64(t / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (-2.0 / z) * (x_m / t);
	tmp = 0.0;
	if (t <= -1.1e+92)
		tmp = t_1;
	elseif (t <= -1.8e-95)
		tmp = (2.0 / y) / (z / x_m);
	elseif (t <= -1.75e-126)
		tmp = t_1;
	elseif (t <= 1.3e+75)
		tmp = 2.0 / (y * (z / x_m));
	else
		tmp = (-2.0 / z) / (t / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -1.1e+92], t$95$1, If[LessEqual[t, -1.8e-95], N[(N[(2.0 / y), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.75e-126], t$95$1, If[LessEqual[t, 1.3e+75], N[(2.0 / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / z), $MachinePrecision] / N[(t / x$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < -1.09999999999999996e92 or -1.8e-95 < t < -1.75e-126

   1. Initial program 87.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 87.0%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 89.4%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 89.4%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 87.2%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 87.2%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 87.2%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 87.2%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 87.2%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 87.1%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 87.2%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 87.2%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 87.2%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 87.2%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 87.2%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 87.2%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 87.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 87.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 87.2%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 87.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 87.2%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 89.4%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 95.4%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   6. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   7. Taylor expanded in t around inf 83.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   8. Step-by-step derivation

      1. associate-*r/ 83.3%

         \[\leadsto \color{blue}{\frac{-2 \cdot x}{t \cdot z}} \]

      2. *-commutative 83.3%

         \[\leadsto \frac{\color{blue}{x \cdot -2}}{t \cdot z} \]

      3. *-commutative 83.3%

         \[\leadsto \frac{x \cdot -2}{\color{blue}{z \cdot t}} \]

      4. times-frac 87.8%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{-2}{t}} \]

      5. associate-*r/ 87.9%

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot -2}{t}} \]

      6. *-commutative 87.9%

         \[\leadsto \frac{\color{blue}{-2 \cdot \frac{x}{z}}}{t} \]

      7. associate-*r/ 87.9%

         \[\leadsto \frac{\color{blue}{\frac{-2 \cdot x}{z}}}{t} \]

   9. Simplified 87.9%

      \[\leadsto \color{blue}{\frac{\frac{-2 \cdot x}{z}}{t}} \]
   10. Step-by-step derivation

       1. associate-/l/ 83.3%

          \[\leadsto \color{blue}{\frac{-2 \cdot x}{t \cdot z}} \]

       2. *-commutative 83.3%

          \[\leadsto \frac{\color{blue}{x \cdot -2}}{t \cdot z} \]

       3. times-frac 88.4%

          \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{-2}{z}} \]

   11. Applied egg-rr 88.4%

       \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{-2}{z}} \]

   if -1.09999999999999996e92 < t < -1.8e-95

   1. Initial program 91.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 91.7%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 94.5%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 94.5%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 91.7%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 91.7%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 91.7%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 91.7%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 91.7%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 91.6%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 91.7%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 91.7%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 91.7%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 91.7%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 91.7%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 91.7%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 91.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 91.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 91.7%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 91.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 91.7%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 94.5%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 96.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   6. Applied egg-rr 96.1%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   7. Taylor expanded in t around 0 64.9%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 64.9%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 69.1%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 69.1%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]
   10. Step-by-step derivation

       1. *-commutative 69.1%

          \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y} \cdot 2} \]

       2. associate-*l/ 69.1%

          \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot 2}{y}} \]

       3. associate-*r/ 69.2%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y}} \]

       4. clear-num 69.2%

          \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{2}{y} \]

       5. associate-*l/ 69.3%

          \[\leadsto \color{blue}{\frac{1 \cdot \frac{2}{y}}{\frac{z}{x}}} \]

       6. *-un-lft-identity 69.3%

          \[\leadsto \frac{\color{blue}{\frac{2}{y}}}{\frac{z}{x}} \]

   11. Applied egg-rr 69.3%

       \[\leadsto \color{blue}{\frac{\frac{2}{y}}{\frac{z}{x}}} \]

   if -1.75e-126 < t < 1.29999999999999992e75

   1. Initial program 91.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 91.3%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 91.3%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 91.3%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 88.7%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 88.7%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 88.7%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 88.7%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 88.7%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 88.2%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 88.7%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 88.7%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 88.7%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 88.7%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 88.7%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 88.7%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 88.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 88.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 88.7%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 88.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 88.7%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 91.3%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 91.4%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   6. Applied egg-rr 91.4%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   7. Taylor expanded in t around 0 72.3%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 72.3%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 75.0%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 75.0%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]
   10. Step-by-step derivation

       1. *-commutative 75.0%

          \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y} \cdot 2} \]

       2. associate-*l/ 75.0%

          \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot 2}{y}} \]

       3. associate-*r/ 74.9%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y}} \]

       4. clear-num 74.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{2}{y} \]

       5. frac-times 75.6%

          \[\leadsto \color{blue}{\frac{1 \cdot 2}{\frac{z}{x} \cdot y}} \]

       6. metadata-eval 75.6%

          \[\leadsto \frac{\color{blue}{2}}{\frac{z}{x} \cdot y} \]

   11. Applied egg-rr 75.6%

       \[\leadsto \color{blue}{\frac{2}{\frac{z}{x} \cdot y}} \]

   if 1.29999999999999992e75 < t

   1. Initial program 85.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 85.0%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 87.0%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 87.0%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 85.0%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 85.0%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 85.0%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 85.0%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 85.0%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 85.0%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 85.0%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 85.0%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 85.0%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 85.0%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 85.0%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 85.0%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 85.0%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 85.0%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 85.0%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 85.0%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 85.0%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 79.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 79.6%

         \[\leadsto \color{blue}{\frac{-2 \cdot x}{t \cdot z}} \]

      2. *-commutative 79.6%

         \[\leadsto \frac{\color{blue}{x \cdot -2}}{t \cdot z} \]

      3. *-commutative 79.6%

         \[\leadsto \frac{x \cdot -2}{\color{blue}{z \cdot t}} \]

      4. associate-/l* 79.6%

         \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]

   7. Simplified 79.6%

      \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]
   8. Step-by-step derivation

      1. associate-*r/ 79.6%

         \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot t}} \]

      2. clear-num 78.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot t}{x \cdot -2}}} \]

      3. *-commutative 78.3%

         \[\leadsto \frac{1}{\frac{z \cdot t}{\color{blue}{-2 \cdot x}}} \]

      4. times-frac 82.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{z}{-2} \cdot \frac{t}{x}}} \]

      5. div-inv 82.2%

         \[\leadsto \frac{1}{\color{blue}{\left(z \cdot \frac{1}{-2}\right)} \cdot \frac{t}{x}} \]

      6. metadata-eval 82.2%

         \[\leadsto \frac{1}{\left(z \cdot \color{blue}{-0.5}\right) \cdot \frac{t}{x}} \]

   9. Applied egg-rr 82.2%

      \[\leadsto \color{blue}{\frac{1}{\left(z \cdot -0.5\right) \cdot \frac{t}{x}}} \]
   10. Step-by-step derivation

       1. associate-/r* 82.4%

          \[\leadsto \color{blue}{\frac{\frac{1}{z \cdot -0.5}}{\frac{t}{x}}} \]

       2. *-commutative 82.4%

          \[\leadsto \frac{\frac{1}{\color{blue}{-0.5 \cdot z}}}{\frac{t}{x}} \]

       3. associate-/r* 82.4%

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{-0.5}}{z}}}{\frac{t}{x}} \]

       4. metadata-eval 82.4%

          \[\leadsto \frac{\frac{\color{blue}{-2}}{z}}{\frac{t}{x}} \]

   11. Simplified 82.4%

       \[\leadsto \color{blue}{\frac{\frac{-2}{z}}{\frac{t}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{+92}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{2}{y}}{\frac{z}{x}}\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-126}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+75}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{z}}{\frac{t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{-2}{z} \cdot \frac{x\_m}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.3 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{z}}{\frac{t}{x\_m}}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (/ -2.0 z) (/ x_m t))))
   (*
    x_s
    (if (<= t -6.3e+84)
      t_1
      (if (<= t -3.8e-94)
        (* 2.0 (/ (/ x_m z) y))
        (if (<= t -3.5e-133)
          t_1
          (if (<= t 2.8e+75)
            (/ 2.0 (* y (/ z x_m)))
            (/ (/ -2.0 z) (/ t x_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (-2.0 / z) * (x_m / t);
	double tmp;
	if (t <= -6.3e+84) {
		tmp = t_1;
	} else if (t <= -3.8e-94) {
		tmp = 2.0 * ((x_m / z) / y);
	} else if (t <= -3.5e-133) {
		tmp = t_1;
	} else if (t <= 2.8e+75) {
		tmp = 2.0 / (y * (z / x_m));
	} else {
		tmp = (-2.0 / z) / (t / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((-2.0d0) / z) * (x_m / t)
    if (t <= (-6.3d+84)) then
        tmp = t_1
    else if (t <= (-3.8d-94)) then
        tmp = 2.0d0 * ((x_m / z) / y)
    else if (t <= (-3.5d-133)) then
        tmp = t_1
    else if (t <= 2.8d+75) then
        tmp = 2.0d0 / (y * (z / x_m))
    else
        tmp = ((-2.0d0) / z) / (t / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (-2.0 / z) * (x_m / t);
	double tmp;
	if (t <= -6.3e+84) {
		tmp = t_1;
	} else if (t <= -3.8e-94) {
		tmp = 2.0 * ((x_m / z) / y);
	} else if (t <= -3.5e-133) {
		tmp = t_1;
	} else if (t <= 2.8e+75) {
		tmp = 2.0 / (y * (z / x_m));
	} else {
		tmp = (-2.0 / z) / (t / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (-2.0 / z) * (x_m / t)
	tmp = 0
	if t <= -6.3e+84:
		tmp = t_1
	elif t <= -3.8e-94:
		tmp = 2.0 * ((x_m / z) / y)
	elif t <= -3.5e-133:
		tmp = t_1
	elif t <= 2.8e+75:
		tmp = 2.0 / (y * (z / x_m))
	else:
		tmp = (-2.0 / z) / (t / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(-2.0 / z) * Float64(x_m / t))
	tmp = 0.0
	if (t <= -6.3e+84)
		tmp = t_1;
	elseif (t <= -3.8e-94)
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / y));
	elseif (t <= -3.5e-133)
		tmp = t_1;
	elseif (t <= 2.8e+75)
		tmp = Float64(2.0 / Float64(y * Float64(z / x_m)));
	else
		tmp = Float64(Float64(-2.0 / z) / Float64(t / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (-2.0 / z) * (x_m / t);
	tmp = 0.0;
	if (t <= -6.3e+84)
		tmp = t_1;
	elseif (t <= -3.8e-94)
		tmp = 2.0 * ((x_m / z) / y);
	elseif (t <= -3.5e-133)
		tmp = t_1;
	elseif (t <= 2.8e+75)
		tmp = 2.0 / (y * (z / x_m));
	else
		tmp = (-2.0 / z) / (t / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -6.3e+84], t$95$1, If[LessEqual[t, -3.8e-94], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.5e-133], t$95$1, If[LessEqual[t, 2.8e+75], N[(2.0 / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / z), $MachinePrecision] / N[(t / x$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < -6.30000000000000013e84 or -3.79999999999999999e-94 < t < -3.50000000000000003e-133

   1. Initial program 85.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 85.7%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 90.1%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 90.1%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 85.9%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 85.9%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 85.9%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 85.9%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 85.9%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 85.8%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 85.9%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 85.9%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 85.9%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 85.9%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 85.9%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 85.9%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 85.9%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 85.9%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 85.9%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 85.9%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 85.9%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 90.1%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 93.7%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   6. Applied egg-rr 93.7%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   7. Taylor expanded in t around inf 80.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   8. Step-by-step derivation

      1. associate-*r/ 80.3%

         \[\leadsto \color{blue}{\frac{-2 \cdot x}{t \cdot z}} \]

      2. *-commutative 80.3%

         \[\leadsto \frac{\color{blue}{x \cdot -2}}{t \cdot z} \]

      3. *-commutative 80.3%

         \[\leadsto \frac{x \cdot -2}{\color{blue}{z \cdot t}} \]

      4. times-frac 84.5%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{-2}{t}} \]

      5. associate-*r/ 84.6%

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot -2}{t}} \]

      6. *-commutative 84.6%

         \[\leadsto \frac{\color{blue}{-2 \cdot \frac{x}{z}}}{t} \]

      7. associate-*r/ 84.6%

         \[\leadsto \frac{\color{blue}{\frac{-2 \cdot x}{z}}}{t} \]

   9. Simplified 84.6%

      \[\leadsto \color{blue}{\frac{\frac{-2 \cdot x}{z}}{t}} \]
   10. Step-by-step derivation

       1. associate-/l/ 80.3%

          \[\leadsto \color{blue}{\frac{-2 \cdot x}{t \cdot z}} \]

       2. *-commutative 80.3%

          \[\leadsto \frac{\color{blue}{x \cdot -2}}{t \cdot z} \]

       3. times-frac 85.1%

          \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{-2}{z}} \]

   11. Applied egg-rr 85.1%

       \[\leadsto \color{blue}{\frac{x}{t} \cdot \frac{-2}{z}} \]

   if -6.30000000000000013e84 < t < -3.79999999999999999e-94

   1. Initial program 94.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 94.2%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 94.2%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 94.2%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 94.2%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 94.2%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 94.2%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 94.2%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 94.2%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 94.1%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 94.2%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 94.2%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 94.2%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 94.2%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 94.2%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 94.2%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 94.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 94.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 94.2%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 94.2%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 94.2%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 95.8%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   6. Applied egg-rr 95.8%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   7. Taylor expanded in t around 0 65.8%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 65.8%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 70.3%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 70.3%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]

   if -3.50000000000000003e-133 < t < 2.80000000000000012e75

   1. Initial program 91.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 91.2%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 91.3%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 91.3%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 88.7%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 88.7%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 88.7%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 88.7%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 88.7%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 88.1%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 88.7%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 88.7%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 88.7%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 88.7%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 88.7%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 88.7%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 88.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 88.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 88.7%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 88.7%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 88.7%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 91.3%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 92.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   6. Applied egg-rr 92.1%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   7. Taylor expanded in t around 0 72.1%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 72.1%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 75.6%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 75.6%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]
   10. Step-by-step derivation

       1. *-commutative 75.6%

          \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y} \cdot 2} \]

       2. associate-*l/ 75.6%

          \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot 2}{y}} \]

       3. associate-*r/ 75.5%

          \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{2}{y}} \]

       4. clear-num 75.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}}} \cdot \frac{2}{y} \]

       5. frac-times 76.2%

          \[\leadsto \color{blue}{\frac{1 \cdot 2}{\frac{z}{x} \cdot y}} \]

       6. metadata-eval 76.2%

          \[\leadsto \frac{\color{blue}{2}}{\frac{z}{x} \cdot y} \]

   11. Applied egg-rr 76.2%

       \[\leadsto \color{blue}{\frac{2}{\frac{z}{x} \cdot y}} \]

   if 2.80000000000000012e75 < t

   1. Initial program 85.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 85.0%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 87.0%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 87.0%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 85.0%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 85.0%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 85.0%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 85.0%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 85.0%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 85.0%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 85.0%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 85.0%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 85.0%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 85.0%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 85.0%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 85.0%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 85.0%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 85.0%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 85.0%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 85.0%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 85.0%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 87.0%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 79.6%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 79.6%

         \[\leadsto \color{blue}{\frac{-2 \cdot x}{t \cdot z}} \]

      2. *-commutative 79.6%

         \[\leadsto \frac{\color{blue}{x \cdot -2}}{t \cdot z} \]

      3. *-commutative 79.6%

         \[\leadsto \frac{x \cdot -2}{\color{blue}{z \cdot t}} \]

      4. associate-/l* 79.6%

         \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]

   7. Simplified 79.6%

      \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]
   8. Step-by-step derivation

      1. associate-*r/ 79.6%

         \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot t}} \]

      2. clear-num 78.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{z \cdot t}{x \cdot -2}}} \]

      3. *-commutative 78.3%

         \[\leadsto \frac{1}{\frac{z \cdot t}{\color{blue}{-2 \cdot x}}} \]

      4. times-frac 82.2%

         \[\leadsto \frac{1}{\color{blue}{\frac{z}{-2} \cdot \frac{t}{x}}} \]

      5. div-inv 82.2%

         \[\leadsto \frac{1}{\color{blue}{\left(z \cdot \frac{1}{-2}\right)} \cdot \frac{t}{x}} \]

      6. metadata-eval 82.2%

         \[\leadsto \frac{1}{\left(z \cdot \color{blue}{-0.5}\right) \cdot \frac{t}{x}} \]

   9. Applied egg-rr 82.2%

      \[\leadsto \color{blue}{\frac{1}{\left(z \cdot -0.5\right) \cdot \frac{t}{x}}} \]
   10. Step-by-step derivation

       1. associate-/r* 82.4%

          \[\leadsto \color{blue}{\frac{\frac{1}{z \cdot -0.5}}{\frac{t}{x}}} \]

       2. *-commutative 82.4%

          \[\leadsto \frac{\frac{1}{\color{blue}{-0.5 \cdot z}}}{\frac{t}{x}} \]

       3. associate-/r* 82.4%

          \[\leadsto \frac{\color{blue}{\frac{\frac{1}{-0.5}}{z}}}{\frac{t}{x}} \]

       4. metadata-eval 82.4%

          \[\leadsto \frac{\frac{\color{blue}{-2}}{z}}{\frac{t}{x}} \]

   11. Simplified 82.4%

       \[\leadsto \color{blue}{\frac{\frac{-2}{z}}{\frac{t}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.3 \cdot 10^{+84}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-94}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+75}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2}{z}}{\frac{t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{\frac{-2}{z}}{t}\\ t_2 := 2 \cdot \frac{\frac{x\_m}{z}}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+99}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-35}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ (/ -2.0 z) t))) (t_2 (* 2.0 (/ (/ x_m z) y))))
   (*
    x_s
    (if (<= y -5.4e+99)
      t_2
      (if (<= y -9.5e+51)
        t_1
        (if (<= y -8e-35)
          t_2
          (if (<= y 1.56e-55) t_1 (* (/ 2.0 z) (/ x_m y)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((-2.0 / z) / t);
	double t_2 = 2.0 * ((x_m / z) / y);
	double tmp;
	if (y <= -5.4e+99) {
		tmp = t_2;
	} else if (y <= -9.5e+51) {
		tmp = t_1;
	} else if (y <= -8e-35) {
		tmp = t_2;
	} else if (y <= 1.56e-55) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    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_m * (((-2.0d0) / z) / t)
    t_2 = 2.0d0 * ((x_m / z) / y)
    if (y <= (-5.4d+99)) then
        tmp = t_2
    else if (y <= (-9.5d+51)) then
        tmp = t_1
    else if (y <= (-8d-35)) then
        tmp = t_2
    else if (y <= 1.56d-55) then
        tmp = t_1
    else
        tmp = (2.0d0 / z) * (x_m / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((-2.0 / z) / t);
	double t_2 = 2.0 * ((x_m / z) / y);
	double tmp;
	if (y <= -5.4e+99) {
		tmp = t_2;
	} else if (y <= -9.5e+51) {
		tmp = t_1;
	} else if (y <= -8e-35) {
		tmp = t_2;
	} else if (y <= 1.56e-55) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * ((-2.0 / z) / t)
	t_2 = 2.0 * ((x_m / z) / y)
	tmp = 0
	if y <= -5.4e+99:
		tmp = t_2
	elif y <= -9.5e+51:
		tmp = t_1
	elif y <= -8e-35:
		tmp = t_2
	elif y <= 1.56e-55:
		tmp = t_1
	else:
		tmp = (2.0 / z) * (x_m / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(Float64(-2.0 / z) / t))
	t_2 = Float64(2.0 * Float64(Float64(x_m / z) / y))
	tmp = 0.0
	if (y <= -5.4e+99)
		tmp = t_2;
	elseif (y <= -9.5e+51)
		tmp = t_1;
	elseif (y <= -8e-35)
		tmp = t_2;
	elseif (y <= 1.56e-55)
		tmp = t_1;
	else
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * ((-2.0 / z) / t);
	t_2 = 2.0 * ((x_m / z) / y);
	tmp = 0.0;
	if (y <= -5.4e+99)
		tmp = t_2;
	elseif (y <= -9.5e+51)
		tmp = t_1;
	elseif (y <= -8e-35)
		tmp = t_2;
	elseif (y <= 1.56e-55)
		tmp = t_1;
	else
		tmp = (2.0 / z) * (x_m / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(N[(-2.0 / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -5.4e+99], t$95$2, If[LessEqual[y, -9.5e+51], t$95$1, If[LessEqual[y, -8e-35], t$95$2, If[LessEqual[y, 1.56e-55], t$95$1, N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.39999999999999978e99 or -9.4999999999999999e51 < y < -8.00000000000000006e-35

   1. Initial program 85.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 85.3%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 86.9%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 86.9%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 85.3%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 85.3%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 85.3%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 85.3%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 85.3%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 85.2%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 85.3%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 85.3%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 85.3%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 85.3%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 85.3%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 85.3%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 85.3%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 85.3%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 85.3%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 85.3%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 85.3%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-/r* 96.8%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   6. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

   7. Taylor expanded in t around 0 78.9%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   8. Step-by-step derivation

      1. *-commutative 78.9%

         \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

      2. associate-/r* 87.7%

         \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

   9. Simplified 87.7%

      \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]

   if -5.39999999999999978e99 < y < -9.4999999999999999e51 or -8.00000000000000006e-35 < y < 1.56e-55

   1. Initial program 93.8%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 93.8%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 94.7%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 94.7%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 92.3%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 92.3%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 92.3%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 92.3%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 92.3%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 91.8%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 92.3%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 92.3%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 92.3%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 92.3%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 92.3%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 92.3%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 92.3%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 92.3%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 92.3%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 92.3%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 92.3%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 94.7%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 80.2%

      \[\leadsto \color{blue}{-2 \cdot \frac{x}{t \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 80.2%

         \[\leadsto \color{blue}{\frac{-2 \cdot x}{t \cdot z}} \]

      2. *-commutative 80.2%

         \[\leadsto \frac{\color{blue}{x \cdot -2}}{t \cdot z} \]

      3. *-commutative 80.2%

         \[\leadsto \frac{x \cdot -2}{\color{blue}{z \cdot t}} \]

      4. associate-/l* 79.8%

         \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]

   7. Simplified 79.8%

      \[\leadsto \color{blue}{x \cdot \frac{-2}{z \cdot t}} \]
   8. Step-by-step derivation

      1. associate-/r* 80.1%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{-2}{z}}{t}} \]

   9. Applied egg-rr 80.1%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{-2}{z}}{t}} \]

   if 1.56e-55 < y

   1. Initial program 84.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 84.6%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 86.3%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 86.3%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 83.1%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 83.1%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 83.1%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 83.1%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 83.1%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 82.9%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 83.1%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 83.1%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 83.1%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 83.1%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 83.1%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 83.1%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 83.1%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 83.1%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 83.1%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 83.1%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 83.1%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 66.2%

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 66.2%

         \[\leadsto \color{blue}{\frac{2 \cdot x}{y \cdot z}} \]

      2. *-commutative 66.2%

         \[\leadsto \frac{2 \cdot x}{\color{blue}{z \cdot y}} \]

      3. times-frac 67.5%

         \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]

   7. Simplified 67.5%

      \[\leadsto \color{blue}{\frac{2}{z} \cdot \frac{x}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-88}:\\ \;\;\;\;\frac{x\_m \cdot -2}{z \cdot \left(t - y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - y} \cdot \frac{-2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 5e-88)
    (/ (* x_m -2.0) (* z (- t y)))
    (* (/ x_m (- t y)) (/ -2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-88) {
		tmp = (x_m * -2.0) / (z * (t - y));
	} else {
		tmp = (x_m / (t - y)) * (-2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-88) then
        tmp = (x_m * (-2.0d0)) / (z * (t - y))
    else
        tmp = (x_m / (t - y)) * ((-2.0d0) / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-88) {
		tmp = (x_m * -2.0) / (z * (t - y));
	} else {
		tmp = (x_m / (t - y)) * (-2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-88:
		tmp = (x_m * -2.0) / (z * (t - y))
	else:
		tmp = (x_m / (t - y)) * (-2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-88)
		tmp = Float64(Float64(x_m * -2.0) / Float64(z * Float64(t - y)));
	else
		tmp = Float64(Float64(x_m / Float64(t - y)) * Float64(-2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-88)
		tmp = (x_m * -2.0) / (z * (t - y));
	else
		tmp = (x_m / (t - y)) * (-2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-88], N[(N[(x$95$m * -2.0), $MachinePrecision] / N[(z * N[(t - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(t - y), $MachinePrecision]), $MachinePrecision] * N[(-2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 5.00000000000000009e-88

   1. Initial program 88.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 88.6%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 89.8%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 89.8%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 88.6%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 88.6%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 88.6%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 88.6%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 88.6%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 88.5%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 88.6%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 88.6%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 88.6%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 88.6%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 88.6%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 88.6%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 88.6%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 88.6%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 88.6%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 88.6%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 88.6%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 89.8%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing

   if 5.00000000000000009e-88 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 90.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 90.1%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 91.5%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 91.5%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 87.1%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 87.1%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 87.1%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 87.1%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 87.1%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 86.5%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 87.1%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 87.1%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 87.1%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 87.1%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 87.1%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 87.1%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 87.1%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 87.1%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 87.1%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 87.1%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 87.1%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 91.5%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 91.5%

         \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(t - y\right) \cdot z}} \]

      2. times-frac 98.6%

         \[\leadsto \color{blue}{\frac{x}{t - y} \cdot \frac{-2}{z}} \]

   6. Applied egg-rr 98.6%

      \[\leadsto \color{blue}{\frac{x}{t - y} \cdot \frac{-2}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 10^{-69}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t - y} \cdot \frac{-2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 1e-69)
    (* x_m (/ (/ 2.0 z) (- y t)))
    (* (/ x_m (- t y)) (/ -2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 1e-69) {
		tmp = x_m * ((2.0 / z) / (y - t));
	} else {
		tmp = (x_m / (t - y)) * (-2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 1d-69) then
        tmp = x_m * ((2.0d0 / z) / (y - t))
    else
        tmp = (x_m / (t - y)) * ((-2.0d0) / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 1e-69) {
		tmp = x_m * ((2.0 / z) / (y - t));
	} else {
		tmp = (x_m / (t - y)) * (-2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 1e-69:
		tmp = x_m * ((2.0 / z) / (y - t))
	else:
		tmp = (x_m / (t - y)) * (-2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 1e-69)
		tmp = Float64(x_m * Float64(Float64(2.0 / z) / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(t - y)) * Float64(-2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 1e-69)
		tmp = x_m * ((2.0 / z) / (y - t));
	else
		tmp = (x_m / (t - y)) * (-2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 1e-69], N[(x$95$m * N[(N[(2.0 / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(t - y), $MachinePrecision]), $MachinePrecision] * N[(-2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 9.9999999999999996e-70

   1. Initial program 88.9%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* 88.8%

         \[\leadsto \color{blue}{x \cdot \frac{2}{y \cdot z - t \cdot z}} \]

      2. *-commutative 88.8%

         \[\leadsto \color{blue}{\frac{2}{y \cdot z - t \cdot z} \cdot x} \]

      3. distribute-rgt-out-- 90.0%

         \[\leadsto \frac{2}{\color{blue}{z \cdot \left(y - t\right)}} \cdot x \]

      4. associate-/r* 90.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{z}}{y - t}} \cdot x \]

   4. Applied egg-rr 90.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{z}}{y - t} \cdot x} \]

   if 9.9999999999999996e-70 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 89.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   2. Step-by-step derivation

      1. remove-double-neg 89.6%

         \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

      2. distribute-rgt-out-- 91.0%

         \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

      3. distribute-lft-neg-in 91.0%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

      4. distribute-rgt-out-- 86.4%

         \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      5. remove-double-neg 86.4%

         \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      6. distribute-lft-neg-out 86.4%

         \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      7. distribute-rgt-neg-in 86.4%

         \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

      8. neg-mul-1 86.4%

         \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

      9. times-frac 85.8%

         \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      10. associate-*r/ 86.4%

          \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

      11. distribute-frac-neg 86.4%

          \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      12. distribute-neg-frac2 86.4%

          \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      13. metadata-eval 86.4%

          \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      14. /-rgt-identity 86.4%

          \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      15. metadata-eval 86.4%

          \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

      16. distribute-rgt-neg-out 86.4%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

      17. distribute-lft-neg-out 86.4%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

      18. *-commutative 86.4%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

      19. cancel-sign-sub 86.4%

          \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

      20. *-commutative 86.4%

          \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

   3. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 91.0%

         \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(t - y\right) \cdot z}} \]

      2. times-frac 98.5%

         \[\leadsto \color{blue}{\frac{x}{t - y} \cdot \frac{-2}{z}} \]

   6. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{x}{t - y} \cdot \frac{-2}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 10^{-69}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t - y} \cdot \frac{-2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 92.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(\frac{x\_m}{t - y} \cdot \frac{-2}{z}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (* (/ x_m (- t y)) (/ -2.0 z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * ((x_m / (t - y)) * (-2.0 / z));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * ((x_m / (t - y)) * ((-2.0d0) / z))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * ((x_m / (t - y)) * (-2.0 / z));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * ((x_m / (t - y)) * (-2.0 / z))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(Float64(x_m / Float64(t - y)) * Float64(-2.0 / z)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * ((x_m / (t - y)) * (-2.0 / z));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(N[(x$95$m / N[(t - y), $MachinePrecision]), $MachinePrecision] * N[(-2.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
2. Step-by-step derivation

   1. remove-double-neg 89.2%

      \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

   2. distribute-rgt-out-- 90.4%

      \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

   3. distribute-lft-neg-in 90.4%

      \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

   4. distribute-rgt-out-- 88.0%

      \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

   5. remove-double-neg 88.0%

      \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

   6. distribute-lft-neg-out 88.0%

      \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

   7. distribute-rgt-neg-in 88.0%

      \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

   8. neg-mul-1 88.0%

      \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

   9. times-frac 87.8%

      \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

   10. associate-*r/ 88.0%

       \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

   11. distribute-frac-neg 88.0%

       \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

   12. distribute-neg-frac2 88.0%

       \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

   13. metadata-eval 88.0%

       \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

   14. /-rgt-identity 88.0%

       \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

   15. metadata-eval 88.0%

       \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

   16. distribute-rgt-neg-out 88.0%

       \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

   17. distribute-lft-neg-out 88.0%

       \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

   18. *-commutative 88.0%

       \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

   19. cancel-sign-sub 88.0%

       \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

   20. *-commutative 88.0%

       \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

3. Simplified 90.4%

   \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 90.4%

      \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(t - y\right) \cdot z}} \]

   2. times-frac 92.4%

      \[\leadsto \color{blue}{\frac{x}{t - y} \cdot \frac{-2}{z}} \]

6. Applied egg-rr 92.4%

   \[\leadsto \color{blue}{\frac{x}{t - y} \cdot \frac{-2}{z}} \]
7. Add Preprocessing

Alternative 16: 55.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(2 \cdot \frac{\frac{x\_m}{z}}{y}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (* 2.0 (/ (/ x_m z) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (2.0 * ((x_m / z) / y));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (2.0d0 * ((x_m / z) / y))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (2.0 * ((x_m / z) / y));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	return x_s * (2.0 * ((x_m / z) / y))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(2.0 * Float64(Float64(x_m / z) / y)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (2.0 * ((x_m / z) / y));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
2. Step-by-step derivation

   1. remove-double-neg 89.2%

      \[\leadsto \frac{x \cdot 2}{\color{blue}{-\left(-\left(y \cdot z - t \cdot z\right)\right)}} \]

   2. distribute-rgt-out-- 90.4%

      \[\leadsto \frac{x \cdot 2}{-\left(-\color{blue}{z \cdot \left(y - t\right)}\right)} \]

   3. distribute-lft-neg-in 90.4%

      \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(-z\right) \cdot \left(y - t\right)}} \]

   4. distribute-rgt-out-- 88.0%

      \[\leadsto \frac{x \cdot 2}{-\color{blue}{\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

   5. remove-double-neg 88.0%

      \[\leadsto \frac{\color{blue}{-\left(-x \cdot 2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

   6. distribute-lft-neg-out 88.0%

      \[\leadsto \frac{-\color{blue}{\left(-x\right) \cdot 2}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

   7. distribute-rgt-neg-in 88.0%

      \[\leadsto \frac{\color{blue}{\left(-x\right) \cdot \left(-2\right)}}{-\left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)} \]

   8. neg-mul-1 88.0%

      \[\leadsto \frac{\left(-x\right) \cdot \left(-2\right)}{\color{blue}{-1 \cdot \left(y \cdot \left(-z\right) - t \cdot \left(-z\right)\right)}} \]

   9. times-frac 87.8%

      \[\leadsto \color{blue}{\frac{-x}{-1} \cdot \frac{-2}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

   10. associate-*r/ 88.0%

       \[\leadsto \color{blue}{\frac{\frac{-x}{-1} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)}} \]

   11. distribute-frac-neg 88.0%

       \[\leadsto \frac{\color{blue}{\left(-\frac{x}{-1}\right)} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

   12. distribute-neg-frac2 88.0%

       \[\leadsto \frac{\color{blue}{\frac{x}{--1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

   13. metadata-eval 88.0%

       \[\leadsto \frac{\frac{x}{\color{blue}{1}} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

   14. /-rgt-identity 88.0%

       \[\leadsto \frac{\color{blue}{x} \cdot \left(-2\right)}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

   15. metadata-eval 88.0%

       \[\leadsto \frac{x \cdot \color{blue}{-2}}{y \cdot \left(-z\right) - t \cdot \left(-z\right)} \]

   16. distribute-rgt-neg-out 88.0%

       \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y \cdot z\right)} - t \cdot \left(-z\right)} \]

   17. distribute-lft-neg-out 88.0%

       \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z} - t \cdot \left(-z\right)} \]

   18. *-commutative 88.0%

       \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z - \color{blue}{\left(-z\right) \cdot t}} \]

   19. cancel-sign-sub 88.0%

       \[\leadsto \frac{x \cdot -2}{\color{blue}{\left(-y\right) \cdot z + z \cdot t}} \]

   20. *-commutative 88.0%

       \[\leadsto \frac{x \cdot -2}{\left(-y\right) \cdot z + \color{blue}{t \cdot z}} \]

3. Simplified 90.4%

   \[\leadsto \color{blue}{\frac{x \cdot -2}{z \cdot \left(t - y\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-/r* 93.9%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

6. Applied egg-rr 93.9%

   \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t - y}} \]

7. Taylor expanded in t around 0 50.2%

   \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
8. Step-by-step derivation

   1. *-commutative 50.2%

      \[\leadsto 2 \cdot \frac{x}{\color{blue}{z \cdot y}} \]

   2. associate-/r* 53.2%

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{x}{z}}{y}} \]

9. Simplified 53.2%

   \[\leadsto \color{blue}{2 \cdot \frac{\frac{x}{z}}{y}} \]
10. Add Preprocessing

Developer target: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ t_2 := \frac{x \cdot 2}{y \cdot z - t \cdot z}\\ \mathbf{if}\;t\_2 < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x (* (- y t) z)) 2.0))
        (t_2 (/ (* x 2.0) (- (* y z) (* t z)))))
   (if (< t_2 -2.559141628295061e-13)
     t_1
     (if (< t_2 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / ((y - t) * z)) * 2.0;
	double t_2 = (x * 2.0) / ((y * z) - (t * z));
	double tmp;
	if (t_2 < -2.559141628295061e-13) {
		tmp = t_1;
	} else if (t_2 < 1.045027827330126e-269) {
		tmp = ((x / z) * 2.0) / (y - 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) :: t_2
    real(8) :: tmp
    t_1 = (x / ((y - t) * z)) * 2.0d0
    t_2 = (x * 2.0d0) / ((y * z) - (t * z))
    if (t_2 < (-2.559141628295061d-13)) then
        tmp = t_1
    else if (t_2 < 1.045027827330126d-269) then
        tmp = ((x / z) * 2.0d0) / (y - 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 = (x / ((y - t) * z)) * 2.0;
	double t_2 = (x * 2.0) / ((y * z) - (t * z));
	double tmp;
	if (t_2 < -2.559141628295061e-13) {
		tmp = t_1;
	} else if (t_2 < 1.045027827330126e-269) {
		tmp = ((x / z) * 2.0) / (y - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / ((y - t) * z)) * 2.0
	t_2 = (x * 2.0) / ((y * z) - (t * z))
	tmp = 0
	if t_2 < -2.559141628295061e-13:
		tmp = t_1
	elif t_2 < 1.045027827330126e-269:
		tmp = ((x / z) * 2.0) / (y - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(Float64(y - t) * z)) * 2.0)
	t_2 = Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
	tmp = 0.0
	if (t_2 < -2.559141628295061e-13)
		tmp = t_1;
	elseif (t_2 < 1.045027827330126e-269)
		tmp = Float64(Float64(Float64(x / z) * 2.0) / Float64(y - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / ((y - t) * z)) * 2.0;
	t_2 = (x * 2.0) / ((y * z) - (t * z));
	tmp = 0.0;
	if (t_2 < -2.559141628295061e-13)
		tmp = t_1;
	elseif (t_2 < 1.045027827330126e-269)
		tmp = ((x / z) * 2.0) / (y - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -2.559141628295061e-13], t$95$1, If[Less[t$95$2, 1.045027827330126e-269], N[(N[(N[(x / z), $MachinePrecision] * 2.0), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024096 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))