Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.7% → 97.7%

Time: 12.5s

Alternatives: 13

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: 89.7% 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: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot 2}{y \cdot z\_m - z\_m \cdot t}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-314}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-27}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* x_m 2.0) (- (* y z_m) (* z_m t)))))
   (*
    z_s
    (*
     x_s
     (if (<= t_1 -5e-314)
       (/ (* x_m 2.0) (* z_m (- y t)))
       (if (<= t_1 4e-27)
         (* (/ x_m z_m) (/ 2.0 (- y t)))
         (* (/ x_m (- y t)) (/ 2.0 z_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m * 2.0) / ((y * z_m) - (z_m * t));
	double tmp;
	if (t_1 <= -5e-314) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else if (t_1 <= 4e-27) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * 2.0d0) / ((y * z_m) - (z_m * t))
    if (t_1 <= (-5d-314)) then
        tmp = (x_m * 2.0d0) / (z_m * (y - t))
    else if (t_1 <= 4d-27) then
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m * 2.0) / ((y * z_m) - (z_m * t));
	double tmp;
	if (t_1 <= -5e-314) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else if (t_1 <= 4e-27) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = (x_m * 2.0) / ((y * z_m) - (z_m * t))
	tmp = 0
	if t_1 <= -5e-314:
		tmp = (x_m * 2.0) / (z_m * (y - t))
	elif t_1 <= 4e-27:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z_m)
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(Float64(x_m * 2.0) / Float64(Float64(y * z_m) - Float64(z_m * t)))
	tmp = 0.0
	if (t_1 <= -5e-314)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z_m * Float64(y - t)));
	elseif (t_1 <= 4e-27)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = (x_m * 2.0) / ((y * z_m) - (z_m * t));
	tmp = 0.0;
	if (t_1 <= -5e-314)
		tmp = (x_m * 2.0) / (z_m * (y - t));
	elseif (t_1 <= 4e-27)
		tmp = (x_m / z_m) * (2.0 / (y - t));
	else
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[t$95$1, -5e-314], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-27], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z))) < -4.99999999982e-314

   1. Initial program 95.9%

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

      1. distribute-rgt-out-- 95.9%

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

   3. Simplified 95.9%

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

   if -4.99999999982e-314 < (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z))) < 4.0000000000000002e-27

   1. Initial program 85.3%

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

      1. distribute-rgt-out-- 85.3%

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

      2. times-frac 97.5%

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

   3. Simplified 97.5%

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

   if 4.0000000000000002e-27 < (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 75.5%

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

      1. distribute-rgt-out-- 89.6%

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

   3. Simplified 89.6%

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

      1. *-commutative 89.6%

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

      2. times-frac 96.8%

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

   6. Applied egg-rr 96.8%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -5 \cdot 10^{-314}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq 4 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := -2 \cdot \frac{x\_m}{z\_m \cdot t}\\ t_2 := 2 \cdot \frac{\frac{x\_m}{y}}{z\_m}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+101}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-71}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-134} \lor \neg \left(y \leq 1.3 \cdot 10^{+65}\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{t}}{z\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* -2.0 (/ x_m (* z_m t)))) (t_2 (* 2.0 (/ (/ x_m y) z_m))))
   (*
    z_s
    (*
     x_s
     (if (<= y -5e+127)
       t_2
       (if (<= y -2.1e+101)
         t_1
         (if (<= y -4.2e-71)
           t_2
           (if (<= y -1.32e-104)
             t_1
             (if (or (<= y -1.4e-134) (not (<= y 1.3e+65)))
               t_2
               (* -2.0 (/ (/ x_m t) z_m)))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = -2.0 * (x_m / (z_m * t));
	double t_2 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_2;
	} else if (y <= -2.1e+101) {
		tmp = t_1;
	} else if (y <= -4.2e-71) {
		tmp = t_2;
	} else if (y <= -1.32e-104) {
		tmp = t_1;
	} else if ((y <= -1.4e-134) || !(y <= 1.3e+65)) {
		tmp = t_2;
	} else {
		tmp = -2.0 * ((x_m / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (-2.0d0) * (x_m / (z_m * t))
    t_2 = 2.0d0 * ((x_m / y) / z_m)
    if (y <= (-5d+127)) then
        tmp = t_2
    else if (y <= (-2.1d+101)) then
        tmp = t_1
    else if (y <= (-4.2d-71)) then
        tmp = t_2
    else if (y <= (-1.32d-104)) then
        tmp = t_1
    else if ((y <= (-1.4d-134)) .or. (.not. (y <= 1.3d+65))) then
        tmp = t_2
    else
        tmp = (-2.0d0) * ((x_m / t) / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = -2.0 * (x_m / (z_m * t));
	double t_2 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_2;
	} else if (y <= -2.1e+101) {
		tmp = t_1;
	} else if (y <= -4.2e-71) {
		tmp = t_2;
	} else if (y <= -1.32e-104) {
		tmp = t_1;
	} else if ((y <= -1.4e-134) || !(y <= 1.3e+65)) {
		tmp = t_2;
	} else {
		tmp = -2.0 * ((x_m / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = -2.0 * (x_m / (z_m * t))
	t_2 = 2.0 * ((x_m / y) / z_m)
	tmp = 0
	if y <= -5e+127:
		tmp = t_2
	elif y <= -2.1e+101:
		tmp = t_1
	elif y <= -4.2e-71:
		tmp = t_2
	elif y <= -1.32e-104:
		tmp = t_1
	elif (y <= -1.4e-134) or not (y <= 1.3e+65):
		tmp = t_2
	else:
		tmp = -2.0 * ((x_m / t) / z_m)
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(-2.0 * Float64(x_m / Float64(z_m * t)))
	t_2 = Float64(2.0 * Float64(Float64(x_m / y) / z_m))
	tmp = 0.0
	if (y <= -5e+127)
		tmp = t_2;
	elseif (y <= -2.1e+101)
		tmp = t_1;
	elseif (y <= -4.2e-71)
		tmp = t_2;
	elseif (y <= -1.32e-104)
		tmp = t_1;
	elseif ((y <= -1.4e-134) || !(y <= 1.3e+65))
		tmp = t_2;
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = -2.0 * (x_m / (z_m * t));
	t_2 = 2.0 * ((x_m / y) / z_m);
	tmp = 0.0;
	if (y <= -5e+127)
		tmp = t_2;
	elseif (y <= -2.1e+101)
		tmp = t_1;
	elseif (y <= -4.2e-71)
		tmp = t_2;
	elseif (y <= -1.32e-104)
		tmp = t_1;
	elseif ((y <= -1.4e-134) || ~((y <= 1.3e+65)))
		tmp = t_2;
	else
		tmp = -2.0 * ((x_m / t) / z_m);
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -5e+127], t$95$2, If[LessEqual[y, -2.1e+101], t$95$1, If[LessEqual[y, -4.2e-71], t$95$2, If[LessEqual[y, -1.32e-104], t$95$1, If[Or[LessEqual[y, -1.4e-134], N[Not[LessEqual[y, 1.3e+65]], $MachinePrecision]], t$95$2, N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.0000000000000004e127 or -2.1e101 < y < -4.2000000000000002e-71 or -1.3199999999999999e-104 < y < -1.3999999999999999e-134 or 1.30000000000000001e65 < y

   1. Initial program 83.5%

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

      1. distribute-rgt-out-- 88.0%

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

   3. Simplified 88.0%

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

      1. *-commutative 88.0%

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

      2. times-frac 93.3%

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

   6. Applied egg-rr 93.3%

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

   7. Taylor expanded in y around inf 77.7%

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

      1. associate-/r* 82.2%

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

   9. Simplified 82.2%

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

   if -5.0000000000000004e127 < y < -2.1e101 or -4.2000000000000002e-71 < y < -1.3199999999999999e-104

   1. Initial program 92.2%

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

      1. *-commutative 92.2%

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

      2. associate-*l/ 92.1%

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

      3. *-commutative 92.1%

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

      4. distribute-rgt-out-- 99.8%

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

      5. associate-/l/ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 97.4%

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

      1. *-commutative 97.4%

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

   7. Simplified 97.4%

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

   if -1.3999999999999999e-134 < y < 1.30000000000000001e65

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l/ 87.4%

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

      3. *-commutative 87.4%

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

      4. distribute-rgt-out-- 89.2%

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

      5. associate-/l/ 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in y around 0 77.2%

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

      1. associate-/r* 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-71}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-104}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-134} \lor \neg \left(y \leq 1.3 \cdot 10^{+65}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := 2 \cdot \frac{\frac{x\_m}{y}}{z\_m}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+96}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-104}:\\ \;\;\;\;x\_m \cdot \frac{-2}{z\_m \cdot t}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-135} \lor \neg \left(y \leq 7.4 \cdot 10^{+57}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{t}}{z\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (/ x_m y) z_m))))
   (*
    z_s
    (*
     x_s
     (if (<= y -5e+127)
       t_1
       (if (<= y -3.1e+96)
         (* -2.0 (/ x_m (* z_m t)))
         (if (<= y -8.2e-72)
           t_1
           (if (<= y -1.4e-104)
             (* x_m (/ -2.0 (* z_m t)))
             (if (or (<= y -8.5e-135) (not (<= y 7.4e+57)))
               t_1
               (* -2.0 (/ (/ x_m t) z_m)))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_1;
	} else if (y <= -3.1e+96) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -8.2e-72) {
		tmp = t_1;
	} else if (y <= -1.4e-104) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else if ((y <= -8.5e-135) || !(y <= 7.4e+57)) {
		tmp = t_1;
	} else {
		tmp = -2.0 * ((x_m / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * ((x_m / y) / z_m)
    if (y <= (-5d+127)) then
        tmp = t_1
    else if (y <= (-3.1d+96)) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else if (y <= (-8.2d-72)) then
        tmp = t_1
    else if (y <= (-1.4d-104)) then
        tmp = x_m * ((-2.0d0) / (z_m * t))
    else if ((y <= (-8.5d-135)) .or. (.not. (y <= 7.4d+57))) then
        tmp = t_1
    else
        tmp = (-2.0d0) * ((x_m / t) / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_1;
	} else if (y <= -3.1e+96) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -8.2e-72) {
		tmp = t_1;
	} else if (y <= -1.4e-104) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else if ((y <= -8.5e-135) || !(y <= 7.4e+57)) {
		tmp = t_1;
	} else {
		tmp = -2.0 * ((x_m / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = 2.0 * ((x_m / y) / z_m)
	tmp = 0
	if y <= -5e+127:
		tmp = t_1
	elif y <= -3.1e+96:
		tmp = -2.0 * (x_m / (z_m * t))
	elif y <= -8.2e-72:
		tmp = t_1
	elif y <= -1.4e-104:
		tmp = x_m * (-2.0 / (z_m * t))
	elif (y <= -8.5e-135) or not (y <= 7.4e+57):
		tmp = t_1
	else:
		tmp = -2.0 * ((x_m / t) / z_m)
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(2.0 * Float64(Float64(x_m / y) / z_m))
	tmp = 0.0
	if (y <= -5e+127)
		tmp = t_1;
	elseif (y <= -3.1e+96)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	elseif (y <= -8.2e-72)
		tmp = t_1;
	elseif (y <= -1.4e-104)
		tmp = Float64(x_m * Float64(-2.0 / Float64(z_m * t)));
	elseif ((y <= -8.5e-135) || !(y <= 7.4e+57))
		tmp = t_1;
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = 2.0 * ((x_m / y) / z_m);
	tmp = 0.0;
	if (y <= -5e+127)
		tmp = t_1;
	elseif (y <= -3.1e+96)
		tmp = -2.0 * (x_m / (z_m * t));
	elseif (y <= -8.2e-72)
		tmp = t_1;
	elseif (y <= -1.4e-104)
		tmp = x_m * (-2.0 / (z_m * t));
	elseif ((y <= -8.5e-135) || ~((y <= 7.4e+57)))
		tmp = t_1;
	else
		tmp = -2.0 * ((x_m / t) / z_m);
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -5e+127], t$95$1, If[LessEqual[y, -3.1e+96], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -8.2e-72], t$95$1, If[LessEqual[y, -1.4e-104], N[(x$95$m * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -8.5e-135], N[Not[LessEqual[y, 7.4e+57]], $MachinePrecision]], t$95$1, N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -5.0000000000000004e127 or -3.0999999999999998e96 < y < -8.20000000000000007e-72 or -1.4e-104 < y < -8.49999999999999942e-135 or 7.40000000000000011e57 < y

   1. Initial program 83.5%

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

      1. distribute-rgt-out-- 88.0%

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

   3. Simplified 88.0%

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

      1. *-commutative 88.0%

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

      2. times-frac 93.3%

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

   6. Applied egg-rr 93.3%

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

   7. Taylor expanded in y around inf 77.7%

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

      1. associate-/r* 82.2%

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

   9. Simplified 82.2%

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

   if -5.0000000000000004e127 < y < -3.0999999999999998e96

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-*l/ 85.7%

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

      3. *-commutative 85.7%

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

      4. distribute-rgt-out-- 100.0%

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

      5. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -8.20000000000000007e-72 < y < -1.4e-104

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-out-- 99.5%

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

      5. associate-/l/ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 94.4%

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

      1. *-commutative 94.4%

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

   7. Simplified 94.4%

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

   if -8.49999999999999942e-135 < y < 7.40000000000000011e57

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l/ 87.4%

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

      3. *-commutative 87.4%

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

      4. distribute-rgt-out-- 89.2%

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

      5. associate-/l/ 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in y around 0 77.2%

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

      1. associate-/r* 78.4%

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

   7. Simplified 78.4%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+96}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-104}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-135} \lor \neg \left(y \leq 7.4 \cdot 10^{+57}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := 2 \cdot \frac{\frac{x\_m}{y}}{z\_m}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+96}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-104}:\\ \;\;\;\;x\_m \cdot \frac{-2}{z\_m \cdot t}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-134}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z\_m}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+60}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{t}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (/ x_m y) z_m))))
   (*
    z_s
    (*
     x_s
     (if (<= y -5e+127)
       t_1
       (if (<= y -3.1e+96)
         (* -2.0 (/ x_m (* z_m t)))
         (if (<= y -3.6e-71)
           t_1
           (if (<= y -1.32e-104)
             (* x_m (/ -2.0 (* z_m t)))
             (if (<= y -1.4e-134)
               (* x_m (/ (/ 2.0 y) z_m))
               (if (<= y 2.05e+60) (* -2.0 (/ (/ x_m t) z_m)) t_1))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_1;
	} else if (y <= -3.1e+96) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -3.6e-71) {
		tmp = t_1;
	} else if (y <= -1.32e-104) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else if (y <= -1.4e-134) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else if (y <= 2.05e+60) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * ((x_m / y) / z_m)
    if (y <= (-5d+127)) then
        tmp = t_1
    else if (y <= (-3.1d+96)) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else if (y <= (-3.6d-71)) then
        tmp = t_1
    else if (y <= (-1.32d-104)) then
        tmp = x_m * ((-2.0d0) / (z_m * t))
    else if (y <= (-1.4d-134)) then
        tmp = x_m * ((2.0d0 / y) / z_m)
    else if (y <= 2.05d+60) then
        tmp = (-2.0d0) * ((x_m / t) / z_m)
    else
        tmp = t_1
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_1;
	} else if (y <= -3.1e+96) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -3.6e-71) {
		tmp = t_1;
	} else if (y <= -1.32e-104) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else if (y <= -1.4e-134) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else if (y <= 2.05e+60) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = 2.0 * ((x_m / y) / z_m)
	tmp = 0
	if y <= -5e+127:
		tmp = t_1
	elif y <= -3.1e+96:
		tmp = -2.0 * (x_m / (z_m * t))
	elif y <= -3.6e-71:
		tmp = t_1
	elif y <= -1.32e-104:
		tmp = x_m * (-2.0 / (z_m * t))
	elif y <= -1.4e-134:
		tmp = x_m * ((2.0 / y) / z_m)
	elif y <= 2.05e+60:
		tmp = -2.0 * ((x_m / t) / z_m)
	else:
		tmp = t_1
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(2.0 * Float64(Float64(x_m / y) / z_m))
	tmp = 0.0
	if (y <= -5e+127)
		tmp = t_1;
	elseif (y <= -3.1e+96)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	elseif (y <= -3.6e-71)
		tmp = t_1;
	elseif (y <= -1.32e-104)
		tmp = Float64(x_m * Float64(-2.0 / Float64(z_m * t)));
	elseif (y <= -1.4e-134)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z_m));
	elseif (y <= 2.05e+60)
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z_m));
	else
		tmp = t_1;
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = 2.0 * ((x_m / y) / z_m);
	tmp = 0.0;
	if (y <= -5e+127)
		tmp = t_1;
	elseif (y <= -3.1e+96)
		tmp = -2.0 * (x_m / (z_m * t));
	elseif (y <= -3.6e-71)
		tmp = t_1;
	elseif (y <= -1.32e-104)
		tmp = x_m * (-2.0 / (z_m * t));
	elseif (y <= -1.4e-134)
		tmp = x_m * ((2.0 / y) / z_m);
	elseif (y <= 2.05e+60)
		tmp = -2.0 * ((x_m / t) / z_m);
	else
		tmp = t_1;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -5e+127], t$95$1, If[LessEqual[y, -3.1e+96], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.6e-71], t$95$1, If[LessEqual[y, -1.32e-104], N[(x$95$m * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.4e-134], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e+60], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if y < -5.0000000000000004e127 or -3.0999999999999998e96 < y < -3.6e-71 or 2.05e60 < y

   1. Initial program 83.9%

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

      1. distribute-rgt-out-- 87.2%

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

   3. Simplified 87.2%

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

      1. *-commutative 87.2%

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

      2. times-frac 93.6%

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

   6. Applied egg-rr 93.6%

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

   7. Taylor expanded in y around inf 76.8%

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

      1. associate-/r* 82.3%

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

   9. Simplified 82.3%

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

   if -5.0000000000000004e127 < y < -3.0999999999999998e96

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-*l/ 85.7%

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

      3. *-commutative 85.7%

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

      4. distribute-rgt-out-- 100.0%

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

      5. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -3.6e-71 < y < -1.3199999999999999e-104

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l/ 99.5%

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

      3. *-commutative 99.5%

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

      4. distribute-rgt-out-- 99.5%

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

      5. associate-/l/ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 94.4%

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

      1. *-commutative 94.4%

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

   7. Simplified 94.4%

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

   if -1.3199999999999999e-104 < y < -1.3999999999999999e-134

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*l/ 77.6%

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

      3. *-commutative 77.6%

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

      4. distribute-rgt-out-- 99.7%

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

      5. associate-/l/ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 90.6%

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

      1. associate-/r* 90.2%

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

   7. Simplified 90.2%

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

   if -1.3999999999999999e-134 < y < 2.05e60

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l/ 87.4%

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

      3. *-commutative 87.4%

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

      4. distribute-rgt-out-- 89.2%

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

      5. associate-/l/ 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in y around 0 77.2%

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

      1. associate-/r* 78.4%

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

   7. Simplified 78.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{\frac{x}{t}}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+96}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-71}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-104}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-134}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+60}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := 2 \cdot \frac{\frac{x\_m}{y}}{z\_m}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-104}:\\ \;\;\;\;\frac{x\_m}{t} \cdot \frac{-2}{z\_m}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-134}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z\_m}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+57}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{t}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (/ x_m y) z_m))))
   (*
    z_s
    (*
     x_s
     (if (<= y -5e+127)
       t_1
       (if (<= y -2.1e+101)
         (* -2.0 (/ x_m (* z_m t)))
         (if (<= y -4.2e-71)
           t_1
           (if (<= y -1.32e-104)
             (* (/ x_m t) (/ -2.0 z_m))
             (if (<= y -1.4e-134)
               (* x_m (/ (/ 2.0 y) z_m))
               (if (<= y 6e+57) (* -2.0 (/ (/ x_m t) z_m)) t_1))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_1;
	} else if (y <= -2.1e+101) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -4.2e-71) {
		tmp = t_1;
	} else if (y <= -1.32e-104) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else if (y <= -1.4e-134) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else if (y <= 6e+57) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * ((x_m / y) / z_m)
    if (y <= (-5d+127)) then
        tmp = t_1
    else if (y <= (-2.1d+101)) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else if (y <= (-4.2d-71)) then
        tmp = t_1
    else if (y <= (-1.32d-104)) then
        tmp = (x_m / t) * ((-2.0d0) / z_m)
    else if (y <= (-1.4d-134)) then
        tmp = x_m * ((2.0d0 / y) / z_m)
    else if (y <= 6d+57) then
        tmp = (-2.0d0) * ((x_m / t) / z_m)
    else
        tmp = t_1
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_1;
	} else if (y <= -2.1e+101) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -4.2e-71) {
		tmp = t_1;
	} else if (y <= -1.32e-104) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else if (y <= -1.4e-134) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else if (y <= 6e+57) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = 2.0 * ((x_m / y) / z_m)
	tmp = 0
	if y <= -5e+127:
		tmp = t_1
	elif y <= -2.1e+101:
		tmp = -2.0 * (x_m / (z_m * t))
	elif y <= -4.2e-71:
		tmp = t_1
	elif y <= -1.32e-104:
		tmp = (x_m / t) * (-2.0 / z_m)
	elif y <= -1.4e-134:
		tmp = x_m * ((2.0 / y) / z_m)
	elif y <= 6e+57:
		tmp = -2.0 * ((x_m / t) / z_m)
	else:
		tmp = t_1
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(2.0 * Float64(Float64(x_m / y) / z_m))
	tmp = 0.0
	if (y <= -5e+127)
		tmp = t_1;
	elseif (y <= -2.1e+101)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	elseif (y <= -4.2e-71)
		tmp = t_1;
	elseif (y <= -1.32e-104)
		tmp = Float64(Float64(x_m / t) * Float64(-2.0 / z_m));
	elseif (y <= -1.4e-134)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z_m));
	elseif (y <= 6e+57)
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z_m));
	else
		tmp = t_1;
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = 2.0 * ((x_m / y) / z_m);
	tmp = 0.0;
	if (y <= -5e+127)
		tmp = t_1;
	elseif (y <= -2.1e+101)
		tmp = -2.0 * (x_m / (z_m * t));
	elseif (y <= -4.2e-71)
		tmp = t_1;
	elseif (y <= -1.32e-104)
		tmp = (x_m / t) * (-2.0 / z_m);
	elseif (y <= -1.4e-134)
		tmp = x_m * ((2.0 / y) / z_m);
	elseif (y <= 6e+57)
		tmp = -2.0 * ((x_m / t) / z_m);
	else
		tmp = t_1;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -5e+127], t$95$1, If[LessEqual[y, -2.1e+101], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4.2e-71], t$95$1, If[LessEqual[y, -1.32e-104], N[(N[(x$95$m / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.4e-134], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+57], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if y < -5.0000000000000004e127 or -2.1e101 < y < -4.2000000000000002e-71 or 5.9999999999999999e57 < y

   1. Initial program 83.9%

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

      1. distribute-rgt-out-- 87.2%

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

   3. Simplified 87.2%

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

      1. *-commutative 87.2%

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

      2. times-frac 93.6%

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

   6. Applied egg-rr 93.6%

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

   7. Taylor expanded in y around inf 76.8%

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

      1. associate-/r* 82.3%

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

   9. Simplified 82.3%

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

   if -5.0000000000000004e127 < y < -2.1e101

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-*l/ 85.7%

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

      3. *-commutative 85.7%

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

      4. distribute-rgt-out-- 100.0%

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

      5. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -4.2000000000000002e-71 < y < -1.3199999999999999e-104

   1. Initial program 99.7%

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

      1. distribute-rgt-out-- 99.7%

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

   3. Simplified 99.7%

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

      1. *-commutative 99.7%

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

      2. times-frac 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 94.4%

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

      1. associate-*r/ 94.4%

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

      2. *-commutative 94.4%

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

      3. times-frac 94.6%

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

   9. Simplified 94.6%

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

   if -1.3199999999999999e-104 < y < -1.3999999999999999e-134

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*l/ 77.6%

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

      3. *-commutative 77.6%

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

      4. distribute-rgt-out-- 99.7%

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

      5. associate-/l/ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 90.6%

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

      1. associate-/r* 90.2%

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

   7. Simplified 90.2%

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

   if -1.3999999999999999e-134 < y < 5.9999999999999999e57

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l/ 87.4%

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

      3. *-commutative 87.4%

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

      4. distribute-rgt-out-- 89.2%

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

      5. associate-/l/ 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in y around 0 77.2%

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

      1. associate-/r* 78.4%

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

   7. Simplified 78.4%

      \[\leadsto \color{blue}{-2 \cdot \frac{\frac{x}{t}}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-71}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-134}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+57}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{z\_m} \cdot \frac{-2}{t}\\ t_2 := 2 \cdot \frac{\frac{x\_m}{y}}{z\_m}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -5.1 \cdot 10^{+100}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{elif}\;y \leq -2100000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-134}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z\_m}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x_m z_m) (/ -2.0 t))) (t_2 (* 2.0 (/ (/ x_m y) z_m))))
   (*
    z_s
    (*
     x_s
     (if (<= y -5e+127)
       t_2
       (if (<= y -5.1e+100)
         (* -2.0 (/ x_m (* z_m t)))
         (if (<= y -2100000000000.0)
           t_2
           (if (<= y -1.32e-104)
             t_1
             (if (<= y -1.4e-134)
               (* x_m (/ (/ 2.0 y) z_m))
               (if (<= y 6e+57) t_1 t_2))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m / z_m) * (-2.0 / t);
	double t_2 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_2;
	} else if (y <= -5.1e+100) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -2100000000000.0) {
		tmp = t_2;
	} else if (y <= -1.32e-104) {
		tmp = t_1;
	} else if (y <= -1.4e-134) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else if (y <= 6e+57) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x_m / z_m) * ((-2.0d0) / t)
    t_2 = 2.0d0 * ((x_m / y) / z_m)
    if (y <= (-5d+127)) then
        tmp = t_2
    else if (y <= (-5.1d+100)) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else if (y <= (-2100000000000.0d0)) then
        tmp = t_2
    else if (y <= (-1.32d-104)) then
        tmp = t_1
    else if (y <= (-1.4d-134)) then
        tmp = x_m * ((2.0d0 / y) / z_m)
    else if (y <= 6d+57) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m / z_m) * (-2.0 / t);
	double t_2 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_2;
	} else if (y <= -5.1e+100) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -2100000000000.0) {
		tmp = t_2;
	} else if (y <= -1.32e-104) {
		tmp = t_1;
	} else if (y <= -1.4e-134) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else if (y <= 6e+57) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = (x_m / z_m) * (-2.0 / t)
	t_2 = 2.0 * ((x_m / y) / z_m)
	tmp = 0
	if y <= -5e+127:
		tmp = t_2
	elif y <= -5.1e+100:
		tmp = -2.0 * (x_m / (z_m * t))
	elif y <= -2100000000000.0:
		tmp = t_2
	elif y <= -1.32e-104:
		tmp = t_1
	elif y <= -1.4e-134:
		tmp = x_m * ((2.0 / y) / z_m)
	elif y <= 6e+57:
		tmp = t_1
	else:
		tmp = t_2
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(Float64(x_m / z_m) * Float64(-2.0 / t))
	t_2 = Float64(2.0 * Float64(Float64(x_m / y) / z_m))
	tmp = 0.0
	if (y <= -5e+127)
		tmp = t_2;
	elseif (y <= -5.1e+100)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	elseif (y <= -2100000000000.0)
		tmp = t_2;
	elseif (y <= -1.32e-104)
		tmp = t_1;
	elseif (y <= -1.4e-134)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z_m));
	elseif (y <= 6e+57)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = (x_m / z_m) * (-2.0 / t);
	t_2 = 2.0 * ((x_m / y) / z_m);
	tmp = 0.0;
	if (y <= -5e+127)
		tmp = t_2;
	elseif (y <= -5.1e+100)
		tmp = -2.0 * (x_m / (z_m * t));
	elseif (y <= -2100000000000.0)
		tmp = t_2;
	elseif (y <= -1.32e-104)
		tmp = t_1;
	elseif (y <= -1.4e-134)
		tmp = x_m * ((2.0 / y) / z_m);
	elseif (y <= 6e+57)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -5e+127], t$95$2, If[LessEqual[y, -5.1e+100], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2100000000000.0], t$95$2, If[LessEqual[y, -1.32e-104], t$95$1, If[LessEqual[y, -1.4e-134], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+57], t$95$1, t$95$2]]]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.0000000000000004e127 or -5.1000000000000001e100 < y < -2.1e12 or 5.9999999999999999e57 < y

   1. Initial program 83.0%

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

      1. distribute-rgt-out-- 86.9%

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

   3. Simplified 86.9%

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

      1. *-commutative 86.9%

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

      2. times-frac 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in y around inf 80.3%

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

      1. associate-/r* 86.7%

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

   9. Simplified 86.7%

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

   if -5.0000000000000004e127 < y < -5.1000000000000001e100

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-*l/ 85.7%

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

      3. *-commutative 85.7%

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

      4. distribute-rgt-out-- 100.0%

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

      5. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -2.1e12 < y < -1.3199999999999999e-104 or -1.3999999999999999e-134 < y < 5.9999999999999999e57

   1. Initial program 89.5%

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

      1. distribute-rgt-out-- 91.1%

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

      2. times-frac 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in y around 0 77.1%

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

   if -1.3199999999999999e-104 < y < -1.3999999999999999e-134

   1. Initial program 77.6%

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

      1. *-commutative 77.6%

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

      2. associate-*l/ 77.6%

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

      3. *-commutative 77.6%

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

      4. distribute-rgt-out-- 99.7%

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

      5. associate-/l/ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 90.6%

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

      1. associate-/r* 90.2%

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

   7. Simplified 90.2%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -5.1 \cdot 10^{+100}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq -2100000000000:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -1.32 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-134}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{z\_m} \cdot \frac{-2}{t}\\ t_2 := 2 \cdot \frac{\frac{x\_m}{y}}{z\_m}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+100}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{elif}\;y \leq -38000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x_m z_m) (/ -2.0 t))) (t_2 (* 2.0 (/ (/ x_m y) z_m))))
   (*
    z_s
    (*
     x_s
     (if (<= y -5e+127)
       t_2
       (if (<= y -5.2e+100)
         (* -2.0 (/ x_m (* z_m t)))
         (if (<= y -38000000000.0)
           t_2
           (if (<= y -4.5e-104)
             t_1
             (if (<= y -1.4e-134)
               (/ (* x_m 2.0) (* y z_m))
               (if (<= y 7.5e+57) t_1 t_2))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m / z_m) * (-2.0 / t);
	double t_2 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_2;
	} else if (y <= -5.2e+100) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -38000000000.0) {
		tmp = t_2;
	} else if (y <= -4.5e-104) {
		tmp = t_1;
	} else if (y <= -1.4e-134) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 7.5e+57) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x_m / z_m) * ((-2.0d0) / t)
    t_2 = 2.0d0 * ((x_m / y) / z_m)
    if (y <= (-5d+127)) then
        tmp = t_2
    else if (y <= (-5.2d+100)) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else if (y <= (-38000000000.0d0)) then
        tmp = t_2
    else if (y <= (-4.5d-104)) then
        tmp = t_1
    else if (y <= (-1.4d-134)) then
        tmp = (x_m * 2.0d0) / (y * z_m)
    else if (y <= 7.5d+57) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m / z_m) * (-2.0 / t);
	double t_2 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_2;
	} else if (y <= -5.2e+100) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -38000000000.0) {
		tmp = t_2;
	} else if (y <= -4.5e-104) {
		tmp = t_1;
	} else if (y <= -1.4e-134) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 7.5e+57) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = (x_m / z_m) * (-2.0 / t)
	t_2 = 2.0 * ((x_m / y) / z_m)
	tmp = 0
	if y <= -5e+127:
		tmp = t_2
	elif y <= -5.2e+100:
		tmp = -2.0 * (x_m / (z_m * t))
	elif y <= -38000000000.0:
		tmp = t_2
	elif y <= -4.5e-104:
		tmp = t_1
	elif y <= -1.4e-134:
		tmp = (x_m * 2.0) / (y * z_m)
	elif y <= 7.5e+57:
		tmp = t_1
	else:
		tmp = t_2
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(Float64(x_m / z_m) * Float64(-2.0 / t))
	t_2 = Float64(2.0 * Float64(Float64(x_m / y) / z_m))
	tmp = 0.0
	if (y <= -5e+127)
		tmp = t_2;
	elseif (y <= -5.2e+100)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	elseif (y <= -38000000000.0)
		tmp = t_2;
	elseif (y <= -4.5e-104)
		tmp = t_1;
	elseif (y <= -1.4e-134)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	elseif (y <= 7.5e+57)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = (x_m / z_m) * (-2.0 / t);
	t_2 = 2.0 * ((x_m / y) / z_m);
	tmp = 0.0;
	if (y <= -5e+127)
		tmp = t_2;
	elseif (y <= -5.2e+100)
		tmp = -2.0 * (x_m / (z_m * t));
	elseif (y <= -38000000000.0)
		tmp = t_2;
	elseif (y <= -4.5e-104)
		tmp = t_1;
	elseif (y <= -1.4e-134)
		tmp = (x_m * 2.0) / (y * z_m);
	elseif (y <= 7.5e+57)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -5e+127], t$95$2, If[LessEqual[y, -5.2e+100], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -38000000000.0], t$95$2, If[LessEqual[y, -4.5e-104], t$95$1, If[LessEqual[y, -1.4e-134], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+57], t$95$1, t$95$2]]]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.0000000000000004e127 or -5.2000000000000003e100 < y < -3.8e10 or 7.5000000000000006e57 < y

   1. Initial program 83.0%

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

      1. distribute-rgt-out-- 86.9%

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

   3. Simplified 86.9%

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

      1. *-commutative 86.9%

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

      2. times-frac 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in y around inf 80.3%

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

      1. associate-/r* 86.7%

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

   9. Simplified 86.7%

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

   if -5.0000000000000004e127 < y < -5.2000000000000003e100

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-*l/ 85.7%

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

      3. *-commutative 85.7%

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

      4. distribute-rgt-out-- 100.0%

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

      5. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -3.8e10 < y < -4.4999999999999997e-104 or -1.3999999999999999e-134 < y < 7.5000000000000006e57

   1. Initial program 89.5%

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

      1. distribute-rgt-out-- 91.1%

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

      2. times-frac 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in y around 0 77.1%

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

   if -4.4999999999999997e-104 < y < -1.3999999999999999e-134

   1. Initial program 77.6%

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

      1. distribute-rgt-out-- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 90.4%

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

      1. *-commutative 90.4%

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

   7. Simplified 90.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+100}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq -38000000000:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -4.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-134}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := 2 \cdot \frac{\frac{x\_m}{y}}{z\_m}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{elif}\;y \leq -2700000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.45 \cdot 10^{-104}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{-2}{t}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-135}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* 2.0 (/ (/ x_m y) z_m))))
   (*
    z_s
    (*
     x_s
     (if (<= y -5e+127)
       t_1
       (if (<= y -2.1e+101)
         (* -2.0 (/ x_m (* z_m t)))
         (if (<= y -2700000000000.0)
           t_1
           (if (<= y -2.45e-104)
             (* (/ x_m z_m) (/ -2.0 t))
             (if (<= y -5e-135)
               (/ (* x_m 2.0) (* y z_m))
               (if (<= y 2.9e+58) (/ (/ (* x_m -2.0) z_m) t) t_1))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_1;
	} else if (y <= -2.1e+101) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -2700000000000.0) {
		tmp = t_1;
	} else if (y <= -2.45e-104) {
		tmp = (x_m / z_m) * (-2.0 / t);
	} else if (y <= -5e-135) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 2.9e+58) {
		tmp = ((x_m * -2.0) / z_m) / t;
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 * ((x_m / y) / z_m)
    if (y <= (-5d+127)) then
        tmp = t_1
    else if (y <= (-2.1d+101)) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else if (y <= (-2700000000000.0d0)) then
        tmp = t_1
    else if (y <= (-2.45d-104)) then
        tmp = (x_m / z_m) * ((-2.0d0) / t)
    else if (y <= (-5d-135)) then
        tmp = (x_m * 2.0d0) / (y * z_m)
    else if (y <= 2.9d+58) then
        tmp = ((x_m * (-2.0d0)) / z_m) / t
    else
        tmp = t_1
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 * ((x_m / y) / z_m);
	double tmp;
	if (y <= -5e+127) {
		tmp = t_1;
	} else if (y <= -2.1e+101) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -2700000000000.0) {
		tmp = t_1;
	} else if (y <= -2.45e-104) {
		tmp = (x_m / z_m) * (-2.0 / t);
	} else if (y <= -5e-135) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 2.9e+58) {
		tmp = ((x_m * -2.0) / z_m) / t;
	} else {
		tmp = t_1;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = 2.0 * ((x_m / y) / z_m)
	tmp = 0
	if y <= -5e+127:
		tmp = t_1
	elif y <= -2.1e+101:
		tmp = -2.0 * (x_m / (z_m * t))
	elif y <= -2700000000000.0:
		tmp = t_1
	elif y <= -2.45e-104:
		tmp = (x_m / z_m) * (-2.0 / t)
	elif y <= -5e-135:
		tmp = (x_m * 2.0) / (y * z_m)
	elif y <= 2.9e+58:
		tmp = ((x_m * -2.0) / z_m) / t
	else:
		tmp = t_1
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(2.0 * Float64(Float64(x_m / y) / z_m))
	tmp = 0.0
	if (y <= -5e+127)
		tmp = t_1;
	elseif (y <= -2.1e+101)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	elseif (y <= -2700000000000.0)
		tmp = t_1;
	elseif (y <= -2.45e-104)
		tmp = Float64(Float64(x_m / z_m) * Float64(-2.0 / t));
	elseif (y <= -5e-135)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	elseif (y <= 2.9e+58)
		tmp = Float64(Float64(Float64(x_m * -2.0) / z_m) / t);
	else
		tmp = t_1;
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = 2.0 * ((x_m / y) / z_m);
	tmp = 0.0;
	if (y <= -5e+127)
		tmp = t_1;
	elseif (y <= -2.1e+101)
		tmp = -2.0 * (x_m / (z_m * t));
	elseif (y <= -2700000000000.0)
		tmp = t_1;
	elseif (y <= -2.45e-104)
		tmp = (x_m / z_m) * (-2.0 / t);
	elseif (y <= -5e-135)
		tmp = (x_m * 2.0) / (y * z_m);
	elseif (y <= 2.9e+58)
		tmp = ((x_m * -2.0) / z_m) / t;
	else
		tmp = t_1;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[(x$95$m / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -5e+127], t$95$1, If[LessEqual[y, -2.1e+101], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2700000000000.0], t$95$1, If[LessEqual[y, -2.45e-104], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5e-135], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+58], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / z$95$m), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if y < -5.0000000000000004e127 or -2.1e101 < y < -2.7e12 or 2.90000000000000002e58 < y

   1. Initial program 83.0%

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

      1. distribute-rgt-out-- 86.9%

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

   3. Simplified 86.9%

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

      1. *-commutative 86.9%

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

      2. times-frac 95.1%

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

   6. Applied egg-rr 95.1%

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

   7. Taylor expanded in y around inf 80.3%

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

      1. associate-/r* 86.7%

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

   9. Simplified 86.7%

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

   if -5.0000000000000004e127 < y < -2.1e101

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-*l/ 85.7%

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

      3. *-commutative 85.7%

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

      4. distribute-rgt-out-- 100.0%

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

      5. associate-/l/ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. *-commutative 100.0%

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

   7. Simplified 100.0%

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

   if -2.7e12 < y < -2.4500000000000001e-104

   1. Initial program 91.8%

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

      1. distribute-rgt-out-- 91.9%

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

      2. times-frac 92.2%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in y around 0 66.7%

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

   if -2.4500000000000001e-104 < y < -5.0000000000000002e-135

   1. Initial program 77.6%

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

      1. distribute-rgt-out-- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 90.4%

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

      1. *-commutative 90.4%

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

   7. Simplified 90.4%

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

   if -5.0000000000000002e-135 < y < 2.90000000000000002e58

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

      2. associate-*l/ 87.4%

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

      3. *-commutative 87.4%

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

      4. distribute-rgt-out-- 89.2%

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

      5. associate-/l/ 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in y around 0 76.0%

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

      1. *-commutative 76.0%

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

   7. Simplified 76.0%

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

      1. associate-*r/ 77.2%

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

      2. associate-/r* 79.5%

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

   9. Applied egg-rr 79.5%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot -2}{z}}{t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+127}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{+101}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq -2700000000000:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \mathbf{elif}\;y \leq -2.45 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-135}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 5e-61)
     (* (/ x_m z_m) (/ 2.0 (- y t)))
     (* (/ x_m (- y t)) (/ 2.0 z_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-61) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-61) then
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-61) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-61:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z_m)
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-61)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-61)
		tmp = (x_m / z_m) * (2.0 / (y - t));
	else
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-61], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 2) < 4.9999999999999999e-61

   1. Initial program 87.6%

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

      1. distribute-rgt-out-- 89.9%

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

      2. times-frac 90.3%

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

   3. Simplified 90.3%

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

   if 4.9999999999999999e-61 < (*.f64 x 2)

   1. Initial program 83.5%

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

      1. distribute-rgt-out-- 89.7%

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

   3. Simplified 89.7%

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

      1. *-commutative 89.7%

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

      2. times-frac 96.1%

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

   6. Applied egg-rr 96.1%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{2}{y - t}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.2 \cdot 10^{+27}:\\ \;\;\;\;x\_m \cdot \frac{t\_1}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot t\_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (- y t))))
   (*
    z_s
    (* x_s (if (<= z_m 5.2e+27) (* x_m (/ t_1 z_m)) (* (/ x_m z_m) t_1))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 / (y - t);
	double tmp;
	if (z_m <= 5.2e+27) {
		tmp = x_m * (t_1 / z_m);
	} else {
		tmp = (x_m / z_m) * t_1;
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (y - t)
    if (z_m <= 5.2d+27) then
        tmp = x_m * (t_1 / z_m)
    else
        tmp = (x_m / z_m) * t_1
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 / (y - t);
	double tmp;
	if (z_m <= 5.2e+27) {
		tmp = x_m * (t_1 / z_m);
	} else {
		tmp = (x_m / z_m) * t_1;
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = 2.0 / (y - t)
	tmp = 0
	if z_m <= 5.2e+27:
		tmp = x_m * (t_1 / z_m)
	else:
		tmp = (x_m / z_m) * t_1
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(2.0 / Float64(y - t))
	tmp = 0.0
	if (z_m <= 5.2e+27)
		tmp = Float64(x_m * Float64(t_1 / z_m));
	else
		tmp = Float64(Float64(x_m / z_m) * t_1);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = 2.0 / (y - t);
	tmp = 0.0;
	if (z_m <= 5.2e+27)
		tmp = x_m * (t_1 / z_m);
	else
		tmp = (x_m / z_m) * t_1;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 5.2e+27], N[(x$95$m * N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.20000000000000018e27

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

      2. associate-*l/ 86.6%

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

      3. *-commutative 86.6%

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

      4. distribute-rgt-out-- 90.7%

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

      5. associate-/l/ 91.0%

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

   3. Simplified 91.0%

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

   if 5.20000000000000018e27 < z

   1. Initial program 83.9%

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

      1. distribute-rgt-out-- 85.7%

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

      2. times-frac 96.5%

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

   3. Simplified 96.5%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+27}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9.2 \cdot 10^{-94}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{t}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 9.2e-94)
     (* -2.0 (/ x_m (* z_m t)))
     (* -2.0 (/ (/ x_m t) z_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 9.2e-94) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = -2.0 * ((x_m / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 9.2d-94) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else
        tmp = (-2.0d0) * ((x_m / t) / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 9.2e-94) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = -2.0 * ((x_m / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if z_m <= 9.2e-94:
		tmp = -2.0 * (x_m / (z_m * t))
	else:
		tmp = -2.0 * ((x_m / t) / z_m)
	return z_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (z_m <= 9.2e-94)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 9.2e-94)
		tmp = -2.0 * (x_m / (z_m * t));
	else
		tmp = -2.0 * ((x_m / t) / z_m);
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 9.2e-94], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.1999999999999997e-94

   1. Initial program 85.6%

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

      1. *-commutative 85.6%

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

      2. associate-*l/ 85.1%

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

      3. *-commutative 85.1%

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

      4. distribute-rgt-out-- 89.7%

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

      5. associate-/l/ 90.0%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in y around 0 50.1%

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

      1. *-commutative 50.1%

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

   7. Simplified 50.1%

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

   if 9.1999999999999997e-94 < z

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. associate-*l/ 87.8%

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

      3. *-commutative 87.8%

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

      4. distribute-rgt-out-- 89.1%

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

      5. associate-/l/ 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in y around 0 45.8%

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

      1. associate-/r* 49.3%

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

   7. Simplified 49.3%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.2 \cdot 10^{-94}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \left(x\_m \cdot \frac{\frac{2}{y - t}}{z\_m}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (* z_s (* x_s (* x_m (/ (/ 2.0 (- y t)) z_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	return z_s * (x_s * (x_m * ((2.0 / (y - t)) / z_m)));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    code = z_s * (x_s * (x_m * ((2.0d0 / (y - t)) / z_m)))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	return z_s * (x_s * (x_m * ((2.0 / (y - t)) / z_m)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	return z_s * (x_s * (x_m * ((2.0 / (y - t)) / z_m)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	return Float64(z_s * Float64(x_s * Float64(x_m * Float64(Float64(2.0 / Float64(y - t)) / z_m))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, x_s, x_m, y, z_m, t)
	tmp = z_s * (x_s * (x_m * ((2.0 / (y - t)) / z_m)));
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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * N[(x$95$m * N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

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

   1. *-commutative 86.3%

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

   2. associate-*l/ 85.9%

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

   3. *-commutative 85.9%

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

   4. distribute-rgt-out-- 89.5%

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

   5. associate-/l/ 89.7%

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

3. Simplified 89.7%

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

5. Final simplification 89.7%

   \[\leadsto x \cdot \frac{\frac{2}{y - t}}{z} \]
6. Add Preprocessing

Alternative 13: 53.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \left(-2 \cdot \frac{x\_m}{z\_m \cdot t}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (* z_s (* x_s (* -2.0 (/ x_m (* z_m t))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	return z_s * (x_s * (-2.0 * (x_m / (z_m * t))));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    code = z_s * (x_s * ((-2.0d0) * (x_m / (z_m * t))))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	return z_s * (x_s * (-2.0 * (x_m / (z_m * t))));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	return z_s * (x_s * (-2.0 * (x_m / (z_m * t))))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	return Float64(z_s * Float64(x_s * Float64(-2.0 * Float64(x_m / Float64(z_m * t)))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, x_s, x_m, y, z_m, t)
	tmp = z_s * (x_s * (-2.0 * (x_m / (z_m * t))));
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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.3%

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

   1. *-commutative 86.3%

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

   2. associate-*l/ 85.9%

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

   3. *-commutative 85.9%

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

   4. distribute-rgt-out-- 89.5%

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

   5. associate-/l/ 89.7%

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

3. Simplified 89.7%

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

5. Taylor expanded in y around 0 48.7%

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

   1. *-commutative 48.7%

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

7. Simplified 48.7%

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

8. Final simplification 48.7%

   \[\leadsto -2 \cdot \frac{x}{z \cdot t} \]
9. Add Preprocessing

Developer target: 97.0% 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 2024027 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (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))))