Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.3% → 96.8%

Time: 11.0s

Alternatives: 16

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * 2.0d0) / ((y * z) - (t * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x * 2.0d0) / ((y * z) - (t * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \sqrt{x\_m \cdot 2}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot 2}{y \cdot z\_m - z\_m \cdot t} \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{y - t} \cdot \frac{t\_1}{z\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (sqrt (* x_m 2.0))))
   (*
    z_s
    (*
     x_s
     (if (<= (/ (* x_m 2.0) (- (* y z_m) (* z_m t))) -5e-310)
       (/ (* x_m 2.0) (* z_m (- y t)))
       (* (/ t_1 (- y t)) (/ t_1 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 = sqrt((x_m * 2.0));
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -5e-310) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (t_1 / (y - t)) * (t_1 / 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 = sqrt((x_m * 2.0d0))
    if (((x_m * 2.0d0) / ((y * z_m) - (z_m * t))) <= (-5d-310)) then
        tmp = (x_m * 2.0d0) / (z_m * (y - t))
    else
        tmp = (t_1 / (y - t)) * (t_1 / 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 = Math.sqrt((x_m * 2.0));
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -5e-310) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (t_1 / (y - t)) * (t_1 / 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 = math.sqrt((x_m * 2.0))
	tmp = 0
	if ((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -5e-310:
		tmp = (x_m * 2.0) / (z_m * (y - t))
	else:
		tmp = (t_1 / (y - t)) * (t_1 / 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 = sqrt(Float64(x_m * 2.0))
	tmp = 0.0
	if (Float64(Float64(x_m * 2.0) / Float64(Float64(y * z_m) - Float64(z_m * t))) <= -5e-310)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(t_1 / Float64(y - t)) * Float64(t_1 / 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 = sqrt((x_m * 2.0));
	tmp = 0.0;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -5e-310)
		tmp = (x_m * 2.0) / (z_m * (y - t));
	else
		tmp = (t_1 / (y - t)) * (t_1 / 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[Sqrt[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-310], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -4.999999999999985e-310

   1. Initial program 98.4%

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

      1. distribute-rgt-out-- 98.5%

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

   3. Simplified 98.5%

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

   if -4.999999999999985e-310 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 84.6%

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

      1. distribute-rgt-out-- 86.5%

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

   3. Simplified 86.5%

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

      1. add-sqr-sqrt 50.5%

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

      2. *-commutative 50.5%

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

      3. times-frac 53.7%

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

   6. Applied egg-rr 53.7%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x \cdot 2}}{y - t} \cdot \frac{\sqrt{x \cdot 2}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{-2}{t \cdot \frac{z\_m}{x\_m}}\\ t_2 := \frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+99}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-55}:\\ \;\;\;\;\frac{x\_m}{t} \cdot \frac{-2}{z\_m}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ -2.0 (* t (/ z_m x_m)))) (t_2 (* (/ x_m z_m) (/ 2.0 y))))
   (*
    z_s
    (*
     x_s
     (if (<= y -5e+99)
       t_2
       (if (<= y -5.5e+51)
         t_1
         (if (<= y -1.1e-31)
           (/ (* x_m 2.0) (* y z_m))
           (if (<= y 3.9e-55)
             (* (/ x_m t) (/ -2.0 z_m))
             (if (<= y 3.8e-6)
               t_2
               (if (<= y 7.8e+78) t_1 (* (/ 2.0 z_m) (/ x_m y))))))))))))

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 / (t * (z_m / x_m));
	double t_2 = (x_m / z_m) * (2.0 / y);
	double tmp;
	if (y <= -5e+99) {
		tmp = t_2;
	} else if (y <= -5.5e+51) {
		tmp = t_1;
	} else if (y <= -1.1e-31) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 3.9e-55) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else if (y <= 3.8e-6) {
		tmp = t_2;
	} else if (y <= 7.8e+78) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	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) / (t * (z_m / x_m))
    t_2 = (x_m / z_m) * (2.0d0 / y)
    if (y <= (-5d+99)) then
        tmp = t_2
    else if (y <= (-5.5d+51)) then
        tmp = t_1
    else if (y <= (-1.1d-31)) then
        tmp = (x_m * 2.0d0) / (y * z_m)
    else if (y <= 3.9d-55) then
        tmp = (x_m / t) * ((-2.0d0) / z_m)
    else if (y <= 3.8d-6) then
        tmp = t_2
    else if (y <= 7.8d+78) then
        tmp = t_1
    else
        tmp = (2.0d0 / z_m) * (x_m / y)
    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 / (t * (z_m / x_m));
	double t_2 = (x_m / z_m) * (2.0 / y);
	double tmp;
	if (y <= -5e+99) {
		tmp = t_2;
	} else if (y <= -5.5e+51) {
		tmp = t_1;
	} else if (y <= -1.1e-31) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 3.9e-55) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else if (y <= 3.8e-6) {
		tmp = t_2;
	} else if (y <= 7.8e+78) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	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 / (t * (z_m / x_m))
	t_2 = (x_m / z_m) * (2.0 / y)
	tmp = 0
	if y <= -5e+99:
		tmp = t_2
	elif y <= -5.5e+51:
		tmp = t_1
	elif y <= -1.1e-31:
		tmp = (x_m * 2.0) / (y * z_m)
	elif y <= 3.9e-55:
		tmp = (x_m / t) * (-2.0 / z_m)
	elif y <= 3.8e-6:
		tmp = t_2
	elif y <= 7.8e+78:
		tmp = t_1
	else:
		tmp = (2.0 / z_m) * (x_m / y)
	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(t * Float64(z_m / x_m)))
	t_2 = Float64(Float64(x_m / z_m) * Float64(2.0 / y))
	tmp = 0.0
	if (y <= -5e+99)
		tmp = t_2;
	elseif (y <= -5.5e+51)
		tmp = t_1;
	elseif (y <= -1.1e-31)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	elseif (y <= 3.9e-55)
		tmp = Float64(Float64(x_m / t) * Float64(-2.0 / z_m));
	elseif (y <= 3.8e-6)
		tmp = t_2;
	elseif (y <= 7.8e+78)
		tmp = t_1;
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	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 / (t * (z_m / x_m));
	t_2 = (x_m / z_m) * (2.0 / y);
	tmp = 0.0;
	if (y <= -5e+99)
		tmp = t_2;
	elseif (y <= -5.5e+51)
		tmp = t_1;
	elseif (y <= -1.1e-31)
		tmp = (x_m * 2.0) / (y * z_m);
	elseif (y <= 3.9e-55)
		tmp = (x_m / t) * (-2.0 / z_m);
	elseif (y <= 3.8e-6)
		tmp = t_2;
	elseif (y <= 7.8e+78)
		tmp = t_1;
	else
		tmp = (2.0 / z_m) * (x_m / y);
	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[(t * N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -5e+99], t$95$2, If[LessEqual[y, -5.5e+51], t$95$1, If[LessEqual[y, -1.1e-31], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e-55], N[(N[(x$95$m / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-6], t$95$2, If[LessEqual[y, 7.8e+78], t$95$1, N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if y < -5.00000000000000008e99 or 3.9e-55 < y < 3.8e-6

   1. Initial program 82.0%

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

      1. distribute-rgt-out-- 83.7%

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

   3. Simplified 83.7%

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

      1. add-sqr-sqrt 43.2%

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

      2. *-commutative 43.2%

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

      3. times-frac 47.9%

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

   6. Applied egg-rr 47.9%

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

   7. Taylor expanded in y around inf 77.4%

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

      1. associate-/r* 81.7%

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

      2. unpow2 81.7%

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

      3. rem-square-sqrt 82.5%

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

      4. associate-/l* 82.4%

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

      5. associate-*l/ 87.8%

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

   9. Simplified 87.8%

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

   if -5.00000000000000008e99 < y < -5.5e51 or 3.8e-6 < y < 7.8000000000000008e78

   1. Initial program 84.9%

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

      1. distribute-rgt-out-- 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in y around 0 70.4%

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

      1. *-commutative 70.4%

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

   7. Simplified 70.4%

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

      1. clear-num 70.4%

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

      2. un-div-inv 70.4%

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

      3. *-commutative 70.4%

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

      4. associate-/l* 81.0%

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

   9. Applied egg-rr 81.0%

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

   if -5.5e51 < y < -1.10000000000000005e-31

   1. Initial program 99.8%

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

      1. distribute-rgt-out-- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 87.7%

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

      1. *-commutative 87.7%

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

   7. Simplified 87.7%

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

   if -1.10000000000000005e-31 < y < 3.9e-55

   1. Initial program 94.1%

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

      1. distribute-rgt-out-- 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in y around 0 80.8%

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

      1. *-commutative 80.8%

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

   7. Simplified 80.8%

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

      1. clear-num 79.9%

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

      2. un-div-inv 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-/l* 80.3%

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

   9. Applied egg-rr 80.3%

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

       1. associate-/r* 80.9%

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

       2. div-inv 80.8%

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

       3. clear-num 81.0%

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

       4. times-frac 80.8%

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

       5. *-commutative 80.8%

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

       6. times-frac 81.4%

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

   11. Applied egg-rr 81.4%

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

   if 7.8000000000000008e78 < y

   1. Initial program 85.0%

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

      1. distribute-rgt-out-- 85.3%

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

   3. Simplified 85.3%

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

      1. *-commutative 85.3%

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

      2. times-frac 89.8%

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

   6. Applied egg-rr 89.8%

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

   7. Taylor expanded in y around inf 77.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+51}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{-2}{t \cdot \frac{z\_m}{x\_m}}\\ t_2 := \frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+100}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-32}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-56}:\\ \;\;\;\;\frac{x\_m}{t} \cdot \frac{-2}{z\_m}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-8}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ -2.0 (* t (/ z_m x_m)))) (t_2 (* (/ x_m z_m) (/ 2.0 y))))
   (*
    z_s
    (*
     x_s
     (if (<= y -2.5e+100)
       t_2
       (if (<= y -1.05e+52)
         t_1
         (if (<= y -1.95e-32)
           t_2
           (if (<= y 7.3e-56)
             (* (/ x_m t) (/ -2.0 z_m))
             (if (<= y 5.8e-8)
               t_2
               (if (<= y 7.8e+78) t_1 (* (/ 2.0 z_m) (/ x_m y))))))))))))

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 / (t * (z_m / x_m));
	double t_2 = (x_m / z_m) * (2.0 / y);
	double tmp;
	if (y <= -2.5e+100) {
		tmp = t_2;
	} else if (y <= -1.05e+52) {
		tmp = t_1;
	} else if (y <= -1.95e-32) {
		tmp = t_2;
	} else if (y <= 7.3e-56) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else if (y <= 5.8e-8) {
		tmp = t_2;
	} else if (y <= 7.8e+78) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	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) / (t * (z_m / x_m))
    t_2 = (x_m / z_m) * (2.0d0 / y)
    if (y <= (-2.5d+100)) then
        tmp = t_2
    else if (y <= (-1.05d+52)) then
        tmp = t_1
    else if (y <= (-1.95d-32)) then
        tmp = t_2
    else if (y <= 7.3d-56) then
        tmp = (x_m / t) * ((-2.0d0) / z_m)
    else if (y <= 5.8d-8) then
        tmp = t_2
    else if (y <= 7.8d+78) then
        tmp = t_1
    else
        tmp = (2.0d0 / z_m) * (x_m / y)
    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 / (t * (z_m / x_m));
	double t_2 = (x_m / z_m) * (2.0 / y);
	double tmp;
	if (y <= -2.5e+100) {
		tmp = t_2;
	} else if (y <= -1.05e+52) {
		tmp = t_1;
	} else if (y <= -1.95e-32) {
		tmp = t_2;
	} else if (y <= 7.3e-56) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else if (y <= 5.8e-8) {
		tmp = t_2;
	} else if (y <= 7.8e+78) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	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 / (t * (z_m / x_m))
	t_2 = (x_m / z_m) * (2.0 / y)
	tmp = 0
	if y <= -2.5e+100:
		tmp = t_2
	elif y <= -1.05e+52:
		tmp = t_1
	elif y <= -1.95e-32:
		tmp = t_2
	elif y <= 7.3e-56:
		tmp = (x_m / t) * (-2.0 / z_m)
	elif y <= 5.8e-8:
		tmp = t_2
	elif y <= 7.8e+78:
		tmp = t_1
	else:
		tmp = (2.0 / z_m) * (x_m / y)
	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(t * Float64(z_m / x_m)))
	t_2 = Float64(Float64(x_m / z_m) * Float64(2.0 / y))
	tmp = 0.0
	if (y <= -2.5e+100)
		tmp = t_2;
	elseif (y <= -1.05e+52)
		tmp = t_1;
	elseif (y <= -1.95e-32)
		tmp = t_2;
	elseif (y <= 7.3e-56)
		tmp = Float64(Float64(x_m / t) * Float64(-2.0 / z_m));
	elseif (y <= 5.8e-8)
		tmp = t_2;
	elseif (y <= 7.8e+78)
		tmp = t_1;
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	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 / (t * (z_m / x_m));
	t_2 = (x_m / z_m) * (2.0 / y);
	tmp = 0.0;
	if (y <= -2.5e+100)
		tmp = t_2;
	elseif (y <= -1.05e+52)
		tmp = t_1;
	elseif (y <= -1.95e-32)
		tmp = t_2;
	elseif (y <= 7.3e-56)
		tmp = (x_m / t) * (-2.0 / z_m);
	elseif (y <= 5.8e-8)
		tmp = t_2;
	elseif (y <= 7.8e+78)
		tmp = t_1;
	else
		tmp = (2.0 / z_m) * (x_m / y);
	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[(t * N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -2.5e+100], t$95$2, If[LessEqual[y, -1.05e+52], t$95$1, If[LessEqual[y, -1.95e-32], t$95$2, If[LessEqual[y, 7.3e-56], N[(N[(x$95$m / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.8e-8], t$95$2, If[LessEqual[y, 7.8e+78], t$95$1, N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.4999999999999999e100 or -1.05e52 < y < -1.9500000000000001e-32 or 7.30000000000000045e-56 < y < 5.8000000000000003e-8

   1. Initial program 85.9%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. add-sqr-sqrt 47.3%

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

      2. *-commutative 47.3%

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

      3. times-frac 51.0%

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

   6. Applied egg-rr 51.0%

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

   7. Taylor expanded in y around inf 79.5%

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

      1. associate-/r* 81.6%

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

      2. unpow2 81.6%

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

      3. rem-square-sqrt 82.3%

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

      4. associate-/l* 82.2%

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

      5. associate-*l/ 87.8%

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

   9. Simplified 87.8%

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

   if -2.4999999999999999e100 < y < -1.05e52 or 5.8000000000000003e-8 < y < 7.8000000000000008e78

   1. Initial program 84.9%

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

      1. distribute-rgt-out-- 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in y around 0 70.4%

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

      1. *-commutative 70.4%

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

   7. Simplified 70.4%

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

      1. clear-num 70.4%

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

      2. un-div-inv 70.4%

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

      3. *-commutative 70.4%

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

      4. associate-/l* 81.0%

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

   9. Applied egg-rr 81.0%

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

   if -1.9500000000000001e-32 < y < 7.30000000000000045e-56

   1. Initial program 94.1%

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

      1. distribute-rgt-out-- 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in y around 0 80.8%

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

      1. *-commutative 80.8%

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

   7. Simplified 80.8%

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

      1. clear-num 79.9%

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

      2. un-div-inv 79.9%

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

      3. *-commutative 79.9%

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

      4. associate-/l* 80.3%

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

   9. Applied egg-rr 80.3%

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

       1. associate-/r* 80.9%

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

       2. div-inv 80.8%

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

       3. clear-num 81.0%

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

       4. times-frac 80.8%

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

       5. *-commutative 80.8%

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

       6. times-frac 81.4%

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

   11. Applied egg-rr 81.4%

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

   if 7.8000000000000008e78 < y

   1. Initial program 85.0%

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

      1. distribute-rgt-out-- 85.3%

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

   3. Simplified 85.3%

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

      1. *-commutative 85.3%

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

      2. times-frac 89.8%

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

   6. Applied egg-rr 89.8%

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

   7. Taylor expanded in y around inf 77.6%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+100}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{+52}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;y \leq -1.95 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 7.3 \cdot 10^{-56}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{t} \cdot \frac{-2}{z\_m}\\ t_2 := \frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{+101}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{+51}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{-32}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.057:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x_m t) (/ -2.0 z_m))) (t_2 (* (/ x_m z_m) (/ 2.0 y))))
   (*
    z_s
    (*
     x_s
     (if (<= y -2.75e+101)
       t_2
       (if (<= y -9.8e+51)
         (* -2.0 (/ x_m (* z_m t)))
         (if (<= y -7.6e-32)
           t_2
           (if (<= y 3.25e-55)
             t_1
             (if (<= y 0.057)
               t_2
               (if (<= y 7.8e+78) t_1 (* (/ 2.0 z_m) (/ x_m y))))))))))))

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 / t) * (-2.0 / z_m);
	double t_2 = (x_m / z_m) * (2.0 / y);
	double tmp;
	if (y <= -2.75e+101) {
		tmp = t_2;
	} else if (y <= -9.8e+51) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -7.6e-32) {
		tmp = t_2;
	} else if (y <= 3.25e-55) {
		tmp = t_1;
	} else if (y <= 0.057) {
		tmp = t_2;
	} else if (y <= 7.8e+78) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	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 / t) * ((-2.0d0) / z_m)
    t_2 = (x_m / z_m) * (2.0d0 / y)
    if (y <= (-2.75d+101)) then
        tmp = t_2
    else if (y <= (-9.8d+51)) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else if (y <= (-7.6d-32)) then
        tmp = t_2
    else if (y <= 3.25d-55) then
        tmp = t_1
    else if (y <= 0.057d0) then
        tmp = t_2
    else if (y <= 7.8d+78) then
        tmp = t_1
    else
        tmp = (2.0d0 / z_m) * (x_m / y)
    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 / t) * (-2.0 / z_m);
	double t_2 = (x_m / z_m) * (2.0 / y);
	double tmp;
	if (y <= -2.75e+101) {
		tmp = t_2;
	} else if (y <= -9.8e+51) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -7.6e-32) {
		tmp = t_2;
	} else if (y <= 3.25e-55) {
		tmp = t_1;
	} else if (y <= 0.057) {
		tmp = t_2;
	} else if (y <= 7.8e+78) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	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 / t) * (-2.0 / z_m)
	t_2 = (x_m / z_m) * (2.0 / y)
	tmp = 0
	if y <= -2.75e+101:
		tmp = t_2
	elif y <= -9.8e+51:
		tmp = -2.0 * (x_m / (z_m * t))
	elif y <= -7.6e-32:
		tmp = t_2
	elif y <= 3.25e-55:
		tmp = t_1
	elif y <= 0.057:
		tmp = t_2
	elif y <= 7.8e+78:
		tmp = t_1
	else:
		tmp = (2.0 / z_m) * (x_m / y)
	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 / t) * Float64(-2.0 / z_m))
	t_2 = Float64(Float64(x_m / z_m) * Float64(2.0 / y))
	tmp = 0.0
	if (y <= -2.75e+101)
		tmp = t_2;
	elseif (y <= -9.8e+51)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	elseif (y <= -7.6e-32)
		tmp = t_2;
	elseif (y <= 3.25e-55)
		tmp = t_1;
	elseif (y <= 0.057)
		tmp = t_2;
	elseif (y <= 7.8e+78)
		tmp = t_1;
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	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 / t) * (-2.0 / z_m);
	t_2 = (x_m / z_m) * (2.0 / y);
	tmp = 0.0;
	if (y <= -2.75e+101)
		tmp = t_2;
	elseif (y <= -9.8e+51)
		tmp = -2.0 * (x_m / (z_m * t));
	elseif (y <= -7.6e-32)
		tmp = t_2;
	elseif (y <= 3.25e-55)
		tmp = t_1;
	elseif (y <= 0.057)
		tmp = t_2;
	elseif (y <= 7.8e+78)
		tmp = t_1;
	else
		tmp = (2.0 / z_m) * (x_m / y);
	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 / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -2.75e+101], t$95$2, If[LessEqual[y, -9.8e+51], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -7.6e-32], t$95$2, If[LessEqual[y, 3.25e-55], t$95$1, If[LessEqual[y, 0.057], t$95$2, If[LessEqual[y, 7.8e+78], t$95$1, N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.75000000000000009e101 or -9.79999999999999967e51 < y < -7.60000000000000015e-32 or 3.25000000000000003e-55 < y < 0.0570000000000000021

   1. Initial program 85.9%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. add-sqr-sqrt 47.3%

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

      2. *-commutative 47.3%

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

      3. times-frac 51.0%

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

   6. Applied egg-rr 51.0%

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

   7. Taylor expanded in y around inf 79.5%

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

      1. associate-/r* 81.6%

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

      2. unpow2 81.6%

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

      3. rem-square-sqrt 82.3%

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

      4. associate-/l* 82.2%

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

      5. associate-*l/ 87.8%

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

   9. Simplified 87.8%

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

   if -2.75000000000000009e101 < y < -9.79999999999999967e51

   1. Initial program 91.2%

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

      1. distribute-rgt-out-- 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in y around 0 74.1%

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

      1. *-commutative 74.1%

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

   7. Simplified 74.1%

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

   if -7.60000000000000015e-32 < y < 3.25000000000000003e-55 or 0.0570000000000000021 < y < 7.8000000000000008e78

   1. Initial program 92.4%

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

      1. distribute-rgt-out-- 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around 0 79.2%

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

      1. *-commutative 79.2%

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

   7. Simplified 79.2%

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

      1. clear-num 78.4%

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

      2. un-div-inv 78.4%

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

      3. *-commutative 78.4%

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

      4. associate-/l* 80.3%

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

   9. Applied egg-rr 80.3%

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

       1. associate-/r* 80.8%

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

       2. div-inv 80.7%

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

       3. clear-num 80.8%

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

       4. times-frac 79.2%

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

       5. *-commutative 79.2%

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

       6. times-frac 79.8%

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

   11. Applied egg-rr 79.8%

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

   if 7.8000000000000008e78 < y

   1. Initial program 85.0%

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

      1. distribute-rgt-out-- 85.3%

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

   3. Simplified 85.3%

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

      1. *-commutative 85.3%

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

      2. times-frac 89.8%

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

   6. Applied egg-rr 89.8%

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

   7. Taylor expanded in y around inf 77.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.75 \cdot 10^{+101}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{+51}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq 0.057:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{t} \cdot \frac{-2}{z\_m}\\ t_2 := \frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ t_3 := x\_m \cdot \frac{2}{y \cdot z\_m}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+99}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{+57}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-34}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (* (/ x_m t) (/ -2.0 z_m)))
        (t_2 (* (/ 2.0 z_m) (/ x_m y)))
        (t_3 (* x_m (/ 2.0 (* y z_m)))))
   (*
    z_s
    (*
     x_s
     (if (<= y -4.8e+99)
       t_2
       (if (<= y -7.6e+57)
         (* -2.0 (/ x_m (* z_m t)))
         (if (<= y -1.55e-34)
           t_3
           (if (<= y 4.2e-55)
             t_1
             (if (<= y 1.25e-5) t_3 (if (<= y 7.8e+78) 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 / t) * (-2.0 / z_m);
	double t_2 = (2.0 / z_m) * (x_m / y);
	double t_3 = x_m * (2.0 / (y * z_m));
	double tmp;
	if (y <= -4.8e+99) {
		tmp = t_2;
	} else if (y <= -7.6e+57) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -1.55e-34) {
		tmp = t_3;
	} else if (y <= 4.2e-55) {
		tmp = t_1;
	} else if (y <= 1.25e-5) {
		tmp = t_3;
	} else if (y <= 7.8e+78) {
		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) :: t_3
    real(8) :: tmp
    t_1 = (x_m / t) * ((-2.0d0) / z_m)
    t_2 = (2.0d0 / z_m) * (x_m / y)
    t_3 = x_m * (2.0d0 / (y * z_m))
    if (y <= (-4.8d+99)) then
        tmp = t_2
    else if (y <= (-7.6d+57)) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else if (y <= (-1.55d-34)) then
        tmp = t_3
    else if (y <= 4.2d-55) then
        tmp = t_1
    else if (y <= 1.25d-5) then
        tmp = t_3
    else if (y <= 7.8d+78) 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 / t) * (-2.0 / z_m);
	double t_2 = (2.0 / z_m) * (x_m / y);
	double t_3 = x_m * (2.0 / (y * z_m));
	double tmp;
	if (y <= -4.8e+99) {
		tmp = t_2;
	} else if (y <= -7.6e+57) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (y <= -1.55e-34) {
		tmp = t_3;
	} else if (y <= 4.2e-55) {
		tmp = t_1;
	} else if (y <= 1.25e-5) {
		tmp = t_3;
	} else if (y <= 7.8e+78) {
		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 / t) * (-2.0 / z_m)
	t_2 = (2.0 / z_m) * (x_m / y)
	t_3 = x_m * (2.0 / (y * z_m))
	tmp = 0
	if y <= -4.8e+99:
		tmp = t_2
	elif y <= -7.6e+57:
		tmp = -2.0 * (x_m / (z_m * t))
	elif y <= -1.55e-34:
		tmp = t_3
	elif y <= 4.2e-55:
		tmp = t_1
	elif y <= 1.25e-5:
		tmp = t_3
	elif y <= 7.8e+78:
		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 / t) * Float64(-2.0 / z_m))
	t_2 = Float64(Float64(2.0 / z_m) * Float64(x_m / y))
	t_3 = Float64(x_m * Float64(2.0 / Float64(y * z_m)))
	tmp = 0.0
	if (y <= -4.8e+99)
		tmp = t_2;
	elseif (y <= -7.6e+57)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	elseif (y <= -1.55e-34)
		tmp = t_3;
	elseif (y <= 4.2e-55)
		tmp = t_1;
	elseif (y <= 1.25e-5)
		tmp = t_3;
	elseif (y <= 7.8e+78)
		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 / t) * (-2.0 / z_m);
	t_2 = (2.0 / z_m) * (x_m / y);
	t_3 = x_m * (2.0 / (y * z_m));
	tmp = 0.0;
	if (y <= -4.8e+99)
		tmp = t_2;
	elseif (y <= -7.6e+57)
		tmp = -2.0 * (x_m / (z_m * t));
	elseif (y <= -1.55e-34)
		tmp = t_3;
	elseif (y <= 4.2e-55)
		tmp = t_1;
	elseif (y <= 1.25e-5)
		tmp = t_3;
	elseif (y <= 7.8e+78)
		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 / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x$95$m * N[(2.0 / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -4.8e+99], t$95$2, If[LessEqual[y, -7.6e+57], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.55e-34], t$95$3, If[LessEqual[y, 4.2e-55], t$95$1, If[LessEqual[y, 1.25e-5], t$95$3, If[LessEqual[y, 7.8e+78], t$95$1, t$95$2]]]]]]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.8000000000000002e99 or 7.8000000000000008e78 < y

   1. Initial program 82.7%

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

      1. distribute-rgt-out-- 83.9%

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

   3. Simplified 83.9%

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

      1. *-commutative 83.9%

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

      2. times-frac 90.6%

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

   6. Applied egg-rr 90.6%

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

   7. Taylor expanded in y around inf 79.5%

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

   if -4.8000000000000002e99 < y < -7.5999999999999997e57

   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. Taylor expanded in y around 0 80.8%

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

      1. *-commutative 80.8%

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

   7. Simplified 80.8%

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

   if -7.5999999999999997e57 < y < -1.5499999999999999e-34 or 4.2000000000000003e-55 < y < 1.25000000000000006e-5

   1. Initial program 92.7%

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

      1. distribute-rgt-out-- 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in y around inf 85.1%

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

      1. *-commutative 85.1%

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

   7. Simplified 85.1%

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

      1. associate-/l* 85.0%

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

      2. *-commutative 85.0%

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

      3. *-commutative 85.0%

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

   9. Applied egg-rr 85.0%

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

   if -1.5499999999999999e-34 < y < 4.2000000000000003e-55 or 1.25000000000000006e-5 < y < 7.8000000000000008e78

   1. Initial program 92.4%

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

      1. distribute-rgt-out-- 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around 0 79.2%

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

      1. *-commutative 79.2%

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

   7. Simplified 79.2%

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

      1. clear-num 78.4%

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

      2. un-div-inv 78.4%

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

      3. *-commutative 78.4%

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

      4. associate-/l* 80.3%

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

   9. Applied egg-rr 80.3%

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

       1. associate-/r* 80.8%

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

       2. div-inv 80.7%

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

       3. clear-num 80.8%

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

       4. times-frac 79.2%

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

       5. *-commutative 79.2%

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

       6. times-frac 79.8%

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

   11. Applied egg-rr 79.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq -7.6 \cdot 10^{+57}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{-34}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{+85} \lor \neg \left(t \leq -2.6 \cdot 10^{-95}\right) \land \left(t \leq -1.22 \cdot 10^{-126} \lor \neg \left(t \leq 1.05 \cdot 10^{+75}\right)\right):\\ \;\;\;\;\frac{x\_m}{t} \cdot \frac{-2}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{z\_m}}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (or (<= t -2.45e+85)
           (and (not (<= t -2.6e-95))
                (or (<= t -1.22e-126) (not (<= t 1.05e+75)))))
     (* (/ x_m t) (/ -2.0 z_m))
     (* x_m (/ (/ 2.0 z_m) y))))))

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 ((t <= -2.45e+85) || (!(t <= -2.6e-95) && ((t <= -1.22e-126) || !(t <= 1.05e+75)))) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else {
		tmp = x_m * ((2.0 / z_m) / y);
	}
	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 ((t <= (-2.45d+85)) .or. (.not. (t <= (-2.6d-95))) .and. (t <= (-1.22d-126)) .or. (.not. (t <= 1.05d+75))) then
        tmp = (x_m / t) * ((-2.0d0) / z_m)
    else
        tmp = x_m * ((2.0d0 / z_m) / y)
    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 ((t <= -2.45e+85) || (!(t <= -2.6e-95) && ((t <= -1.22e-126) || !(t <= 1.05e+75)))) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else {
		tmp = x_m * ((2.0 / z_m) / y);
	}
	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 (t <= -2.45e+85) or (not (t <= -2.6e-95) and ((t <= -1.22e-126) or not (t <= 1.05e+75))):
		tmp = (x_m / t) * (-2.0 / z_m)
	else:
		tmp = x_m * ((2.0 / z_m) / y)
	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 ((t <= -2.45e+85) || (!(t <= -2.6e-95) && ((t <= -1.22e-126) || !(t <= 1.05e+75))))
		tmp = Float64(Float64(x_m / t) * Float64(-2.0 / z_m));
	else
		tmp = Float64(x_m * Float64(Float64(2.0 / z_m) / y));
	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 ((t <= -2.45e+85) || (~((t <= -2.6e-95)) && ((t <= -1.22e-126) || ~((t <= 1.05e+75)))))
		tmp = (x_m / t) * (-2.0 / z_m);
	else
		tmp = x_m * ((2.0 / z_m) / y);
	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[Or[LessEqual[t, -2.45e+85], And[N[Not[LessEqual[t, -2.6e-95]], $MachinePrecision], Or[LessEqual[t, -1.22e-126], N[Not[LessEqual[t, 1.05e+75]], $MachinePrecision]]]], N[(N[(x$95$m / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(2.0 / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -2.4499999999999998e85 or -2.60000000000000001e-95 < t < -1.21999999999999996e-126 or 1.04999999999999999e75 < t

   1. Initial program 85.9%

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

      1. distribute-rgt-out-- 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in y around 0 81.2%

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

      1. *-commutative 81.2%

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

   7. Simplified 81.2%

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

      1. clear-num 79.8%

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

      2. un-div-inv 79.8%

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

      3. *-commutative 79.8%

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

      4. associate-/l* 85.3%

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

   9. Applied egg-rr 85.3%

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

       1. associate-/r* 86.7%

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

       2. div-inv 86.6%

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

       3. clear-num 86.6%

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

       4. times-frac 81.2%

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

       5. *-commutative 81.2%

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

       6. times-frac 84.9%

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

   11. Applied egg-rr 84.9%

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

   if -2.4499999999999998e85 < t < -2.60000000000000001e-95 or -1.21999999999999996e-126 < t < 1.04999999999999999e75

   1. Initial program 91.4%

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

      1. distribute-rgt-out-- 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in x around 0 92.1%

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

      1. associate-/r* 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in y around inf 70.6%

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

      1. associate-*r/ 70.6%

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

      2. *-commutative 70.6%

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

      3. associate-*r/ 70.5%

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

      4. associate-/l/ 70.5%

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

   10. Simplified 70.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{+85} \lor \neg \left(t \leq -2.6 \cdot 10^{-95}\right) \land \left(t \leq -1.22 \cdot 10^{-126} \lor \neg \left(t \leq 1.05 \cdot 10^{+75}\right)\right):\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+99} \lor \neg \left(y \leq -1.05 \cdot 10^{+52}\right) \land \left(y \leq -1.9 \cdot 10^{-32} \lor \neg \left(y \leq 1.5 \cdot 10^{-55}\right)\right):\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (or (<= y -4.9e+99)
           (and (not (<= y -1.05e+52))
                (or (<= y -1.9e-32) (not (<= y 1.5e-55)))))
     (* x_m (/ (/ 2.0 z_m) y))
     (* -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) {
	double tmp;
	if ((y <= -4.9e+99) || (!(y <= -1.05e+52) && ((y <= -1.9e-32) || !(y <= 1.5e-55)))) {
		tmp = x_m * ((2.0 / z_m) / y);
	} else {
		tmp = -2.0 * (x_m / (z_m * t));
	}
	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 ((y <= (-4.9d+99)) .or. (.not. (y <= (-1.05d+52))) .and. (y <= (-1.9d-32)) .or. (.not. (y <= 1.5d-55))) then
        tmp = x_m * ((2.0d0 / z_m) / y)
    else
        tmp = (-2.0d0) * (x_m / (z_m * t))
    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 ((y <= -4.9e+99) || (!(y <= -1.05e+52) && ((y <= -1.9e-32) || !(y <= 1.5e-55)))) {
		tmp = x_m * ((2.0 / z_m) / y);
	} else {
		tmp = -2.0 * (x_m / (z_m * t));
	}
	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 (y <= -4.9e+99) or (not (y <= -1.05e+52) and ((y <= -1.9e-32) or not (y <= 1.5e-55))):
		tmp = x_m * ((2.0 / z_m) / y)
	else:
		tmp = -2.0 * (x_m / (z_m * t))
	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 ((y <= -4.9e+99) || (!(y <= -1.05e+52) && ((y <= -1.9e-32) || !(y <= 1.5e-55))))
		tmp = Float64(x_m * Float64(Float64(2.0 / z_m) / y));
	else
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	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 ((y <= -4.9e+99) || (~((y <= -1.05e+52)) && ((y <= -1.9e-32) || ~((y <= 1.5e-55)))))
		tmp = x_m * ((2.0 / z_m) / y);
	else
		tmp = -2.0 * (x_m / (z_m * t));
	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[Or[LessEqual[y, -4.9e+99], And[N[Not[LessEqual[y, -1.05e+52]], $MachinePrecision], Or[LessEqual[y, -1.9e-32], N[Not[LessEqual[y, 1.5e-55]], $MachinePrecision]]]], N[(x$95$m * N[(N[(2.0 / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -4.8999999999999997e99 or -1.05e52 < y < -1.90000000000000004e-32 or 1.50000000000000008e-55 < y

   1. Initial program 84.9%

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

      1. distribute-rgt-out-- 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in x around 0 86.5%

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

      1. associate-/r* 95.4%

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

   7. Simplified 95.4%

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

   8. Taylor expanded in y around inf 72.3%

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

      1. associate-*r/ 72.3%

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

      2. *-commutative 72.3%

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

      3. associate-*r/ 72.2%

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

      4. associate-/l/ 72.3%

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

   10. Simplified 72.3%

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

   if -4.8999999999999997e99 < y < -1.05e52 or -1.90000000000000004e-32 < y < 1.50000000000000008e-55

   1. Initial program 93.8%

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

      1. distribute-rgt-out-- 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in y around 0 80.2%

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

      1. *-commutative 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.9 \cdot 10^{+99} \lor \neg \left(y \leq -1.05 \cdot 10^{+52}\right) \land \left(y \leq -1.9 \cdot 10^{-32} \lor \neg \left(y \leq 1.5 \cdot 10^{-55}\right)\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{\frac{x\_m \cdot -2}{z\_m}}{t}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-32}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ (/ (* x_m -2.0) z_m) t)))
   (*
    z_s
    (*
     x_s
     (if (<= y -4.8e+99)
       (* (/ x_m z_m) (/ 2.0 y))
       (if (<= y -5.2e+51)
         t_1
         (if (<= y -9e-32)
           (/ (* x_m 2.0) (* y z_m))
           (if (<= y 7.8e+78) t_1 (* (/ 2.0 z_m) (/ x_m y))))))))))

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) / z_m) / t;
	double tmp;
	if (y <= -4.8e+99) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else if (y <= -5.2e+51) {
		tmp = t_1;
	} else if (y <= -9e-32) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 7.8e+78) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	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)) / z_m) / t
    if (y <= (-4.8d+99)) then
        tmp = (x_m / z_m) * (2.0d0 / y)
    else if (y <= (-5.2d+51)) then
        tmp = t_1
    else if (y <= (-9d-32)) then
        tmp = (x_m * 2.0d0) / (y * z_m)
    else if (y <= 7.8d+78) then
        tmp = t_1
    else
        tmp = (2.0d0 / z_m) * (x_m / y)
    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) / z_m) / t;
	double tmp;
	if (y <= -4.8e+99) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else if (y <= -5.2e+51) {
		tmp = t_1;
	} else if (y <= -9e-32) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 7.8e+78) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	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) / z_m) / t
	tmp = 0
	if y <= -4.8e+99:
		tmp = (x_m / z_m) * (2.0 / y)
	elif y <= -5.2e+51:
		tmp = t_1
	elif y <= -9e-32:
		tmp = (x_m * 2.0) / (y * z_m)
	elif y <= 7.8e+78:
		tmp = t_1
	else:
		tmp = (2.0 / z_m) * (x_m / y)
	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(Float64(x_m * -2.0) / z_m) / t)
	tmp = 0.0
	if (y <= -4.8e+99)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / y));
	elseif (y <= -5.2e+51)
		tmp = t_1;
	elseif (y <= -9e-32)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	elseif (y <= 7.8e+78)
		tmp = t_1;
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	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) / z_m) / t;
	tmp = 0.0;
	if (y <= -4.8e+99)
		tmp = (x_m / z_m) * (2.0 / y);
	elseif (y <= -5.2e+51)
		tmp = t_1;
	elseif (y <= -9e-32)
		tmp = (x_m * 2.0) / (y * z_m);
	elseif (y <= 7.8e+78)
		tmp = t_1;
	else
		tmp = (2.0 / z_m) * (x_m / y);
	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[(N[(x$95$m * -2.0), $MachinePrecision] / z$95$m), $MachinePrecision] / t), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -4.8e+99], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.2e+51], t$95$1, If[LessEqual[y, -9e-32], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.8e+78], t$95$1, N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -4.8000000000000002e99

   1. Initial program 80.6%

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

      1. distribute-rgt-out-- 82.6%

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

   3. Simplified 82.6%

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

      1. add-sqr-sqrt 40.8%

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

      2. *-commutative 40.8%

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

      3. times-frac 44.6%

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

   6. Applied egg-rr 44.6%

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

   7. Taylor expanded in y around inf 75.4%

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

      1. associate-/r* 80.5%

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

      2. unpow2 80.5%

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

      3. rem-square-sqrt 81.3%

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

      4. associate-/l* 81.2%

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

      5. associate-*l/ 87.6%

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

   9. Simplified 87.6%

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

   if -4.8000000000000002e99 < y < -5.2000000000000002e51 or -9.00000000000000009e-32 < y < 7.8000000000000008e78

   1. Initial program 92.2%

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

      1. distribute-rgt-out-- 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in y around 0 74.8%

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

      1. *-commutative 74.8%

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

   7. Simplified 74.8%

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

      1. *-commutative 74.8%

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

      2. associate-*l/ 74.8%

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

      3. metadata-eval 74.8%

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

      4. distribute-rgt-neg-in 74.8%

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

      5. associate-/r* 77.5%

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

      6. distribute-rgt-neg-in 77.5%

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

      7. metadata-eval 77.5%

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

   9. Applied egg-rr 77.5%

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

   if -5.2000000000000002e51 < y < -9.00000000000000009e-32

   1. Initial program 99.8%

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

      1. distribute-rgt-out-- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 87.7%

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

      1. *-commutative 87.7%

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

   7. Simplified 87.7%

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

   if 7.8000000000000008e78 < y

   1. Initial program 85.0%

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

      1. distribute-rgt-out-- 85.3%

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

   3. Simplified 85.3%

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

      1. *-commutative 85.3%

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

      2. times-frac 89.8%

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

   6. Applied egg-rr 89.8%

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

   7. Taylor expanded in y around inf 77.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{\frac{-2}{\frac{z\_m}{x\_m}}}{t}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-32}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ (/ -2.0 (/ z_m x_m)) t)))
   (*
    z_s
    (*
     x_s
     (if (<= y -7.4e+99)
       (* (/ x_m z_m) (/ 2.0 y))
       (if (<= y -2.6e+51)
         t_1
         (if (<= y -2.1e-32)
           (/ (* x_m 2.0) (* y z_m))
           (if (<= y 6.5e+80) t_1 (* (/ 2.0 z_m) (/ x_m y))))))))))

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 / (z_m / x_m)) / t;
	double tmp;
	if (y <= -7.4e+99) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else if (y <= -2.6e+51) {
		tmp = t_1;
	} else if (y <= -2.1e-32) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 6.5e+80) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	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) / (z_m / x_m)) / t
    if (y <= (-7.4d+99)) then
        tmp = (x_m / z_m) * (2.0d0 / y)
    else if (y <= (-2.6d+51)) then
        tmp = t_1
    else if (y <= (-2.1d-32)) then
        tmp = (x_m * 2.0d0) / (y * z_m)
    else if (y <= 6.5d+80) then
        tmp = t_1
    else
        tmp = (2.0d0 / z_m) * (x_m / y)
    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 / (z_m / x_m)) / t;
	double tmp;
	if (y <= -7.4e+99) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else if (y <= -2.6e+51) {
		tmp = t_1;
	} else if (y <= -2.1e-32) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 6.5e+80) {
		tmp = t_1;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	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 / (z_m / x_m)) / t
	tmp = 0
	if y <= -7.4e+99:
		tmp = (x_m / z_m) * (2.0 / y)
	elif y <= -2.6e+51:
		tmp = t_1
	elif y <= -2.1e-32:
		tmp = (x_m * 2.0) / (y * z_m)
	elif y <= 6.5e+80:
		tmp = t_1
	else:
		tmp = (2.0 / z_m) * (x_m / y)
	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(-2.0 / Float64(z_m / x_m)) / t)
	tmp = 0.0
	if (y <= -7.4e+99)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / y));
	elseif (y <= -2.6e+51)
		tmp = t_1;
	elseif (y <= -2.1e-32)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	elseif (y <= 6.5e+80)
		tmp = t_1;
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	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 / (z_m / x_m)) / t;
	tmp = 0.0;
	if (y <= -7.4e+99)
		tmp = (x_m / z_m) * (2.0 / y);
	elseif (y <= -2.6e+51)
		tmp = t_1;
	elseif (y <= -2.1e-32)
		tmp = (x_m * 2.0) / (y * z_m);
	elseif (y <= 6.5e+80)
		tmp = t_1;
	else
		tmp = (2.0 / z_m) * (x_m / y);
	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[(-2.0 / N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -7.4e+99], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.6e+51], t$95$1, If[LessEqual[y, -2.1e-32], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+80], t$95$1, N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -7.4000000000000002e99

   1. Initial program 80.6%

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

      1. distribute-rgt-out-- 82.6%

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

   3. Simplified 82.6%

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

      1. add-sqr-sqrt 40.8%

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

      2. *-commutative 40.8%

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

      3. times-frac 44.6%

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

   6. Applied egg-rr 44.6%

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

   7. Taylor expanded in y around inf 75.4%

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

      1. associate-/r* 80.5%

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

      2. unpow2 80.5%

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

      3. rem-square-sqrt 81.3%

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

      4. associate-/l* 81.2%

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

      5. associate-*l/ 87.6%

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

   9. Simplified 87.6%

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

   if -7.4000000000000002e99 < y < -2.6000000000000001e51 or -2.0999999999999999e-32 < y < 6.4999999999999998e80

   1. Initial program 92.2%

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

      1. distribute-rgt-out-- 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in y around 0 74.8%

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

      1. associate-*r/ 74.8%

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

      2. *-commutative 74.8%

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

      3. *-commutative 74.8%

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

      4. associate-/l* 74.4%

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

   7. Simplified 74.4%

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

      1. *-commutative 74.4%

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

      2. associate-/r/ 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*r/ 76.9%

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

      5. *-commutative 76.9%

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

      6. associate-/r* 77.4%

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

   9. Applied egg-rr 77.4%

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

   if -2.6000000000000001e51 < y < -2.0999999999999999e-32

   1. Initial program 99.8%

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

      1. distribute-rgt-out-- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 87.7%

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

      1. *-commutative 87.7%

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

   7. Simplified 87.7%

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

   if 6.4999999999999998e80 < y

   1. Initial program 85.0%

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

      1. distribute-rgt-out-- 85.3%

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

   3. Simplified 85.3%

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

      1. *-commutative 85.3%

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

      2. times-frac 89.8%

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

   6. Applied egg-rr 89.8%

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

   7. Taylor expanded in y around inf 77.6%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{-2}{\frac{z}{x}}}{t}\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{-2}{\frac{z}{x}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := -2 \cdot \frac{x\_m}{z\_m \cdot t}\\ t_2 := x\_m \cdot \frac{\frac{2}{z\_m}}{y}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -8.3 \cdot 10^{+99}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{+58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-32}:\\ \;\;\;\;x\_m \cdot \frac{2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) 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 (* x_m (/ (/ 2.0 z_m) y))))
   (*
    z_s
    (*
     x_s
     (if (<= y -8.3e+99)
       t_2
       (if (<= y -9.2e+58)
         t_1
         (if (<= y -1.1e-32)
           (* x_m (/ 2.0 (* y z_m)))
           (if (<= y 4.2e-55) 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 = -2.0 * (x_m / (z_m * t));
	double t_2 = x_m * ((2.0 / z_m) / y);
	double tmp;
	if (y <= -8.3e+99) {
		tmp = t_2;
	} else if (y <= -9.2e+58) {
		tmp = t_1;
	} else if (y <= -1.1e-32) {
		tmp = x_m * (2.0 / (y * z_m));
	} else if (y <= 4.2e-55) {
		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 = (-2.0d0) * (x_m / (z_m * t))
    t_2 = x_m * ((2.0d0 / z_m) / y)
    if (y <= (-8.3d+99)) then
        tmp = t_2
    else if (y <= (-9.2d+58)) then
        tmp = t_1
    else if (y <= (-1.1d-32)) then
        tmp = x_m * (2.0d0 / (y * z_m))
    else if (y <= 4.2d-55) 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 = -2.0 * (x_m / (z_m * t));
	double t_2 = x_m * ((2.0 / z_m) / y);
	double tmp;
	if (y <= -8.3e+99) {
		tmp = t_2;
	} else if (y <= -9.2e+58) {
		tmp = t_1;
	} else if (y <= -1.1e-32) {
		tmp = x_m * (2.0 / (y * z_m));
	} else if (y <= 4.2e-55) {
		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 = -2.0 * (x_m / (z_m * t))
	t_2 = x_m * ((2.0 / z_m) / y)
	tmp = 0
	if y <= -8.3e+99:
		tmp = t_2
	elif y <= -9.2e+58:
		tmp = t_1
	elif y <= -1.1e-32:
		tmp = x_m * (2.0 / (y * z_m))
	elif y <= 4.2e-55:
		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(-2.0 * Float64(x_m / Float64(z_m * t)))
	t_2 = Float64(x_m * Float64(Float64(2.0 / z_m) / y))
	tmp = 0.0
	if (y <= -8.3e+99)
		tmp = t_2;
	elseif (y <= -9.2e+58)
		tmp = t_1;
	elseif (y <= -1.1e-32)
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z_m)));
	elseif (y <= 4.2e-55)
		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 = -2.0 * (x_m / (z_m * t));
	t_2 = x_m * ((2.0 / z_m) / y);
	tmp = 0.0;
	if (y <= -8.3e+99)
		tmp = t_2;
	elseif (y <= -9.2e+58)
		tmp = t_1;
	elseif (y <= -1.1e-32)
		tmp = x_m * (2.0 / (y * z_m));
	elseif (y <= 4.2e-55)
		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[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$95$m * N[(N[(2.0 / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[y, -8.3e+99], t$95$2, If[LessEqual[y, -9.2e+58], t$95$1, If[LessEqual[y, -1.1e-32], N[(x$95$m * N[(2.0 / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e-55], t$95$1, t$95$2]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.3e99 or 4.2000000000000003e-55 < y

   1. Initial program 82.9%

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

      1. distribute-rgt-out-- 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 84.7%

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

      1. associate-/r* 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in y around inf 70.3%

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

      1. associate-*r/ 70.3%

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

      2. *-commutative 70.3%

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

      3. associate-*r/ 70.2%

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

      4. associate-/l/ 70.2%

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

   10. Simplified 70.2%

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

   if -8.3e99 < y < -9.2000000000000001e58 or -1.1e-32 < y < 4.2000000000000003e-55

   1. Initial program 94.5%

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

      1. distribute-rgt-out-- 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in y around 0 80.8%

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

      1. *-commutative 80.8%

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

   7. Simplified 80.8%

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

   if -9.2000000000000001e58 < y < -1.1e-32

   1. Initial program 94.3%

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

      1. distribute-rgt-out-- 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in y around inf 82.9%

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

      1. *-commutative 82.9%

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

   7. Simplified 82.9%

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

      1. associate-/l* 82.8%

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

      2. *-commutative 82.8%

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

      3. *-commutative 82.8%

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

   9. Applied egg-rr 82.8%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.3 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{+58}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-55}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.6% accurate, 0.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 5e-88)
     (/ (* x_m 2.0) (* z_m (- y t)))
     (/ (/ (* 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) {
	double tmp;
	if ((x_m * 2.0) <= 5e-88) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = ((x_m * 2.0) / (y - 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 ((x_m * 2.0d0) <= 5d-88) then
        tmp = (x_m * 2.0d0) / (z_m * (y - t))
    else
        tmp = ((x_m * 2.0d0) / (y - 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 ((x_m * 2.0) <= 5e-88) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = ((x_m * 2.0) / (y - 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 (x_m * 2.0) <= 5e-88:
		tmp = (x_m * 2.0) / (z_m * (y - t))
	else:
		tmp = ((x_m * 2.0) / (y - 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 (Float64(x_m * 2.0) <= 5e-88)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(Float64(x_m * 2.0) / Float64(y - 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 ((x_m * 2.0) <= 5e-88)
		tmp = (x_m * 2.0) / (z_m * (y - t));
	else
		tmp = ((x_m * 2.0) / (y - 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[N[(x$95$m * 2.0), $MachinePrecision], 5e-88], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.6%

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

      1. distribute-rgt-out-- 89.8%

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

   3. Simplified 89.8%

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

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

   1. Initial program 90.1%

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

      1. distribute-rgt-out-- 91.5%

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

   3. Simplified 91.5%

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

      1. add-sqr-sqrt 91.2%

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

      2. *-commutative 91.2%

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

      3. times-frac 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. associate-*r/ 98.3%

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

      2. associate-*l/ 98.4%

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

      3. add-sqr-sqrt 98.6%

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

   8. Applied egg-rr 98.6%

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

Alternative 12: 96.6% accurate, 0.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 5e-88)
     (/ (* x_m 2.0) (* z_m (- 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-88) {
		tmp = (x_m * 2.0) / (z_m * (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-88) then
        tmp = (x_m * 2.0d0) / (z_m * (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-88) {
		tmp = (x_m * 2.0) / (z_m * (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-88:
		tmp = (x_m * 2.0) / (z_m * (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-88)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z_m * 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-88)
		tmp = (x_m * 2.0) / (z_m * (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-88], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z$95$m * 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 #s(literal 2 binary64)) < 5.00000000000000009e-88

   1. Initial program 88.6%

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

      1. distribute-rgt-out-- 89.8%

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

   3. Simplified 89.8%

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

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

   1. Initial program 90.1%

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

      1. distribute-rgt-out-- 91.5%

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

   3. Simplified 91.5%

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

      1. *-commutative 91.5%

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

      2. times-frac 98.6%

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

   6. Applied egg-rr 98.6%

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

Alternative 13: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 10^{-69}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{z\_m}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 1e-69)
     (* x_m (/ (/ 2.0 z_m) (- 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) <= 1e-69) {
		tmp = x_m * ((2.0 / z_m) / (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) <= 1d-69) then
        tmp = x_m * ((2.0d0 / z_m) / (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) <= 1e-69) {
		tmp = x_m * ((2.0 / z_m) / (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) <= 1e-69:
		tmp = x_m * ((2.0 / z_m) / (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) <= 1e-69)
		tmp = Float64(x_m * Float64(Float64(2.0 / z_m) / 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) <= 1e-69)
		tmp = x_m * ((2.0 / z_m) / (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], 1e-69], N[(x$95$m * N[(N[(2.0 / z$95$m), $MachinePrecision] / 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 #s(literal 2 binary64)) < 9.9999999999999996e-70

   1. Initial program 88.9%

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

      1. distribute-rgt-out-- 90.1%

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

   3. Simplified 90.1%

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

      1. add-sqr-sqrt 32.9%

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

      2. *-commutative 32.9%

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

      3. times-frac 30.2%

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

   6. Applied egg-rr 30.2%

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

   7. Taylor expanded in x around 0 89.3%

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

      1. associate-/l* 89.3%

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

      2. unpow2 89.3%

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

      3. rem-square-sqrt 90.0%

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

      4. *-commutative 90.0%

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

      5. associate-/l/ 90.4%

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

   9. Simplified 90.4%

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

   if 9.9999999999999996e-70 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 89.6%

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

      1. distribute-rgt-out-- 91.0%

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

   3. Simplified 91.0%

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

      1. *-commutative 91.0%

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

      2. times-frac 98.5%

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

   6. Applied egg-rr 98.5%

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

Alternative 14: 96.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.5 \cdot 10^{+52}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{z\_m}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z\_m}}{y - t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 1.5e+52)
     (* x_m (/ (/ 2.0 z_m) (- y t)))
     (* 2.0 (/ (/ x_m z_m) (- y 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) {
	double tmp;
	if (z_m <= 1.5e+52) {
		tmp = x_m * ((2.0 / z_m) / (y - t));
	} else {
		tmp = 2.0 * ((x_m / z_m) / (y - t));
	}
	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 <= 1.5d+52) then
        tmp = x_m * ((2.0d0 / z_m) / (y - t))
    else
        tmp = 2.0d0 * ((x_m / z_m) / (y - t))
    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 <= 1.5e+52) {
		tmp = x_m * ((2.0 / z_m) / (y - t));
	} else {
		tmp = 2.0 * ((x_m / z_m) / (y - t));
	}
	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 <= 1.5e+52:
		tmp = x_m * ((2.0 / z_m) / (y - t))
	else:
		tmp = 2.0 * ((x_m / z_m) / (y - t))
	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 <= 1.5e+52)
		tmp = Float64(x_m * Float64(Float64(2.0 / z_m) / Float64(y - t)));
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / z_m) / Float64(y - t)));
	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 <= 1.5e+52)
		tmp = x_m * ((2.0 / z_m) / (y - t));
	else
		tmp = 2.0 * ((x_m / z_m) / (y - t));
	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, 1.5e+52], N[(x$95$m * N[(N[(2.0 / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.5e52

   1. Initial program 91.6%

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

      1. distribute-rgt-out-- 92.2%

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

   3. Simplified 92.2%

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

      1. add-sqr-sqrt 56.0%

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

      2. *-commutative 56.0%

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

      3. times-frac 55.7%

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

   6. Applied egg-rr 55.7%

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

   7. Taylor expanded in x around 0 91.3%

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

      1. associate-/l* 91.1%

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

      2. unpow2 91.1%

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

      3. rem-square-sqrt 91.9%

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

      4. *-commutative 91.9%

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

      5. associate-/l/ 92.2%

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

   9. Simplified 92.2%

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

   if 1.5e52 < z

   1. Initial program 79.7%

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

      1. distribute-rgt-out-- 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in x around 0 83.6%

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

      1. associate-/r* 94.9%

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

   7. Simplified 94.9%

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

Alternative 15: 92.2% accurate, 1.2× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (* z_s (* x_s (* 2.0 (/ (/ x_m z_m) (- y 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) / (y - 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) / (y - 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) / (y - 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) / (y - 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(Float64(x_m / z_m) / Float64(y - 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) / (y - 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[(N[(x$95$m / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

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

   1. distribute-rgt-out-- 90.4%

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

3. Simplified 90.4%

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

5. Taylor expanded in x around 0 90.4%

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

   1. associate-/r* 93.9%

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

7. Simplified 93.9%

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

Alternative 16: 52.9% 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 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) 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 89.2%

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

   1. distribute-rgt-out-- 90.4%

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

3. Simplified 90.4%

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

5. Taylor expanded in y around 0 55.0%

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

   1. *-commutative 55.0%

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

7. Simplified 55.0%

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

Developer target: 96.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ t_2 := \frac{x \cdot 2}{y \cdot z - t \cdot z}\\ \mathbf{if}\;t\_2 < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x (* (- y t) z)) 2.0))
        (t_2 (/ (* x 2.0) (- (* y z) (* t z)))))
   (if (< t_2 -2.559141628295061e-13)
     t_1
     (if (< t_2 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / ((y - t) * z)) * 2.0;
	double t_2 = (x * 2.0) / ((y * z) - (t * z));
	double tmp;
	if (t_2 < -2.559141628295061e-13) {
		tmp = t_1;
	} else if (t_2 < 1.045027827330126e-269) {
		tmp = ((x / z) * 2.0) / (y - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x / ((y - t) * z)) * 2.0d0
    t_2 = (x * 2.0d0) / ((y * z) - (t * z))
    if (t_2 < (-2.559141628295061d-13)) then
        tmp = t_1
    else if (t_2 < 1.045027827330126d-269) then
        tmp = ((x / z) * 2.0d0) / (y - t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / ((y - t) * z)) * 2.0;
	double t_2 = (x * 2.0) / ((y * z) - (t * z));
	double tmp;
	if (t_2 < -2.559141628295061e-13) {
		tmp = t_1;
	} else if (t_2 < 1.045027827330126e-269) {
		tmp = ((x / z) * 2.0) / (y - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / ((y - t) * z)) * 2.0
	t_2 = (x * 2.0) / ((y * z) - (t * z))
	tmp = 0
	if t_2 < -2.559141628295061e-13:
		tmp = t_1
	elif t_2 < 1.045027827330126e-269:
		tmp = ((x / z) * 2.0) / (y - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(Float64(y - t) * z)) * 2.0)
	t_2 = Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
	tmp = 0.0
	if (t_2 < -2.559141628295061e-13)
		tmp = t_1;
	elseif (t_2 < 1.045027827330126e-269)
		tmp = Float64(Float64(Float64(x / z) * 2.0) / Float64(y - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / ((y - t) * z)) * 2.0;
	t_2 = (x * 2.0) / ((y * z) - (t * z));
	tmp = 0.0;
	if (t_2 < -2.559141628295061e-13)
		tmp = t_1;
	elseif (t_2 < 1.045027827330126e-269)
		tmp = ((x / z) * 2.0) / (y - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -2.559141628295061e-13], t$95$1, If[Less[t$95$2, 1.045027827330126e-269], N[(N[(N[(x / z), $MachinePrecision] * 2.0), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024096 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))