Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.8% → 97.5%

Time: 13.2s

Alternatives: 13

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \sqrt{x\_m \cdot 2}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{y - t} \cdot \frac{t\_1}{z\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (sqrt (* x_m 2.0))))
   (*
    z_s
    (*
     x_s
     (if (<= z_m 1.5e-133)
       (/ (* 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 (z_m <= 1.5e-133) {
		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 (z_m <= 1.5d-133) 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 (z_m <= 1.5e-133) {
		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 z_m <= 1.5e-133:
		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 (z_m <= 1.5e-133)
		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 (z_m <= 1.5e-133)
		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[z$95$m, 1.5e-133], 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 z < 1.5000000000000001e-133

   1. Initial program 90.1%

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

      1. distribute-rgt-out-- 93.0%

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

   3. Simplified 93.0%

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

   if 1.5000000000000001e-133 < z

   1. Initial program 88.1%

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

      1. distribute-rgt-out-- 88.2%

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

   3. Simplified 88.2%

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

      1. add-sqr-sqrt 51.6%

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

      2. *-commutative 51.6%

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

      3. times-frac 53.9%

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

   6. Applied egg-rr 53.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x \cdot 2}}{y - t} \cdot \frac{\sqrt{x \cdot 2}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{z\_m \cdot 0.5}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-100}:\\ \;\;\;\;x\_m \cdot \frac{-2}{z\_m \cdot t}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{2}{z\_m}}{\frac{-t}{x\_m}}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-22}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z\_m \cdot 0.5}}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -4.5e-72)
     (/ (/ x_m y) (* z_m 0.5))
     (if (<= y -1.8e-100)
       (* x_m (/ -2.0 (* z_m t)))
       (if (<= y -8.5e-115)
         (* (/ x_m z_m) (/ 2.0 y))
         (if (<= y 1.75e-269)
           (/ (/ 2.0 z_m) (/ (- t) x_m))
           (if (<= y 7.2e-22)
             (* -2.0 (/ x_m (* z_m t)))
             (/ (/ x_m (* z_m 0.5)) 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 (y <= -4.5e-72) {
		tmp = (x_m / y) / (z_m * 0.5);
	} else if (y <= -1.8e-100) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else if (y <= -8.5e-115) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else if (y <= 1.75e-269) {
		tmp = (2.0 / z_m) / (-t / x_m);
	} else if (y <= 7.2e-22) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = (x_m / (z_m * 0.5)) / 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 (y <= (-4.5d-72)) then
        tmp = (x_m / y) / (z_m * 0.5d0)
    else if (y <= (-1.8d-100)) then
        tmp = x_m * ((-2.0d0) / (z_m * t))
    else if (y <= (-8.5d-115)) then
        tmp = (x_m / z_m) * (2.0d0 / y)
    else if (y <= 1.75d-269) then
        tmp = (2.0d0 / z_m) / (-t / x_m)
    else if (y <= 7.2d-22) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else
        tmp = (x_m / (z_m * 0.5d0)) / 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 (y <= -4.5e-72) {
		tmp = (x_m / y) / (z_m * 0.5);
	} else if (y <= -1.8e-100) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else if (y <= -8.5e-115) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else if (y <= 1.75e-269) {
		tmp = (2.0 / z_m) / (-t / x_m);
	} else if (y <= 7.2e-22) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = (x_m / (z_m * 0.5)) / 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 y <= -4.5e-72:
		tmp = (x_m / y) / (z_m * 0.5)
	elif y <= -1.8e-100:
		tmp = x_m * (-2.0 / (z_m * t))
	elif y <= -8.5e-115:
		tmp = (x_m / z_m) * (2.0 / y)
	elif y <= 1.75e-269:
		tmp = (2.0 / z_m) / (-t / x_m)
	elif y <= 7.2e-22:
		tmp = -2.0 * (x_m / (z_m * t))
	else:
		tmp = (x_m / (z_m * 0.5)) / 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 (y <= -4.5e-72)
		tmp = Float64(Float64(x_m / y) / Float64(z_m * 0.5));
	elseif (y <= -1.8e-100)
		tmp = Float64(x_m * Float64(-2.0 / Float64(z_m * t)));
	elseif (y <= -8.5e-115)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / y));
	elseif (y <= 1.75e-269)
		tmp = Float64(Float64(2.0 / z_m) / Float64(Float64(-t) / x_m));
	elseif (y <= 7.2e-22)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	else
		tmp = Float64(Float64(x_m / Float64(z_m * 0.5)) / 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 (y <= -4.5e-72)
		tmp = (x_m / y) / (z_m * 0.5);
	elseif (y <= -1.8e-100)
		tmp = x_m * (-2.0 / (z_m * t));
	elseif (y <= -8.5e-115)
		tmp = (x_m / z_m) * (2.0 / y);
	elseif (y <= 1.75e-269)
		tmp = (2.0 / z_m) / (-t / x_m);
	elseif (y <= 7.2e-22)
		tmp = -2.0 * (x_m / (z_m * t));
	else
		tmp = (x_m / (z_m * 0.5)) / 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[LessEqual[y, -4.5e-72], N[(N[(x$95$m / y), $MachinePrecision] / N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.8e-100], N[(x$95$m * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -8.5e-115], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-269], N[(N[(2.0 / z$95$m), $MachinePrecision] / N[((-t) / x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e-22], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 6 regimes

2. if y < -4.5e-72

   1. Initial program 88.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 inf 73.8%

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

      1. *-commutative 73.8%

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

   7. Simplified 73.8%

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

      1. *-commutative 73.8%

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

      2. times-frac 78.2%

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

   9. Applied egg-rr 78.2%

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

       1. clear-num 78.2%

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

       2. un-div-inv 78.2%

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

       3. div-inv 78.2%

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

       4. metadata-eval 78.2%

          \[\leadsto \frac{\frac{x}{y}}{z \cdot \color{blue}{0.5}} \]

   11. Applied egg-rr 78.2%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{z \cdot 0.5}} \]

   if -4.5e-72 < y < -1.7999999999999999e-100

   1. Initial program 99.4%

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

      1. distribute-rgt-out-- 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 88.2%

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

      1. associate-*r/ 88.2%

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

      2. *-commutative 88.2%

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

      3. *-commutative 88.2%

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

      4. associate-/l* 88.5%

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

   7. Simplified 88.5%

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

   if -1.7999999999999999e-100 < y < -8.49999999999999953e-115

   1. Initial program 99.5%

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

      1. distribute-rgt-out-- 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 99.5%

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

      1. *-commutative 99.5%

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

   7. Simplified 99.5%

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

      1. times-frac 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -8.49999999999999953e-115 < y < 1.75000000000000009e-269

   1. Initial program 83.8%

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

      1. distribute-rgt-out-- 89.6%

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

   3. Simplified 89.6%

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

      1. add-sqr-sqrt 47.0%

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

      2. *-commutative 47.0%

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

      3. times-frac 53.4%

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

   6. Applied egg-rr 53.4%

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

      1. frac-times 47.0%

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

      2. add-sqr-sqrt 89.6%

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

      3. frac-times 95.9%

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

      4. *-commutative 95.9%

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

      5. clear-num 96.0%

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

      6. un-div-inv 96.0%

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

   8. Applied egg-rr 96.0%

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

   9. Taylor expanded in y around 0 90.3%

      \[\leadsto \frac{\frac{2}{z}}{\color{blue}{-1 \cdot \frac{t}{x}}} \]
   10. Step-by-step derivation

       1. neg-mul-1 90.3%

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

       2. distribute-neg-frac2 90.3%

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

   11. Simplified 90.3%

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

   if 1.75000000000000009e-269 < y < 7.1999999999999996e-22

   1. Initial program 97.5%

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

      1. distribute-rgt-out-- 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in y around 0 84.6%

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

      1. *-commutative 84.6%

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

   7. Simplified 84.6%

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

   if 7.1999999999999996e-22 < y

   1. Initial program 89.1%

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

      1. distribute-rgt-out-- 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in y around inf 77.6%

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

      1. *-commutative 77.6%

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

   7. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. times-frac 75.4%

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

   9. Applied egg-rr 75.4%

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

       1. associate-*l/ 78.4%

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

       2. clear-num 78.4%

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

       3. un-div-inv 78.5%

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

       4. div-inv 78.5%

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

       5. metadata-eval 78.5%

          \[\leadsto \frac{\frac{x}{z \cdot \color{blue}{0.5}}}{y} \]

   11. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot 0.5}}{y}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{x}{y}}{z \cdot 0.5}\\ \mathbf{elif}\;y \leq -1.8 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{-t}{x}}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-22}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot 0.5}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-71} \lor \neg \left(y \leq 1.1 \cdot 10^{-22}\right):\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (or (<= y -3.15e-71) (not (<= y 1.1e-22)))
     (* 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 <= -3.15e-71) || !(y <= 1.1e-22)) {
		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 <= (-3.15d-71)) .or. (.not. (y <= 1.1d-22))) 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 <= -3.15e-71) || !(y <= 1.1e-22)) {
		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 <= -3.15e-71) or not (y <= 1.1e-22):
		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 <= -3.15e-71) || !(y <= 1.1e-22))
		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 <= -3.15e-71) || ~((y <= 1.1e-22)))
		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, -3.15e-71], N[Not[LessEqual[y, 1.1e-22]], $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 < -3.1500000000000002e-71 or 1.1e-22 < y

   1. Initial program 88.6%

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

      1. distribute-rgt-out-- 89.9%

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

   3. Simplified 89.9%

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

      1. add-sqr-sqrt 43.9%

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

      2. *-commutative 43.9%

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

      3. times-frac 45.5%

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

   6. Applied egg-rr 45.5%

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

   7. Taylor expanded in y around inf 75.2%

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

      1. associate-/l* 74.7%

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

      2. unpow2 74.7%

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

      3. rem-square-sqrt 75.2%

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

      4. *-commutative 75.2%

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

      5. associate-/r* 75.9%

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

   9. Simplified 75.9%

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

   if -3.1500000000000002e-71 < y < 1.1e-22

   1. Initial program 90.7%

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

      1. distribute-rgt-out-- 93.7%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in y around 0 83.0%

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

      1. *-commutative 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-71} \lor \neg \left(y \leq 1.1 \cdot 10^{-22}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-27}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{z\_m}}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -2.7e-72)
     (* (/ 2.0 z_m) (/ x_m y))
     (if (<= y 1.55e-27)
       (* -2.0 (/ x_m (* z_m t)))
       (* 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 (y <= -2.7e-72) {
		tmp = (2.0 / z_m) * (x_m / y);
	} else if (y <= 1.55e-27) {
		tmp = -2.0 * (x_m / (z_m * t));
	} 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 (y <= (-2.7d-72)) then
        tmp = (2.0d0 / z_m) * (x_m / y)
    else if (y <= 1.55d-27) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    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 (y <= -2.7e-72) {
		tmp = (2.0 / z_m) * (x_m / y);
	} else if (y <= 1.55e-27) {
		tmp = -2.0 * (x_m / (z_m * t));
	} 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 y <= -2.7e-72:
		tmp = (2.0 / z_m) * (x_m / y)
	elif y <= 1.55e-27:
		tmp = -2.0 * (x_m / (z_m * t))
	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 (y <= -2.7e-72)
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	elseif (y <= 1.55e-27)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	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 (y <= -2.7e-72)
		tmp = (2.0 / z_m) * (x_m / y);
	elseif (y <= 1.55e-27)
		tmp = -2.0 * (x_m / (z_m * t));
	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[LessEqual[y, -2.7e-72], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e-27], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(2.0 / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.7e-72

   1. Initial program 88.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 inf 73.8%

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

      1. *-commutative 73.8%

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

   7. Simplified 73.8%

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

      1. *-commutative 73.8%

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

      2. times-frac 78.2%

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

   9. Applied egg-rr 78.2%

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

   if -2.7e-72 < y < 1.5499999999999999e-27

   1. Initial program 90.7%

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

      1. distribute-rgt-out-- 93.7%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in y around 0 83.0%

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

      1. *-commutative 83.0%

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

   7. Simplified 83.0%

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

   if 1.5499999999999999e-27 < y

   1. Initial program 89.1%

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

      1. distribute-rgt-out-- 91.6%

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

   3. Simplified 91.6%

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

      1. add-sqr-sqrt 43.7%

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

      2. *-commutative 43.7%

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

      3. times-frac 44.3%

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

   6. Applied egg-rr 44.3%

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

   7. Taylor expanded in y around inf 77.0%

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

      1. associate-/l* 77.0%

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

      2. unpow2 77.0%

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

      3. rem-square-sqrt 77.5%

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

      4. *-commutative 77.5%

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

      5. associate-/r* 78.3%

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

   9. Simplified 78.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-72}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-27}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-64}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -6e-71)
     (* (/ 2.0 z_m) (/ x_m y))
     (if (<= y 1.7e-64)
       (* x_m (/ (/ -2.0 z_m) t))
       (* (/ x_m z_m) (/ 2.0 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 (y <= -6e-71) {
		tmp = (2.0 / z_m) * (x_m / y);
	} else if (y <= 1.7e-64) {
		tmp = x_m * ((-2.0 / z_m) / t);
	} else {
		tmp = (x_m / z_m) * (2.0 / 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 (y <= (-6d-71)) then
        tmp = (2.0d0 / z_m) * (x_m / y)
    else if (y <= 1.7d-64) then
        tmp = x_m * (((-2.0d0) / z_m) / t)
    else
        tmp = (x_m / z_m) * (2.0d0 / 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 (y <= -6e-71) {
		tmp = (2.0 / z_m) * (x_m / y);
	} else if (y <= 1.7e-64) {
		tmp = x_m * ((-2.0 / z_m) / t);
	} else {
		tmp = (x_m / z_m) * (2.0 / 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 y <= -6e-71:
		tmp = (2.0 / z_m) * (x_m / y)
	elif y <= 1.7e-64:
		tmp = x_m * ((-2.0 / z_m) / t)
	else:
		tmp = (x_m / z_m) * (2.0 / 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 (y <= -6e-71)
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	elseif (y <= 1.7e-64)
		tmp = Float64(x_m * Float64(Float64(-2.0 / z_m) / t));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / 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 (y <= -6e-71)
		tmp = (2.0 / z_m) * (x_m / y);
	elseif (y <= 1.7e-64)
		tmp = x_m * ((-2.0 / z_m) / t);
	else
		tmp = (x_m / z_m) * (2.0 / 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[LessEqual[y, -6e-71], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e-64], N[(x$95$m * N[(N[(-2.0 / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -6.0000000000000003e-71

   1. Initial program 88.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 inf 73.8%

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

      1. *-commutative 73.8%

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

   7. Simplified 73.8%

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

      1. *-commutative 73.8%

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

      2. times-frac 78.2%

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

   9. Applied egg-rr 78.2%

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

   if -6.0000000000000003e-71 < y < 1.70000000000000006e-64

   1. Initial program 90.7%

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

      1. distribute-rgt-out-- 93.9%

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

   3. Simplified 93.9%

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

      1. add-sqr-sqrt 50.6%

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

      2. *-commutative 50.6%

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

      3. times-frac 53.4%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in y around 0 84.0%

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

      1. mul-1-neg 84.0%

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

      2. times-frac 81.9%

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

      3. distribute-rgt-neg-in 81.9%

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

      4. unpow2 81.9%

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

      5. rem-square-sqrt 82.5%

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

      6. distribute-neg-frac 82.5%

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

      7. metadata-eval 82.5%

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

      8. times-frac 84.7%

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

      9. associate-/l* 84.7%

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

      10. *-commutative 84.7%

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

      11. associate-/r* 84.7%

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

   9. Simplified 84.7%

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

   if 1.70000000000000006e-64 < y

   1. Initial program 89.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. Taylor expanded in y around inf 76.2%

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

      1. *-commutative 76.2%

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

   7. Simplified 76.2%

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

      1. times-frac 76.9%

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

   9. Applied egg-rr 76.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{-71}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-64}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{z\_m \cdot 0.5}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-67}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -5.8e-71)
     (/ (/ x_m y) (* z_m 0.5))
     (if (<= y 1.45e-67)
       (* x_m (/ (/ -2.0 z_m) t))
       (* (/ x_m z_m) (/ 2.0 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 (y <= -5.8e-71) {
		tmp = (x_m / y) / (z_m * 0.5);
	} else if (y <= 1.45e-67) {
		tmp = x_m * ((-2.0 / z_m) / t);
	} else {
		tmp = (x_m / z_m) * (2.0 / 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 (y <= (-5.8d-71)) then
        tmp = (x_m / y) / (z_m * 0.5d0)
    else if (y <= 1.45d-67) then
        tmp = x_m * (((-2.0d0) / z_m) / t)
    else
        tmp = (x_m / z_m) * (2.0d0 / 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 (y <= -5.8e-71) {
		tmp = (x_m / y) / (z_m * 0.5);
	} else if (y <= 1.45e-67) {
		tmp = x_m * ((-2.0 / z_m) / t);
	} else {
		tmp = (x_m / z_m) * (2.0 / 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 y <= -5.8e-71:
		tmp = (x_m / y) / (z_m * 0.5)
	elif y <= 1.45e-67:
		tmp = x_m * ((-2.0 / z_m) / t)
	else:
		tmp = (x_m / z_m) * (2.0 / 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 (y <= -5.8e-71)
		tmp = Float64(Float64(x_m / y) / Float64(z_m * 0.5));
	elseif (y <= 1.45e-67)
		tmp = Float64(x_m * Float64(Float64(-2.0 / z_m) / t));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / 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 (y <= -5.8e-71)
		tmp = (x_m / y) / (z_m * 0.5);
	elseif (y <= 1.45e-67)
		tmp = x_m * ((-2.0 / z_m) / t);
	else
		tmp = (x_m / z_m) * (2.0 / 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[LessEqual[y, -5.8e-71], N[(N[(x$95$m / y), $MachinePrecision] / N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e-67], N[(x$95$m * N[(N[(-2.0 / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -5.7999999999999997e-71

   1. Initial program 88.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 inf 73.8%

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

      1. *-commutative 73.8%

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

   7. Simplified 73.8%

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

      1. *-commutative 73.8%

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

      2. times-frac 78.2%

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

   9. Applied egg-rr 78.2%

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

       1. clear-num 78.2%

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

       2. un-div-inv 78.2%

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

       3. div-inv 78.2%

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

       4. metadata-eval 78.2%

          \[\leadsto \frac{\frac{x}{y}}{z \cdot \color{blue}{0.5}} \]

   11. Applied egg-rr 78.2%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{z \cdot 0.5}} \]

   if -5.7999999999999997e-71 < y < 1.45000000000000002e-67

   1. Initial program 90.7%

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

      1. distribute-rgt-out-- 93.9%

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

   3. Simplified 93.9%

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

      1. add-sqr-sqrt 50.6%

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

      2. *-commutative 50.6%

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

      3. times-frac 53.4%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in y around 0 84.0%

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

      1. mul-1-neg 84.0%

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

      2. times-frac 81.9%

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

      3. distribute-rgt-neg-in 81.9%

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

      4. unpow2 81.9%

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

      5. rem-square-sqrt 82.5%

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

      6. distribute-neg-frac 82.5%

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

      7. metadata-eval 82.5%

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

      8. times-frac 84.7%

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

      9. associate-/l* 84.7%

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

      10. *-commutative 84.7%

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

      11. associate-/r* 84.7%

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

   9. Simplified 84.7%

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

   if 1.45000000000000002e-67 < y

   1. Initial program 89.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. Taylor expanded in y around inf 76.2%

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

      1. *-commutative 76.2%

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

   7. Simplified 76.2%

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

      1. times-frac 76.9%

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

   9. Applied egg-rr 76.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{x}{y}}{z \cdot 0.5}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-67}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{z\_m \cdot 0.5}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-64}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z\_m \cdot 0.5}}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -7.6e-72)
     (/ (/ x_m y) (* z_m 0.5))
     (if (<= y 3.8e-64)
       (* x_m (/ (/ -2.0 z_m) t))
       (/ (/ x_m (* z_m 0.5)) 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 (y <= -7.6e-72) {
		tmp = (x_m / y) / (z_m * 0.5);
	} else if (y <= 3.8e-64) {
		tmp = x_m * ((-2.0 / z_m) / t);
	} else {
		tmp = (x_m / (z_m * 0.5)) / 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 (y <= (-7.6d-72)) then
        tmp = (x_m / y) / (z_m * 0.5d0)
    else if (y <= 3.8d-64) then
        tmp = x_m * (((-2.0d0) / z_m) / t)
    else
        tmp = (x_m / (z_m * 0.5d0)) / 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 (y <= -7.6e-72) {
		tmp = (x_m / y) / (z_m * 0.5);
	} else if (y <= 3.8e-64) {
		tmp = x_m * ((-2.0 / z_m) / t);
	} else {
		tmp = (x_m / (z_m * 0.5)) / 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 y <= -7.6e-72:
		tmp = (x_m / y) / (z_m * 0.5)
	elif y <= 3.8e-64:
		tmp = x_m * ((-2.0 / z_m) / t)
	else:
		tmp = (x_m / (z_m * 0.5)) / 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 (y <= -7.6e-72)
		tmp = Float64(Float64(x_m / y) / Float64(z_m * 0.5));
	elseif (y <= 3.8e-64)
		tmp = Float64(x_m * Float64(Float64(-2.0 / z_m) / t));
	else
		tmp = Float64(Float64(x_m / Float64(z_m * 0.5)) / 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 (y <= -7.6e-72)
		tmp = (x_m / y) / (z_m * 0.5);
	elseif (y <= 3.8e-64)
		tmp = x_m * ((-2.0 / z_m) / t);
	else
		tmp = (x_m / (z_m * 0.5)) / 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[LessEqual[y, -7.6e-72], N[(N[(x$95$m / y), $MachinePrecision] / N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e-64], N[(x$95$m * N[(N[(-2.0 / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -7.60000000000000004e-72

   1. Initial program 88.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 inf 73.8%

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

      1. *-commutative 73.8%

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

   7. Simplified 73.8%

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

      1. *-commutative 73.8%

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

      2. times-frac 78.2%

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

   9. Applied egg-rr 78.2%

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

       1. clear-num 78.2%

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

       2. un-div-inv 78.2%

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

       3. div-inv 78.2%

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

       4. metadata-eval 78.2%

          \[\leadsto \frac{\frac{x}{y}}{z \cdot \color{blue}{0.5}} \]

   11. Applied egg-rr 78.2%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{z \cdot 0.5}} \]

   if -7.60000000000000004e-72 < y < 3.8000000000000002e-64

   1. Initial program 90.7%

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

      1. distribute-rgt-out-- 93.9%

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

   3. Simplified 93.9%

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

      1. add-sqr-sqrt 50.6%

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

      2. *-commutative 50.6%

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

      3. times-frac 53.4%

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

   6. Applied egg-rr 53.4%

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

   7. Taylor expanded in y around 0 84.0%

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

      1. mul-1-neg 84.0%

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

      2. times-frac 81.9%

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

      3. distribute-rgt-neg-in 81.9%

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

      4. unpow2 81.9%

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

      5. rem-square-sqrt 82.5%

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

      6. distribute-neg-frac 82.5%

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

      7. metadata-eval 82.5%

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

      8. times-frac 84.7%

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

      9. associate-/l* 84.7%

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

      10. *-commutative 84.7%

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

      11. associate-/r* 84.7%

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

   9. Simplified 84.7%

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

   if 3.8000000000000002e-64 < y

   1. Initial program 89.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. Taylor expanded in y around inf 76.2%

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

      1. *-commutative 76.2%

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

   7. Simplified 76.2%

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

      1. *-commutative 76.2%

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

      2. times-frac 74.1%

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

   9. Applied egg-rr 74.1%

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

       1. associate-*l/ 76.9%

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

       2. clear-num 76.9%

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

       3. un-div-inv 77.0%

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

       4. div-inv 77.0%

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

       5. metadata-eval 77.0%

          \[\leadsto \frac{\frac{x}{z \cdot \color{blue}{0.5}}}{y} \]

   11. Applied egg-rr 77.0%

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot 0.5}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{x}{y}}{z \cdot 0.5}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-64}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot 0.5}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{z\_m \cdot 0.5}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z\_m \cdot 0.5}}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -3.5e-71)
     (/ (/ x_m y) (* z_m 0.5))
     (if (<= y 2.9e-16)
       (/ (/ (* x_m -2.0) z_m) t)
       (/ (/ x_m (* z_m 0.5)) 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 (y <= -3.5e-71) {
		tmp = (x_m / y) / (z_m * 0.5);
	} else if (y <= 2.9e-16) {
		tmp = ((x_m * -2.0) / z_m) / t;
	} else {
		tmp = (x_m / (z_m * 0.5)) / 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 (y <= (-3.5d-71)) then
        tmp = (x_m / y) / (z_m * 0.5d0)
    else if (y <= 2.9d-16) then
        tmp = ((x_m * (-2.0d0)) / z_m) / t
    else
        tmp = (x_m / (z_m * 0.5d0)) / 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 (y <= -3.5e-71) {
		tmp = (x_m / y) / (z_m * 0.5);
	} else if (y <= 2.9e-16) {
		tmp = ((x_m * -2.0) / z_m) / t;
	} else {
		tmp = (x_m / (z_m * 0.5)) / 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 y <= -3.5e-71:
		tmp = (x_m / y) / (z_m * 0.5)
	elif y <= 2.9e-16:
		tmp = ((x_m * -2.0) / z_m) / t
	else:
		tmp = (x_m / (z_m * 0.5)) / 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 (y <= -3.5e-71)
		tmp = Float64(Float64(x_m / y) / Float64(z_m * 0.5));
	elseif (y <= 2.9e-16)
		tmp = Float64(Float64(Float64(x_m * -2.0) / z_m) / t);
	else
		tmp = Float64(Float64(x_m / Float64(z_m * 0.5)) / 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 (y <= -3.5e-71)
		tmp = (x_m / y) / (z_m * 0.5);
	elseif (y <= 2.9e-16)
		tmp = ((x_m * -2.0) / z_m) / t;
	else
		tmp = (x_m / (z_m * 0.5)) / 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[LessEqual[y, -3.5e-71], N[(N[(x$95$m / y), $MachinePrecision] / N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e-16], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / z$95$m), $MachinePrecision] / t), $MachinePrecision], N[(N[(x$95$m / N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.4999999999999999e-71

   1. Initial program 88.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 inf 73.8%

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

      1. *-commutative 73.8%

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

   7. Simplified 73.8%

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

      1. *-commutative 73.8%

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

      2. times-frac 78.2%

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

   9. Applied egg-rr 78.2%

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

       1. clear-num 78.2%

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

       2. un-div-inv 78.2%

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

       3. div-inv 78.2%

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

       4. metadata-eval 78.2%

          \[\leadsto \frac{\frac{x}{y}}{z \cdot \color{blue}{0.5}} \]

   11. Applied egg-rr 78.2%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{z \cdot 0.5}} \]

   if -3.4999999999999999e-71 < y < 2.8999999999999998e-16

   1. Initial program 90.7%

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

      1. distribute-rgt-out-- 93.7%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in y around 0 83.0%

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

      1. *-commutative 83.0%

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

   7. Simplified 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*l/ 83.0%

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

      3. metadata-eval 83.0%

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

      4. distribute-rgt-neg-in 83.0%

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

      5. associate-/r* 83.1%

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

      6. distribute-rgt-neg-in 83.1%

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

      7. metadata-eval 83.1%

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

   9. Applied egg-rr 83.1%

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

   if 2.8999999999999998e-16 < y

   1. Initial program 89.1%

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

      1. distribute-rgt-out-- 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in y around inf 77.6%

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

      1. *-commutative 77.6%

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

   7. Simplified 77.6%

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

      1. *-commutative 77.6%

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

      2. times-frac 75.4%

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

   9. Applied egg-rr 75.4%

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

       1. associate-*l/ 78.4%

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

       2. clear-num 78.4%

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

       3. un-div-inv 78.5%

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

       4. div-inv 78.5%

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

       5. metadata-eval 78.5%

          \[\leadsto \frac{\frac{x}{z \cdot \color{blue}{0.5}}}{y} \]

   11. Applied egg-rr 78.5%

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot 0.5}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{x}{y}}{z \cdot 0.5}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot 0.5}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 10^{-51}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z\_m}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y - t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 1e-51)
     (* 2.0 (/ (/ x_m z_m) (- y t)))
     (* (/ 2.0 z_m) (/ x_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 ((x_m * 2.0) <= 1e-51) {
		tmp = 2.0 * ((x_m / z_m) / (y - t));
	} else {
		tmp = (2.0 / z_m) * (x_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 ((x_m * 2.0d0) <= 1d-51) then
        tmp = 2.0d0 * ((x_m / z_m) / (y - t))
    else
        tmp = (2.0d0 / z_m) * (x_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 ((x_m * 2.0) <= 1e-51) {
		tmp = 2.0 * ((x_m / z_m) / (y - t));
	} else {
		tmp = (2.0 / z_m) * (x_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 (x_m * 2.0) <= 1e-51:
		tmp = 2.0 * ((x_m / z_m) / (y - t))
	else:
		tmp = (2.0 / z_m) * (x_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 (Float64(x_m * 2.0) <= 1e-51)
		tmp = Float64(2.0 * Float64(Float64(x_m / z_m) / Float64(y - t)));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_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 ((x_m * 2.0) <= 1e-51)
		tmp = 2.0 * ((x_m / z_m) / (y - t));
	else
		tmp = (2.0 / z_m) * (x_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[N[(x$95$m * 2.0), $MachinePrecision], 1e-51], N[(2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 2) < 1e-51

   1. Initial program 89.2%

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

      1. distribute-rgt-out-- 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in x around 0 90.7%

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

      1. associate-/r* 91.8%

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

   7. Simplified 91.8%

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

   if 1e-51 < (*.f64 x 2)

   1. Initial program 90.1%

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

      1. distribute-rgt-out-- 93.2%

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

   3. Simplified 93.2%

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

      1. *-commutative 93.2%

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

      2. times-frac 98.1%

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

   6. Applied egg-rr 98.1%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 10^{-51}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y - t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 5e+17)
     (/ (* x_m 2.0) (* z_m (- y t)))
     (* (/ 2.0 z_m) (/ x_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 ((x_m * 2.0) <= 5e+17) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (2.0 / z_m) * (x_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 ((x_m * 2.0d0) <= 5d+17) then
        tmp = (x_m * 2.0d0) / (z_m * (y - t))
    else
        tmp = (2.0d0 / z_m) * (x_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 ((x_m * 2.0) <= 5e+17) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (2.0 / z_m) * (x_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 (x_m * 2.0) <= 5e+17:
		tmp = (x_m * 2.0) / (z_m * (y - t))
	else:
		tmp = (2.0 / z_m) * (x_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 (Float64(x_m * 2.0) <= 5e+17)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_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 ((x_m * 2.0) <= 5e+17)
		tmp = (x_m * 2.0) / (z_m * (y - t));
	else
		tmp = (2.0 / z_m) * (x_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[N[(x$95$m * 2.0), $MachinePrecision], 5e+17], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 2) < 5e17

   1. Initial program 89.3%

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

      1. distribute-rgt-out-- 91.3%

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

   3. Simplified 91.3%

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

   if 5e17 < (*.f64 x 2)

   1. Initial program 89.7%

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

      1. distribute-rgt-out-- 91.7%

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

   3. Simplified 91.7%

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

      1. *-commutative 91.7%

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

      2. times-frac 97.8%

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

   6. Applied egg-rr 97.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{+17}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z\_m}}{\frac{y - t}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 7.6e+42)
     (/ (* x_m 2.0) (* z_m (- y t)))
     (/ (/ 2.0 z_m) (/ (- y t) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 7.6e+42) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (2.0 / z_m) / ((y - t) / x_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 7.6d+42) then
        tmp = (x_m * 2.0d0) / (z_m * (y - t))
    else
        tmp = (2.0d0 / z_m) / ((y - t) / x_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 7.6e+42) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (2.0 / z_m) / ((y - t) / x_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if z_m <= 7.6e+42:
		tmp = (x_m * 2.0) / (z_m * (y - t))
	else:
		tmp = (2.0 / z_m) / ((y - t) / x_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (z_m <= 7.6e+42)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(2.0 / z_m) / Float64(Float64(y - t) / x_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 7.6e+42)
		tmp = (x_m * 2.0) / (z_m * (y - t));
	else
		tmp = (2.0 / z_m) / ((y - t) / x_m);
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 7.6e+42], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.5999999999999997e42

   1. Initial program 91.0%

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

      1. distribute-rgt-out-- 93.5%

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

   3. Simplified 93.5%

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

   if 7.5999999999999997e42 < z

   1. Initial program 83.4%

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

      1. add-sqr-sqrt 45.3%

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

      2. *-commutative 45.3%

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

      3. times-frac 47.0%

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

   6. Applied egg-rr 47.0%

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

      1. frac-times 45.3%

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

      2. add-sqr-sqrt 83.6%

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

      3. frac-times 97.7%

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

      4. *-commutative 97.7%

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

      5. clear-num 97.7%

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

      6. un-div-inv 97.9%

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

   8. Applied egg-rr 97.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.6 \cdot 10^{+42}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.9% 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 1 x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (* z_s (* x_s (* 2.0 (/ (/ x_m z_m) (- 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 89.4%

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

   1. distribute-rgt-out-- 91.4%

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

3. Simplified 91.4%

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

5. Taylor expanded in x around 0 91.4%

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

   1. associate-/r* 91.7%

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

7. Simplified 91.7%

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

8. Final simplification 91.7%

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

Alternative 13: 53.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x_s, x_m, y, z_m, t)
	tmp = z_s * (x_s * (-2.0 * (x_m / (z_m * t))));
end

Wolfram

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

Derivation

1. Initial program 89.4%

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

   1. distribute-rgt-out-- 91.4%

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

3. Simplified 91.4%

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

5. Taylor expanded in y around 0 52.6%

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

   1. *-commutative 52.6%

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

7. Simplified 52.6%

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

8. Final simplification 52.6%

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

Developer target: 96.8% 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 2024057 
(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))))