Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.4% → 97.3%

Time: 24.7s

Alternatives: 10

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: 89.4% 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: 97.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \sqrt{x\_m \cdot 2}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.3 \cdot 10^{-120}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{y - t} \cdot \frac{t\_1}{z\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (sqrt (* x_m 2.0))))
   (*
    z_s
    (*
     x_s
     (if (<= z_m 5.3e-120)
       (/ (* 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 <= 5.3e-120) {
		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 <= 5.3d-120) 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 <= 5.3e-120) {
		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 <= 5.3e-120:
		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 <= 5.3e-120)
		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 <= 5.3e-120)
		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, 5.3e-120], 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 < 5.29999999999999997e-120

   1. Initial program 90.2%

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

      1. distribute-rgt-out-- 92.9%

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

   3. Simplified 92.9%

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

   if 5.29999999999999997e-120 < z

   1. Initial program 89.4%

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

      1. distribute-rgt-out-- 90.6%

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

   3. Simplified 90.6%

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

      1. add-sqr-sqrt 47.3%

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

      2. *-commutative 47.3%

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

      3. times-frac 50.4%

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

   6. Applied egg-rr 50.4%

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

Alternative 2: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.1 \cdot 10^{+39} \lor \neg \left(t \leq 42000\right):\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z\_m}}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (or (<= t -5.1e+39) (not (<= t 42000.0)))
     (* -2.0 (/ x_m (* z_m t)))
     (* 2.0 (/ (/ x_m z_m) y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((t <= -5.1e+39) || !(t <= 42000.0)) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = 2.0 * ((x_m / z_m) / y);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-5.1d+39)) .or. (.not. (t <= 42000.0d0))) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else
        tmp = 2.0d0 * ((x_m / z_m) / y)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((t <= -5.1e+39) || !(t <= 42000.0)) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = 2.0 * ((x_m / z_m) / y);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (t <= -5.1e+39) or not (t <= 42000.0):
		tmp = -2.0 * (x_m / (z_m * t))
	else:
		tmp = 2.0 * ((x_m / z_m) / y)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if ((t <= -5.1e+39) || !(t <= 42000.0))
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / z_m) / y));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((t <= -5.1e+39) || ~((t <= 42000.0)))
		tmp = -2.0 * (x_m / (z_m * t));
	else
		tmp = 2.0 * ((x_m / z_m) / y);
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[Or[LessEqual[t, -5.1e+39], N[Not[LessEqual[t, 42000.0]], $MachinePrecision]], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -5.0999999999999998e39 or 42000 < t

   1. Initial program 88.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 76.3%

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

   if -5.0999999999999998e39 < t < 42000

   1. Initial program 91.0%

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

      1. distribute-rgt-out-- 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in y around inf 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-/r* 73.4%

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

   7. Simplified 73.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.1 \cdot 10^{+39} \lor \neg \left(t \leq 42000\right):\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{+39}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{elif}\;t \leq 1.45:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -2.35e+39)
     (* -2.0 (/ x_m (* z_m t)))
     (if (<= t 1.45) (* x_m (/ (/ 2.0 y) z_m)) (* x_m (/ (/ -2.0 t) z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -2.35e+39) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (t <= 1.45) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-2.35d+39)) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else if (t <= 1.45d0) then
        tmp = x_m * ((2.0d0 / y) / z_m)
    else
        tmp = x_m * (((-2.0d0) / t) / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -2.35e+39) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (t <= 1.45) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -2.35e+39:
		tmp = -2.0 * (x_m / (z_m * t))
	elif t <= 1.45:
		tmp = x_m * ((2.0 / y) / z_m)
	else:
		tmp = x_m * ((-2.0 / t) / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -2.35e+39)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	elseif (t <= 1.45)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z_m));
	else
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -2.35e+39)
		tmp = -2.0 * (x_m / (z_m * t));
	elseif (t <= 1.45)
		tmp = x_m * ((2.0 / y) / z_m);
	else
		tmp = x_m * ((-2.0 / t) / z_m);
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[t, -2.35e+39], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -2.35e39

   1. Initial program 89.4%

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

      1. distribute-rgt-out-- 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in y around 0 76.2%

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

   if -2.35e39 < t < 1.44999999999999996

   1. Initial program 91.0%

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

      1. distribute-rgt-out-- 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in x around 0 92.5%

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

      1. associate-*r/ 92.5%

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

      2. *-commutative 92.5%

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

      3. associate-*r/ 92.5%

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

      4. *-commutative 92.5%

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

      5. associate-/r* 92.6%

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

   7. Simplified 92.6%

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

   8. Taylor expanded in y around inf 76.8%

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

   if 1.44999999999999996 < t

   1. Initial program 87.9%

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

      1. distribute-rgt-out-- 89.8%

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

   3. Simplified 89.8%

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

      1. add-sqr-sqrt 44.9%

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

      2. *-commutative 44.9%

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

      3. times-frac 51.5%

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

   6. Applied egg-rr 51.5%

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

   7. Taylor expanded in y around 0 75.7%

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

      1. associate-*r/ 75.7%

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

      2. mul-1-neg 75.7%

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

      3. unpow2 75.7%

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

      4. rem-square-sqrt 76.4%

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

      5. distribute-rgt-neg-in 76.4%

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

      6. metadata-eval 76.4%

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

      7. associate-/l* 76.2%

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

      8. associate-/r* 76.9%

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

   9. Simplified 76.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{+39}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 1.45:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.58 \cdot 10^{+38}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{elif}\;t \leq 13.2:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -1.58e+38)
     (* -2.0 (/ x_m (* z_m t)))
     (if (<= t 13.2) (* 2.0 (/ (/ x_m z_m) y)) (* x_m (/ (/ -2.0 t) z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.58e+38) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (t <= 13.2) {
		tmp = 2.0 * ((x_m / z_m) / y);
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.58d+38)) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else if (t <= 13.2d0) then
        tmp = 2.0d0 * ((x_m / z_m) / y)
    else
        tmp = x_m * (((-2.0d0) / t) / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.58e+38) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (t <= 13.2) {
		tmp = 2.0 * ((x_m / z_m) / y);
	} else {
		tmp = x_m * ((-2.0 / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -1.58e+38:
		tmp = -2.0 * (x_m / (z_m * t))
	elif t <= 13.2:
		tmp = 2.0 * ((x_m / z_m) / y)
	else:
		tmp = x_m * ((-2.0 / t) / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -1.58e+38)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	elseif (t <= 13.2)
		tmp = Float64(2.0 * Float64(Float64(x_m / z_m) / y));
	else
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -1.58e+38)
		tmp = -2.0 * (x_m / (z_m * t));
	elseif (t <= 13.2)
		tmp = 2.0 * ((x_m / z_m) / y);
	else
		tmp = x_m * ((-2.0 / t) / z_m);
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[t, -1.58e+38], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 13.2], N[(2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.58e38

   1. Initial program 89.4%

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

      1. distribute-rgt-out-- 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in y around 0 76.2%

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

   if -1.58e38 < t < 13.199999999999999

   1. Initial program 91.0%

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

      1. distribute-rgt-out-- 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in y around inf 76.7%

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

      1. *-commutative 76.7%

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

      2. associate-/r* 73.4%

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

   7. Simplified 73.4%

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

   if 13.199999999999999 < t

   1. Initial program 87.9%

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

      1. distribute-rgt-out-- 89.8%

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

   3. Simplified 89.8%

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

      1. add-sqr-sqrt 44.9%

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

      2. *-commutative 44.9%

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

      3. times-frac 51.5%

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

   6. Applied egg-rr 51.5%

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

   7. Taylor expanded in y around 0 75.7%

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

      1. associate-*r/ 75.7%

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

      2. mul-1-neg 75.7%

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

      3. unpow2 75.7%

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

      4. rem-square-sqrt 76.4%

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

      5. distribute-rgt-neg-in 76.4%

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

      6. metadata-eval 76.4%

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

      7. associate-/l* 76.2%

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

      8. associate-/r* 76.9%

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

   9. Simplified 76.9%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.58 \cdot 10^{+38}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 13.2:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\left(y - t\right) \cdot 0.5}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 5e-52)
     (* (/ x_m z_m) (/ 2.0 (- y t)))
     (/ (/ x_m (* (- y t) 0.5)) z_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-52) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / ((y - t) * 0.5)) / z_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-52) then
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    else
        tmp = (x_m / ((y - t) * 0.5d0)) / z_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-52) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / ((y - t) * 0.5)) / z_m;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-52:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	else:
		tmp = (x_m / ((y - t) * 0.5)) / z_m
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-52)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(Float64(y - t) * 0.5)) / z_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-52)
		tmp = (x_m / z_m) * (2.0 / (y - t));
	else
		tmp = (x_m / ((y - t) * 0.5)) / z_m;
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-52], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(N[(y - t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 5e-52

   1. Initial program 91.2%

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

      1. times-frac 91.4%

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

   6. Applied egg-rr 91.4%

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

   if 5e-52 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 86.9%

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

      1. distribute-rgt-out-- 88.5%

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

   3. Simplified 88.5%

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

      1. add-sqr-sqrt 88.3%

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

      2. *-commutative 88.3%

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

      3. times-frac 95.5%

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

   6. Applied egg-rr 95.5%

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

      1. frac-times 88.3%

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

      2. add-sqr-sqrt 88.5%

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

      3. associate-/r* 98.5%

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

      4. associate-*r/ 98.4%

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

      5. clear-num 98.4%

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

      6. un-div-inv 98.5%

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

      7. div-inv 98.5%

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

      8. metadata-eval 98.5%

         \[\leadsto \frac{\frac{x}{\left(y - t\right) \cdot \color{blue}{0.5}}}{z} \]

   8. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(y - t\right) \cdot 0.5}}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 5e-52)
     (* (/ x_m z_m) (/ 2.0 (- y t)))
     (* (/ x_m (- y t)) (/ 2.0 z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-52) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-52) then
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-52) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-52:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-52)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-52)
		tmp = (x_m / z_m) * (2.0 / (y - t));
	else
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-52], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 5e-52

   1. Initial program 91.2%

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

      1. times-frac 91.4%

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

   6. Applied egg-rr 91.4%

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

   if 5e-52 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 86.9%

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

      1. distribute-rgt-out-- 88.5%

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

   3. Simplified 88.5%

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

      1. *-commutative 88.5%

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

      2. times-frac 98.4%

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

   6. Applied egg-rr 98.4%

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

Alternative 7: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{2}{y - t}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8.5 \cdot 10^{+49}:\\ \;\;\;\;x\_m \cdot \frac{t\_1}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot t\_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (- y t))))
   (*
    z_s
    (* x_s (if (<= z_m 8.5e+49) (* x_m (/ t_1 z_m)) (* (/ x_m z_m) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 / (y - t);
	double tmp;
	if (z_m <= 8.5e+49) {
		tmp = x_m * (t_1 / z_m);
	} else {
		tmp = (x_m / z_m) * t_1;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (y - t)
    if (z_m <= 8.5d+49) then
        tmp = x_m * (t_1 / z_m)
    else
        tmp = (x_m / z_m) * t_1
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = 2.0 / (y - t);
	double tmp;
	if (z_m <= 8.5e+49) {
		tmp = x_m * (t_1 / z_m);
	} else {
		tmp = (x_m / z_m) * t_1;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = 2.0 / (y - t)
	tmp = 0
	if z_m <= 8.5e+49:
		tmp = x_m * (t_1 / z_m)
	else:
		tmp = (x_m / z_m) * t_1
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = Float64(2.0 / Float64(y - t))
	tmp = 0.0
	if (z_m <= 8.5e+49)
		tmp = Float64(x_m * Float64(t_1 / z_m));
	else
		tmp = Float64(Float64(x_m / z_m) * t_1);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = 2.0 / (y - t);
	tmp = 0.0;
	if (z_m <= 8.5e+49)
		tmp = x_m * (t_1 / z_m);
	else
		tmp = (x_m / z_m) * t_1;
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 8.5e+49], N[(x$95$m * N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8.4999999999999996e49

   1. Initial program 91.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 x around 0 93.7%

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

      1. associate-*r/ 93.7%

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

      2. *-commutative 93.7%

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

      3. associate-*r/ 93.6%

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

      4. *-commutative 93.6%

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

      5. associate-/r* 93.7%

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

   7. Simplified 93.7%

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

   if 8.4999999999999996e49 < z

   1. Initial program 84.5%

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

      1. distribute-rgt-out-- 86.3%

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

   3. Simplified 86.3%

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

      1. times-frac 96.4%

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

   6. Applied egg-rr 96.4%

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

Alternative 8: 57.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+50}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z\_m}}{t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 1e+50) (* -2.0 (/ x_m (* z_m t))) (* -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 (z_m <= 1e+50) {
		tmp = -2.0 * (x_m / (z_m * t));
	} 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 (z_m <= 1d+50) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    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 (z_m <= 1e+50) {
		tmp = -2.0 * (x_m / (z_m * t));
	} 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 z_m <= 1e+50:
		tmp = -2.0 * (x_m / (z_m * t))
	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 (z_m <= 1e+50)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / 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 (z_m <= 1e+50)
		tmp = -2.0 * (x_m / (z_m * t));
	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[LessEqual[z$95$m, 1e+50], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.0000000000000001e50

   1. Initial program 91.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 46.6%

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

   if 1.0000000000000001e50 < z

   1. Initial program 84.5%

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

      1. distribute-rgt-out-- 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in y around 0 48.3%

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

      1. *-commutative 48.3%

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

      2. associate-/r* 52.6%

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

   7. Simplified 52.6%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+50}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. distribute-rgt-out-- 92.1%

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

3. Simplified 92.1%

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

5. Taylor expanded in x around 0 92.1%

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

   1. associate-*r/ 92.1%

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

   2. *-commutative 92.1%

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

   3. associate-*r/ 92.0%

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

   4. *-commutative 92.0%

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

   5. associate-/r* 92.7%

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

7. Simplified 92.7%

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

Alternative 10: 52.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.9%

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

   1. distribute-rgt-out-- 92.1%

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

3. Simplified 92.1%

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

5. Taylor expanded in y around 0 47.0%

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

6. Final simplification 47.0%

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

Developer Target 1: 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 2024143 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (* x 2) (- (* y z) (* t z))) -2559141628295061/10000000000000000000000000000) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 522513913665063/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2))))

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