Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.8% → 97.1%

Time: 12.4s

Alternatives: 12

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: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z\_m}{2}}}{y - t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 3.6e+35)
    (/ (* x 2.0) (* z_m (- y t)))
    (/ (/ x (/ z_m 2.0)) (- y t)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.6e+35) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x / (z_m / 2.0)) / (y - t);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 3.6d+35) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else
        tmp = (x / (z_m / 2.0d0)) / (y - t)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 3.6e+35) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x / (z_m / 2.0)) / (y - t);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 3.6e+35:
		tmp = (x * 2.0) / (z_m * (y - t))
	else:
		tmp = (x / (z_m / 2.0)) / (y - t)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 3.6e+35)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(x / Float64(z_m / 2.0)) / Float64(y - t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 3.6e+35)
		tmp = (x * 2.0) / (z_m * (y - t));
	else
		tmp = (x / (z_m / 2.0)) / (y - t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 3.6e+35], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(z$95$m / 2.0), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.6e35

   1. Initial program 92.7%

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

      1. distribute-rgt-out-- 96.0%

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

   3. Simplified 96.0%

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

   if 3.6e35 < z

   1. Initial program 79.0%

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

      1. distribute-rgt-out-- 83.2%

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

      2. *-commutative 83.2%

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

      3. times-frac 96.8%

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

      4. associate-*l/ 96.9%

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

      5. associate-*r/ 96.8%

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

   3. Simplified 96.8%

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

      1. associate-/l/ 83.1%

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

      2. distribute-rgt-out-- 78.9%

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

      3. associate-*r/ 79.0%

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

      4. *-commutative 79.0%

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

      5. distribute-rgt-out-- 83.2%

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

      6. associate-/r* 98.7%

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

      7. associate-/l* 98.7%

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

   6. Applied egg-rr 98.7%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\frac{z}{2}}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.7 \cdot 10^{-12} \lor \neg \left(t \leq 480000\right):\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= t -5.7e-12) (not (<= t 480000.0)))
    (* (/ x t) (/ -2.0 z_m))
    (* x (/ 2.0 (* z_m y))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((t <= -5.7e-12) || !(t <= 480000.0)) {
		tmp = (x / t) * (-2.0 / z_m);
	} else {
		tmp = x * (2.0 / (z_m * y));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-5.7d-12)) .or. (.not. (t <= 480000.0d0))) then
        tmp = (x / t) * ((-2.0d0) / z_m)
    else
        tmp = x * (2.0d0 / (z_m * y))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((t <= -5.7e-12) || !(t <= 480000.0)) {
		tmp = (x / t) * (-2.0 / z_m);
	} else {
		tmp = x * (2.0 / (z_m * y));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (t <= -5.7e-12) or not (t <= 480000.0):
		tmp = (x / t) * (-2.0 / z_m)
	else:
		tmp = x * (2.0 / (z_m * y))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if ((t <= -5.7e-12) || !(t <= 480000.0))
		tmp = Float64(Float64(x / t) * Float64(-2.0 / z_m));
	else
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((t <= -5.7e-12) || ~((t <= 480000.0)))
		tmp = (x / t) * (-2.0 / z_m);
	else
		tmp = x * (2.0 / (z_m * y));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[Or[LessEqual[t, -5.7e-12], N[Not[LessEqual[t, 480000.0]], $MachinePrecision]], N[(N[(x / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -5.7000000000000003e-12 or 4.8e5 < t

   1. Initial program 86.6%

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

      1. distribute-rgt-out-- 91.0%

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

      2. *-commutative 91.0%

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

      3. times-frac 93.6%

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

      4. associate-*l/ 93.7%

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

      5. associate-*r/ 93.7%

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

   3. Simplified 93.7%

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

   5. Taylor expanded in y around 0 77.3%

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

      1. associate-*r/ 77.3%

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

      2. *-commutative 77.3%

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

      3. times-frac 81.3%

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

   7. Simplified 81.3%

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

   if -5.7000000000000003e-12 < t < 4.8e5

   1. Initial program 92.4%

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

      1. distribute-rgt-out-- 94.9%

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

      2. *-commutative 94.9%

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

      3. times-frac 92.1%

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

      4. associate-*l/ 92.1%

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

      5. associate-*r/ 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in x around 0 94.9%

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

      1. associate-*r/ 94.9%

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

      2. *-commutative 94.9%

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

      3. *-commutative 94.9%

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

      4. associate-*r/ 94.9%

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

      5. associate-/r* 94.8%

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

      6. *-lft-identity 94.8%

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

      7. associate-*l/ 94.8%

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

      8. associate-*r/ 94.8%

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

      9. associate-*l/ 94.8%

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

      10. metadata-eval 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in y around inf 79.8%

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

      1. *-commutative 79.8%

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

   10. Simplified 79.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.7 \cdot 10^{-12} \lor \neg \left(t \leq 480000\right):\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z\_m}\\ \mathbf{elif}\;t \leq 3600000:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{-2}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -5.4e-12)
    (* (/ x t) (/ -2.0 z_m))
    (if (<= t 3600000.0) (* x (/ 2.0 (* z_m y))) (* (/ x z_m) (/ -2.0 t))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -5.4e-12) {
		tmp = (x / t) * (-2.0 / z_m);
	} else if (t <= 3600000.0) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = (x / z_m) * (-2.0 / t);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.4d-12)) then
        tmp = (x / t) * ((-2.0d0) / z_m)
    else if (t <= 3600000.0d0) then
        tmp = x * (2.0d0 / (z_m * y))
    else
        tmp = (x / z_m) * ((-2.0d0) / t)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -5.4e-12) {
		tmp = (x / t) * (-2.0 / z_m);
	} else if (t <= 3600000.0) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = (x / z_m) * (-2.0 / t);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= -5.4e-12:
		tmp = (x / t) * (-2.0 / z_m)
	elif t <= 3600000.0:
		tmp = x * (2.0 / (z_m * y))
	else:
		tmp = (x / z_m) * (-2.0 / t)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -5.4e-12)
		tmp = Float64(Float64(x / t) * Float64(-2.0 / z_m));
	elseif (t <= 3600000.0)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	else
		tmp = Float64(Float64(x / z_m) * Float64(-2.0 / t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= -5.4e-12)
		tmp = (x / t) * (-2.0 / z_m);
	elseif (t <= 3600000.0)
		tmp = x * (2.0 / (z_m * y));
	else
		tmp = (x / z_m) * (-2.0 / t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, -5.4e-12], N[(N[(x / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3600000.0], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -5.39999999999999961e-12

   1. Initial program 91.8%

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

      1. distribute-rgt-out-- 91.9%

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

      2. *-commutative 91.9%

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

      3. times-frac 95.8%

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

      4. associate-*l/ 95.9%

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

      5. associate-*r/ 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in y around 0 77.3%

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

      1. associate-*r/ 77.4%

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

      2. *-commutative 77.4%

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

      3. times-frac 83.7%

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

   7. Simplified 83.7%

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

   if -5.39999999999999961e-12 < t < 3.6e6

   1. Initial program 92.4%

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

      1. distribute-rgt-out-- 94.9%

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

      2. *-commutative 94.9%

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

      3. times-frac 92.1%

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

      4. associate-*l/ 92.1%

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

      5. associate-*r/ 92.1%

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

   3. Simplified 92.1%

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

   5. Taylor expanded in x around 0 94.9%

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

      1. associate-*r/ 94.9%

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

      2. *-commutative 94.9%

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

      3. *-commutative 94.9%

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

      4. associate-*r/ 94.9%

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

      5. associate-/r* 94.8%

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

      6. *-lft-identity 94.8%

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

      7. associate-*l/ 94.8%

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

      8. associate-*r/ 94.8%

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

      9. associate-*l/ 94.8%

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

      10. metadata-eval 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in y around inf 79.8%

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

      1. *-commutative 79.8%

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

   10. Simplified 79.8%

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

   if 3.6e6 < t

   1. Initial program 80.9%

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

      1. distribute-rgt-out-- 89.9%

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

      2. times-frac 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in y around 0 84.9%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;t \leq 3600000:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z\_m}\\ \mathbf{elif}\;t \leq 2400000:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{-2}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -1.6e-18)
    (* (/ x t) (/ -2.0 z_m))
    (if (<= t 2400000.0) (* (/ x z_m) (/ 2.0 y)) (* (/ x z_m) (/ -2.0 t))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.6e-18) {
		tmp = (x / t) * (-2.0 / z_m);
	} else if (t <= 2400000.0) {
		tmp = (x / z_m) * (2.0 / y);
	} else {
		tmp = (x / z_m) * (-2.0 / t);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.6d-18)) then
        tmp = (x / t) * ((-2.0d0) / z_m)
    else if (t <= 2400000.0d0) then
        tmp = (x / z_m) * (2.0d0 / y)
    else
        tmp = (x / z_m) * ((-2.0d0) / t)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -1.6e-18) {
		tmp = (x / t) * (-2.0 / z_m);
	} else if (t <= 2400000.0) {
		tmp = (x / z_m) * (2.0 / y);
	} else {
		tmp = (x / z_m) * (-2.0 / t);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= -1.6e-18:
		tmp = (x / t) * (-2.0 / z_m)
	elif t <= 2400000.0:
		tmp = (x / z_m) * (2.0 / y)
	else:
		tmp = (x / z_m) * (-2.0 / t)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -1.6e-18)
		tmp = Float64(Float64(x / t) * Float64(-2.0 / z_m));
	elseif (t <= 2400000.0)
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	else
		tmp = Float64(Float64(x / z_m) * Float64(-2.0 / t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= -1.6e-18)
		tmp = (x / t) * (-2.0 / z_m);
	elseif (t <= 2400000.0)
		tmp = (x / z_m) * (2.0 / y);
	else
		tmp = (x / z_m) * (-2.0 / t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, -1.6e-18], N[(N[(x / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2400000.0], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.6e-18

   1. Initial program 91.8%

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

      1. distribute-rgt-out-- 91.9%

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

      2. *-commutative 91.9%

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

      3. times-frac 95.8%

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

      4. associate-*l/ 95.9%

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

      5. associate-*r/ 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in y around 0 77.3%

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

      1. associate-*r/ 77.4%

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

      2. *-commutative 77.4%

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

      3. times-frac 83.7%

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

   7. Simplified 83.7%

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

   if -1.6e-18 < t < 2.4e6

   1. Initial program 92.4%

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

      1. distribute-rgt-out-- 94.9%

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

      2. times-frac 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in y around inf 83.2%

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

   if 2.4e6 < t

   1. Initial program 80.9%

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

      1. distribute-rgt-out-- 89.9%

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

      2. times-frac 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in y around 0 84.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;t \leq 2400000:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z\_m}\\ \mathbf{elif}\;t \leq 550000:\\ \;\;\;\;\frac{2}{\frac{y}{\frac{x}{z\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{-2}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -5.5e-30)
    (* (/ x t) (/ -2.0 z_m))
    (if (<= t 550000.0) (/ 2.0 (/ y (/ x z_m))) (* (/ x z_m) (/ -2.0 t))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -5.5e-30) {
		tmp = (x / t) * (-2.0 / z_m);
	} else if (t <= 550000.0) {
		tmp = 2.0 / (y / (x / z_m));
	} else {
		tmp = (x / z_m) * (-2.0 / t);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.5d-30)) then
        tmp = (x / t) * ((-2.0d0) / z_m)
    else if (t <= 550000.0d0) then
        tmp = 2.0d0 / (y / (x / z_m))
    else
        tmp = (x / z_m) * ((-2.0d0) / t)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -5.5e-30) {
		tmp = (x / t) * (-2.0 / z_m);
	} else if (t <= 550000.0) {
		tmp = 2.0 / (y / (x / z_m));
	} else {
		tmp = (x / z_m) * (-2.0 / t);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= -5.5e-30:
		tmp = (x / t) * (-2.0 / z_m)
	elif t <= 550000.0:
		tmp = 2.0 / (y / (x / z_m))
	else:
		tmp = (x / z_m) * (-2.0 / t)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -5.5e-30)
		tmp = Float64(Float64(x / t) * Float64(-2.0 / z_m));
	elseif (t <= 550000.0)
		tmp = Float64(2.0 / Float64(y / Float64(x / z_m)));
	else
		tmp = Float64(Float64(x / z_m) * Float64(-2.0 / t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= -5.5e-30)
		tmp = (x / t) * (-2.0 / z_m);
	elseif (t <= 550000.0)
		tmp = 2.0 / (y / (x / z_m));
	else
		tmp = (x / z_m) * (-2.0 / t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, -5.5e-30], N[(N[(x / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 550000.0], N[(2.0 / N[(y / N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -5.49999999999999976e-30

   1. Initial program 92.1%

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

      1. distribute-rgt-out-- 92.2%

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

      2. *-commutative 92.2%

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

      3. times-frac 95.4%

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

      4. associate-*l/ 95.5%

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

      5. associate-*r/ 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in y around 0 76.7%

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

      1. associate-*r/ 76.7%

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

      2. *-commutative 76.7%

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

      3. times-frac 82.9%

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

   7. Simplified 82.9%

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

   if -5.49999999999999976e-30 < t < 5.5e5

   1. Initial program 92.2%

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

      1. distribute-rgt-out-- 94.8%

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

      2. *-commutative 94.8%

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

      3. times-frac 92.3%

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

      4. associate-*l/ 92.3%

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

      5. associate-*r/ 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in y around inf 78.7%

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

      1. clear-num 78.7%

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

      2. un-div-inv 78.7%

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

      3. div-inv 78.6%

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

      4. clear-num 78.7%

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

   7. Applied egg-rr 78.7%

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

      1. associate-*r/ 80.3%

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

      2. *-commutative 80.3%

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

      3. associate-/l* 83.7%

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

   9. Applied egg-rr 83.7%

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

   if 5.5e5 < t

   1. Initial program 80.9%

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

      1. distribute-rgt-out-- 89.9%

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

      2. times-frac 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in y around 0 84.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-2}{z}\\ \mathbf{elif}\;t \leq 550000:\\ \;\;\;\;\frac{2}{\frac{y}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{t}}{z\_m}\\ \mathbf{elif}\;t \leq 600000:\\ \;\;\;\;\frac{2}{\frac{y}{\frac{x}{z\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{-2}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -4.5e-30)
    (/ (/ (* x -2.0) t) z_m)
    (if (<= t 600000.0) (/ 2.0 (/ y (/ x z_m))) (* (/ x z_m) (/ -2.0 t))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -4.5e-30) {
		tmp = ((x * -2.0) / t) / z_m;
	} else if (t <= 600000.0) {
		tmp = 2.0 / (y / (x / z_m));
	} else {
		tmp = (x / z_m) * (-2.0 / t);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-4.5d-30)) then
        tmp = ((x * (-2.0d0)) / t) / z_m
    else if (t <= 600000.0d0) then
        tmp = 2.0d0 / (y / (x / z_m))
    else
        tmp = (x / z_m) * ((-2.0d0) / t)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -4.5e-30) {
		tmp = ((x * -2.0) / t) / z_m;
	} else if (t <= 600000.0) {
		tmp = 2.0 / (y / (x / z_m));
	} else {
		tmp = (x / z_m) * (-2.0 / t);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= -4.5e-30:
		tmp = ((x * -2.0) / t) / z_m
	elif t <= 600000.0:
		tmp = 2.0 / (y / (x / z_m))
	else:
		tmp = (x / z_m) * (-2.0 / t)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -4.5e-30)
		tmp = Float64(Float64(Float64(x * -2.0) / t) / z_m);
	elseif (t <= 600000.0)
		tmp = Float64(2.0 / Float64(y / Float64(x / z_m)));
	else
		tmp = Float64(Float64(x / z_m) * Float64(-2.0 / t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= -4.5e-30)
		tmp = ((x * -2.0) / t) / z_m;
	elseif (t <= 600000.0)
		tmp = 2.0 / (y / (x / z_m));
	else
		tmp = (x / z_m) * (-2.0 / t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, -4.5e-30], N[(N[(N[(x * -2.0), $MachinePrecision] / t), $MachinePrecision] / z$95$m), $MachinePrecision], If[LessEqual[t, 600000.0], N[(2.0 / N[(y / N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -4.49999999999999967e-30

   1. Initial program 92.1%

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

      1. distribute-rgt-out-- 92.2%

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

      2. times-frac 92.0%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in y around 0 76.6%

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

      1. associate-*l/ 83.0%

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

      2. associate-*r/ 83.0%

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

   7. Applied egg-rr 83.0%

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

   if -4.49999999999999967e-30 < t < 6e5

   1. Initial program 92.2%

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

      1. distribute-rgt-out-- 94.8%

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

      2. *-commutative 94.8%

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

      3. times-frac 92.3%

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

      4. associate-*l/ 92.3%

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

      5. associate-*r/ 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in y around inf 78.7%

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

      1. clear-num 78.7%

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

      2. un-div-inv 78.7%

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

      3. div-inv 78.6%

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

      4. clear-num 78.7%

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

   7. Applied egg-rr 78.7%

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

      1. associate-*r/ 80.3%

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

      2. *-commutative 80.3%

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

      3. associate-/l* 83.7%

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

   9. Applied egg-rr 83.7%

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

   if 6e5 < t

   1. Initial program 80.9%

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

      1. distribute-rgt-out-- 89.9%

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

      2. times-frac 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in y around 0 84.9%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{t}}{z}\\ \mathbf{elif}\;t \leq 600000:\\ \;\;\;\;\frac{2}{\frac{y}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 1000000:\\ \;\;\;\;x \cdot \frac{\frac{2}{z\_m}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y - t}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= (* x 2.0) 1000000.0)
    (* x (/ (/ 2.0 z_m) (- y t)))
    (* 2.0 (/ (/ x (- y t)) z_m)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((x * 2.0) <= 1000000.0) {
		tmp = x * ((2.0 / z_m) / (y - t));
	} else {
		tmp = 2.0 * ((x / (y - t)) / z_m);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x * 2.0d0) <= 1000000.0d0) then
        tmp = x * ((2.0d0 / z_m) / (y - t))
    else
        tmp = 2.0d0 * ((x / (y - t)) / z_m)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((x * 2.0) <= 1000000.0) {
		tmp = x * ((2.0 / z_m) / (y - t));
	} else {
		tmp = 2.0 * ((x / (y - t)) / z_m);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (x * 2.0) <= 1000000.0:
		tmp = x * ((2.0 / z_m) / (y - t))
	else:
		tmp = 2.0 * ((x / (y - t)) / z_m)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (Float64(x * 2.0) <= 1000000.0)
		tmp = Float64(x * Float64(Float64(2.0 / z_m) / Float64(y - t)));
	else
		tmp = Float64(2.0 * Float64(Float64(x / Float64(y - t)) / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((x * 2.0) <= 1000000.0)
		tmp = x * ((2.0 / z_m) / (y - t));
	else
		tmp = 2.0 * ((x / (y - t)) / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[N[(x * 2.0), $MachinePrecision], 1000000.0], N[(x * N[(N[(2.0 / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x 2) < 1e6

   1. Initial program 94.0%

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

      1. distribute-rgt-out-- 98.3%

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

      2. *-commutative 98.3%

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

      3. times-frac 91.5%

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

      4. associate-*l/ 91.6%

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

      5. associate-*r/ 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x around 0 98.3%

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

      1. associate-*r/ 98.3%

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

      2. *-commutative 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*r/ 98.2%

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

      5. associate-/r* 98.2%

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

      6. *-lft-identity 98.2%

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

      7. associate-*l/ 98.2%

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

      8. associate-*r/ 98.2%

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

      9. associate-*l/ 98.2%

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

      10. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

   if 1e6 < (*.f64 x 2)

   1. Initial program 76.0%

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

      1. distribute-rgt-out-- 77.5%

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

      2. *-commutative 77.5%

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

      3. times-frac 96.8%

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

      4. associate-*l/ 96.9%

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

      5. associate-*r/ 96.9%

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

   3. Simplified 96.9%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 1000000:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y - t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.1 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z\_m}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y - t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2.1e+35)
    (* x (/ (/ 2.0 z_m) (- y t)))
    (* (/ x z_m) (/ 2.0 (- y t))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.1e+35) {
		tmp = x * ((2.0 / z_m) / (y - t));
	} else {
		tmp = (x / z_m) * (2.0 / (y - t));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 2.1d+35) then
        tmp = x * ((2.0d0 / z_m) / (y - t))
    else
        tmp = (x / z_m) * (2.0d0 / (y - t))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.1e+35) {
		tmp = x * ((2.0 / z_m) / (y - t));
	} else {
		tmp = (x / z_m) * (2.0 / (y - t));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 2.1e+35:
		tmp = x * ((2.0 / z_m) / (y - t))
	else:
		tmp = (x / z_m) * (2.0 / (y - t))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2.1e+35)
		tmp = Float64(x * Float64(Float64(2.0 / z_m) / Float64(y - t)));
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 2.1e+35)
		tmp = x * ((2.0 / z_m) / (y - t));
	else
		tmp = (x / z_m) * (2.0 / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 2.1e+35], N[(x * N[(N[(2.0 / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.0999999999999999e35

   1. Initial program 92.7%

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

      1. distribute-rgt-out-- 96.0%

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

      2. *-commutative 96.0%

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

      3. times-frac 91.6%

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

      4. associate-*l/ 91.6%

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

      5. associate-*r/ 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in x around 0 96.0%

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

      1. associate-*r/ 96.0%

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

      2. *-commutative 96.0%

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

      3. *-commutative 96.0%

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

      4. associate-*r/ 95.9%

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

      5. associate-/r* 95.9%

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

      6. *-lft-identity 95.9%

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

      7. associate-*l/ 95.9%

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

      8. associate-*r/ 95.9%

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

      9. associate-*l/ 95.9%

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

      10. metadata-eval 95.9%

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

   7. Simplified 95.9%

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

   if 2.0999999999999999e35 < z

   1. Initial program 79.0%

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

      1. distribute-rgt-out-- 83.2%

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

      2. times-frac 98.7%

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

   3. Simplified 98.7%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y - t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2e+35)
    (/ (* x 2.0) (* z_m (- y t)))
    (* (/ x z_m) (/ 2.0 (- y t))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2e+35) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x / z_m) * (2.0 / (y - t));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 2d+35) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else
        tmp = (x / z_m) * (2.0d0 / (y - t))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2e+35) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x / z_m) * (2.0 / (y - t));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 2e+35:
		tmp = (x * 2.0) / (z_m * (y - t))
	else:
		tmp = (x / z_m) * (2.0 / (y - t))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2e+35)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 2e+35)
		tmp = (x * 2.0) / (z_m * (y - t));
	else
		tmp = (x / z_m) * (2.0 / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 2e+35], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.9999999999999999e35

   1. Initial program 92.7%

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

      1. distribute-rgt-out-- 96.0%

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

   3. Simplified 96.0%

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

   if 1.9999999999999999e35 < z

   1. Initial program 79.0%

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

      1. distribute-rgt-out-- 83.2%

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

      2. times-frac 98.7%

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

   3. Simplified 98.7%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (* z_s (if (<= z_m 8e-84) (* x (/ 2.0 (* z_m y))) (* 2.0 (/ (/ x y) z_m)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 8e-84) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = 2.0 * ((x / y) / z_m);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 8d-84) then
        tmp = x * (2.0d0 / (z_m * y))
    else
        tmp = 2.0d0 * ((x / y) / z_m)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 8e-84) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = 2.0 * ((x / y) / z_m);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 8e-84:
		tmp = x * (2.0 / (z_m * y))
	else:
		tmp = 2.0 * ((x / y) / z_m)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 8e-84)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	else
		tmp = Float64(2.0 * Float64(Float64(x / y) / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 8e-84)
		tmp = x * (2.0 / (z_m * y));
	else
		tmp = 2.0 * ((x / y) / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 8e-84], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.0000000000000003e-84

   1. Initial program 91.6%

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

      1. distribute-rgt-out-- 95.4%

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

      2. *-commutative 95.4%

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

      3. times-frac 90.3%

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

      4. associate-*l/ 90.4%

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

      5. associate-*r/ 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in x around 0 95.4%

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

      1. associate-*r/ 95.4%

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

      2. *-commutative 95.4%

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

      3. *-commutative 95.4%

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

      4. associate-*r/ 95.3%

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

      5. associate-/r* 95.3%

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

      6. *-lft-identity 95.3%

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

      7. associate-*l/ 95.3%

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

      8. associate-*r/ 95.3%

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

      9. associate-*l/ 95.3%

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

      10. metadata-eval 95.3%

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

   7. Simplified 95.3%

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

   8. Taylor expanded in y around inf 55.9%

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

      1. *-commutative 55.9%

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

   10. Simplified 55.9%

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

   if 8.0000000000000003e-84 < z

   1. Initial program 85.0%

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

      1. distribute-rgt-out-- 88.0%

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

      2. *-commutative 88.0%

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

      3. times-frac 97.6%

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

      4. associate-*l/ 97.7%

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

      5. associate-*r/ 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in y around inf 51.9%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * N[(2.0 * N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

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

   1. distribute-rgt-out-- 92.8%

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

   2. *-commutative 92.8%

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

   3. times-frac 92.9%

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

   4. associate-*l/ 93.0%

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

   5. associate-*r/ 93.0%

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

3. Simplified 93.0%

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

5. Final simplification 93.0%

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

Alternative 12: 54.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t)
	tmp = z_s * (2.0 * ((x / y) / z_m));
end

Wolfram

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_, y_, z$95$m_, t_] := N[(z$95$s * N[(2.0 * N[(N[(x / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

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

   1. distribute-rgt-out-- 92.8%

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

   2. *-commutative 92.8%

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

   3. times-frac 92.9%

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

   4. associate-*l/ 93.0%

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

   5. associate-*r/ 93.0%

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

3. Simplified 93.0%

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

5. Taylor expanded in y around inf 52.8%

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

6. Final simplification 52.8%

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

Developer target: 97.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024034 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

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

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