Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 90.2% → 96.9%

Time: 14.3s

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: 90.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{2}{y - t}}{z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot x}{y - t}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 4e+14)
    (* (/ (/ 2.0 (- y t)) z_m) x)
    (/ (/ (* 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 (z_m <= 4e+14) {
		tmp = ((2.0 / (y - t)) / z_m) * x;
	} 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 (z_m <= 4d+14) then
        tmp = ((2.0d0 / (y - t)) / z_m) * x
    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 (z_m <= 4e+14) {
		tmp = ((2.0 / (y - t)) / z_m) * x;
	} 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 z_m <= 4e+14:
		tmp = ((2.0 / (y - t)) / z_m) * x
	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 (z_m <= 4e+14)
		tmp = Float64(Float64(Float64(2.0 / Float64(y - t)) / z_m) * x);
	else
		tmp = Float64(Float64(Float64(2.0 * 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 (z_m <= 4e+14)
		tmp = ((2.0 / (y - t)) / z_m) * x;
	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[z$95$m, 4e+14], N[(N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(2.0 * x), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4e14

   1. Initial program 93.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 94.6

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

   4. Applied egg-rr 94.6%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 94.9

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

   6. Applied egg-rr 94.9%

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

   if 4e14 < z

   1. Initial program 87.3%

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 98.5

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

   4. Applied egg-rr 98.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{2}{y - t}}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot x}{y - t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot y - z\_m \cdot t \leq 5 \cdot 10^{+188}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= (- (* z_m y) (* z_m t)) 5e+188)
    (* x (/ 2.0 (* z_m (- y t))))
    (* (/ 2.0 (- y t)) (/ x 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 * y) - (z_m * t)) <= 5e+188) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (2.0 / (y - t)) * (x / 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 * y) - (z_m * t)) <= 5d+188) then
        tmp = x * (2.0d0 / (z_m * (y - t)))
    else
        tmp = (2.0d0 / (y - t)) * (x / 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 * y) - (z_m * t)) <= 5e+188) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (2.0 / (y - t)) * (x / 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 * y) - (z_m * t)) <= 5e+188:
		tmp = x * (2.0 / (z_m * (y - t)))
	else:
		tmp = (2.0 / (y - t)) * (x / 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(Float64(z_m * y) - Float64(z_m * t)) <= 5e+188)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(Float64(2.0 / Float64(y - t)) * Float64(x / 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 * y) - (z_m * t)) <= 5e+188)
		tmp = x * (2.0 / (z_m * (y - t)));
	else
		tmp = (2.0 / (y - t)) * (x / 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[(N[(z$95$m * y), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision], 5e+188], N[(x * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < 5.0000000000000001e188

   1. Initial program 93.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 94.0

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

   4. Applied egg-rr 94.0%

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

   if 5.0000000000000001e188 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 83.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 86.1

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

   4. Applied egg-rr 86.1%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 86.1

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

   6. Applied egg-rr 86.1%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 99.9

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y - z \cdot t \leq 5 \cdot 10^{+188}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot y - z\_m \cdot t \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z\_m \cdot 0.5}}{y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= (- (* z_m y) (* z_m t)) 5e+301)
    (* x (/ 2.0 (* z_m (- y t))))
    (/ (/ x (* z_m 0.5)) 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 (((z_m * y) - (z_m * t)) <= 5e+301) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x / (z_m * 0.5)) / 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 (((z_m * y) - (z_m * t)) <= 5d+301) then
        tmp = x * (2.0d0 / (z_m * (y - t)))
    else
        tmp = (x / (z_m * 0.5d0)) / 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 (((z_m * y) - (z_m * t)) <= 5e+301) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x / (z_m * 0.5)) / 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 ((z_m * y) - (z_m * t)) <= 5e+301:
		tmp = x * (2.0 / (z_m * (y - t)))
	else:
		tmp = (x / (z_m * 0.5)) / 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 (Float64(Float64(z_m * y) - Float64(z_m * t)) <= 5e+301)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(Float64(x / Float64(z_m * 0.5)) / 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 (((z_m * y) - (z_m * t)) <= 5e+301)
		tmp = x * (2.0 / (z_m * (y - t)));
	else
		tmp = (x / (z_m * 0.5)) / 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[LessEqual[N[(N[(z$95$m * y), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision], 5e+301], N[(x * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < 5.0000000000000004e301

   1. Initial program 94.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 94.4

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

   4. Applied egg-rr 94.4%

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

   if 5.0000000000000004e301 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 73.1%

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 50.8

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

   7. Simplified 50.8%

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

      1. associate-/r* N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. *-commutative N/A

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

      5. distribute-neg-frac N/A

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

      6. /-lowering-/.f64 N/A

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

      7. distribute-neg-frac2 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

      13. /-lowering-/.f64 N/A

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

      14. div-inv N/A

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

      15. *-lowering-*.f64 N/A

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

      16. metadata-eval 76.8

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

   9. Applied egg-rr 76.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot 0.5}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y - z \cdot t \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z \cdot 0.5}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot y - z\_m \cdot t \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{y}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= (- (* z_m y) (* z_m t)) 5e+301)
    (* x (/ 2.0 (* z_m (- y t))))
    (/ (* x (/ 2.0 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 * y) - (z_m * t)) <= 5e+301) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x * (2.0 / 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 * y) - (z_m * t)) <= 5d+301) then
        tmp = x * (2.0d0 / (z_m * (y - t)))
    else
        tmp = (x * (2.0d0 / 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 * y) - (z_m * t)) <= 5e+301) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x * (2.0 / 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 * y) - (z_m * t)) <= 5e+301:
		tmp = x * (2.0 / (z_m * (y - t)))
	else:
		tmp = (x * (2.0 / 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 (Float64(Float64(z_m * y) - Float64(z_m * t)) <= 5e+301)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(Float64(x * Float64(2.0 / 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 * y) - (z_m * t)) <= 5e+301)
		tmp = x * (2.0 / (z_m * (y - t)));
	else
		tmp = (x * (2.0 / 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[N[(N[(z$95$m * y), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision], 5e+301], N[(x * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(2.0 / y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < 5.0000000000000004e301

   1. Initial program 94.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 94.4

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

   4. Applied egg-rr 94.4%

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

   if 5.0000000000000004e301 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 73.1%

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

      1. distribute-rgt-out-- N/A

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

      2. times-frac N/A

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

      3. associate-*l/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 69.4%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y - z \cdot t \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{2}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \cdot y - z\_m \cdot t \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= (- (* z_m y) (* z_m t)) 5e+301)
    (* x (/ 2.0 (* z_m (- y t))))
    (/ 2.0 (* z_m (/ y x))))))

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 * y) - (z_m * t)) <= 5e+301) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = 2.0 / (z_m * (y / x));
	}
	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 * y) - (z_m * t)) <= 5d+301) then
        tmp = x * (2.0d0 / (z_m * (y - t)))
    else
        tmp = 2.0d0 / (z_m * (y / x))
    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 * y) - (z_m * t)) <= 5e+301) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = 2.0 / (z_m * (y / x));
	}
	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 * y) - (z_m * t)) <= 5e+301:
		tmp = x * (2.0 / (z_m * (y - t)))
	else:
		tmp = 2.0 / (z_m * (y / x))
	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(Float64(z_m * y) - Float64(z_m * t)) <= 5e+301)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(2.0 / Float64(z_m * Float64(y / x)));
	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 * y) - (z_m * t)) <= 5e+301)
		tmp = x * (2.0 / (z_m * (y - t)));
	else
		tmp = 2.0 / (z_m * (y / x));
	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[(N[(z$95$m * y), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision], 5e+301], N[(x * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(z$95$m * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < 5.0000000000000004e301

   1. Initial program 94.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 94.4

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

   4. Applied egg-rr 94.4%

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

   if 5.0000000000000004e301 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 73.1%

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 50.8

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

   7. Simplified 50.8%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 69.3

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

   9. Applied egg-rr 69.3%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y - z \cdot t \leq 5 \cdot 10^{+301}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.2 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{2}{y - t}}{z\_m} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 4.2e+18)
    (* (/ (/ 2.0 (- y t)) z_m) x)
    (* (/ x (- y t)) (/ 2.0 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 <= 4.2e+18) {
		tmp = ((2.0 / (y - t)) / z_m) * x;
	} else {
		tmp = (x / (y - t)) * (2.0 / 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 <= 4.2d+18) then
        tmp = ((2.0d0 / (y - t)) / z_m) * x
    else
        tmp = (x / (y - t)) * (2.0d0 / 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 <= 4.2e+18) {
		tmp = ((2.0 / (y - t)) / z_m) * x;
	} else {
		tmp = (x / (y - t)) * (2.0 / 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 <= 4.2e+18:
		tmp = ((2.0 / (y - t)) / z_m) * x
	else:
		tmp = (x / (y - t)) * (2.0 / 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 <= 4.2e+18)
		tmp = Float64(Float64(Float64(2.0 / Float64(y - t)) / z_m) * x);
	else
		tmp = Float64(Float64(x / Float64(y - t)) * Float64(2.0 / 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 <= 4.2e+18)
		tmp = ((2.0 / (y - t)) / z_m) * x;
	else
		tmp = (x / (y - t)) * (2.0 / 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, 4.2e+18], N[(N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.2e18

   1. Initial program 93.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 94.7

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

   4. Applied egg-rr 94.7%

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

      1. associate-/l/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. /-lowering-/.f64 N/A

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

      4. --lowering--.f64 94.9

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

   6. Applied egg-rr 94.9%

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

   if 4.2e18 < z

   1. Initial program 87.1%

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 98.4

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

   4. 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: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 9.5e+14)
    (* x (/ 2.0 (* z_m (- y t))))
    (* (/ x (- y t)) (/ 2.0 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 <= 9.5e+14) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x / (y - t)) * (2.0 / 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 <= 9.5d+14) then
        tmp = x * (2.0d0 / (z_m * (y - t)))
    else
        tmp = (x / (y - t)) * (2.0d0 / 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 <= 9.5e+14) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x / (y - t)) * (2.0 / 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 <= 9.5e+14:
		tmp = x * (2.0 / (z_m * (y - t)))
	else:
		tmp = (x / (y - t)) * (2.0 / 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 <= 9.5e+14)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(Float64(x / Float64(y - t)) * Float64(2.0 / 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 <= 9.5e+14)
		tmp = x * (2.0 / (z_m * (y - t)));
	else
		tmp = (x / (y - t)) * (2.0 / 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, 9.5e+14], N[(x * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.5e14

   1. Initial program 93.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 94.6

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

   4. Applied egg-rr 94.6%

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

   if 9.5e14 < z

   1. Initial program 87.3%

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. /-lowering-/.f64 98.4

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

   4. Applied egg-rr 98.4%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+72}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+39}:\\ \;\;\;\;\frac{x \cdot -2}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{z\_m \cdot y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y -1.2e+72)
    (* x (/ 2.0 (* z_m y)))
    (if (<= y 1.25e+39) (/ (* x -2.0) (* z_m t)) (/ (* 2.0 x) (* 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 (y <= -1.2e+72) {
		tmp = x * (2.0 / (z_m * y));
	} else if (y <= 1.25e+39) {
		tmp = (x * -2.0) / (z_m * t);
	} else {
		tmp = (2.0 * x) / (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 (y <= (-1.2d+72)) then
        tmp = x * (2.0d0 / (z_m * y))
    else if (y <= 1.25d+39) then
        tmp = (x * (-2.0d0)) / (z_m * t)
    else
        tmp = (2.0d0 * x) / (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 (y <= -1.2e+72) {
		tmp = x * (2.0 / (z_m * y));
	} else if (y <= 1.25e+39) {
		tmp = (x * -2.0) / (z_m * t);
	} else {
		tmp = (2.0 * x) / (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 y <= -1.2e+72:
		tmp = x * (2.0 / (z_m * y))
	elif y <= 1.25e+39:
		tmp = (x * -2.0) / (z_m * t)
	else:
		tmp = (2.0 * x) / (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 (y <= -1.2e+72)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	elseif (y <= 1.25e+39)
		tmp = Float64(Float64(x * -2.0) / Float64(z_m * t));
	else
		tmp = Float64(Float64(2.0 * x) / 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 (y <= -1.2e+72)
		tmp = x * (2.0 / (z_m * y));
	elseif (y <= 1.25e+39)
		tmp = (x * -2.0) / (z_m * t);
	else
		tmp = (2.0 * x) / (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[LessEqual[y, -1.2e+72], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+39], N[(N[(x * -2.0), $MachinePrecision] / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * x), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.20000000000000005e72

   1. Initial program 92.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 94.9

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

   4. Applied egg-rr 94.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 80.9%

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

   if -1.20000000000000005e72 < y < 1.25000000000000004e39

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 76.7

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

   5. Simplified 76.7%

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

   if 1.25000000000000004e39 < y

   1. Initial program 84.1%

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 88.7

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

   4. Applied egg-rr 88.7%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 76.6

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

   7. Simplified 76.6%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+72}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+39}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+73}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{-2}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{z\_m \cdot y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y -1.7e+73)
    (* x (/ 2.0 (* z_m y)))
    (if (<= y 1.28e+39) (* x (/ -2.0 (* z_m t))) (/ (* 2.0 x) (* 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 (y <= -1.7e+73) {
		tmp = x * (2.0 / (z_m * y));
	} else if (y <= 1.28e+39) {
		tmp = x * (-2.0 / (z_m * t));
	} else {
		tmp = (2.0 * x) / (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 (y <= (-1.7d+73)) then
        tmp = x * (2.0d0 / (z_m * y))
    else if (y <= 1.28d+39) then
        tmp = x * ((-2.0d0) / (z_m * t))
    else
        tmp = (2.0d0 * x) / (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 (y <= -1.7e+73) {
		tmp = x * (2.0 / (z_m * y));
	} else if (y <= 1.28e+39) {
		tmp = x * (-2.0 / (z_m * t));
	} else {
		tmp = (2.0 * x) / (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 y <= -1.7e+73:
		tmp = x * (2.0 / (z_m * y))
	elif y <= 1.28e+39:
		tmp = x * (-2.0 / (z_m * t))
	else:
		tmp = (2.0 * x) / (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 (y <= -1.7e+73)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	elseif (y <= 1.28e+39)
		tmp = Float64(x * Float64(-2.0 / Float64(z_m * t)));
	else
		tmp = Float64(Float64(2.0 * x) / 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 (y <= -1.7e+73)
		tmp = x * (2.0 / (z_m * y));
	elseif (y <= 1.28e+39)
		tmp = x * (-2.0 / (z_m * t));
	else
		tmp = (2.0 * x) / (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[LessEqual[y, -1.7e+73], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.28e+39], N[(x * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * x), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.7000000000000001e73

   1. Initial program 92.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 94.9

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

   4. Applied egg-rr 94.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 80.9%

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

   if -1.7000000000000001e73 < y < 1.27999999999999994e39

   1. Initial program 94.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 95.0

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

   4. Applied egg-rr 95.0%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 76.7

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

   7. Simplified 76.7%

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

   if 1.27999999999999994e39 < y

   1. Initial program 84.1%

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

      1. distribute-rgt-out-- N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 88.7

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

   4. Applied egg-rr 88.7%

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

   5. Taylor expanded in y around inf

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

      1. associate-*r/ N/A

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

      2. /-lowering-/.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 76.6

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

   7. Simplified 76.6%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+73}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;y \leq 1.28 \cdot 10^{+39}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot x}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := x \cdot \frac{2}{z\_m \cdot y}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 10^{+39}:\\ \;\;\;\;x \cdot \frac{-2}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (* x (/ 2.0 (* z_m y)))))
   (*
    z_s
    (if (<= y -1.1e+72) t_1 (if (<= y 1e+39) (* x (/ -2.0 (* z_m t))) t_1)))))

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 t_1 = x * (2.0 / (z_m * y));
	double tmp;
	if (y <= -1.1e+72) {
		tmp = t_1;
	} else if (y <= 1e+39) {
		tmp = x * (-2.0 / (z_m * t));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x * (2.0d0 / (z_m * y))
    if (y <= (-1.1d+72)) then
        tmp = t_1
    else if (y <= 1d+39) then
        tmp = x * ((-2.0d0) / (z_m * t))
    else
        tmp = t_1
    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 t_1 = x * (2.0 / (z_m * y));
	double tmp;
	if (y <= -1.1e+72) {
		tmp = t_1;
	} else if (y <= 1e+39) {
		tmp = x * (-2.0 / (z_m * t));
	} else {
		tmp = t_1;
	}
	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):
	t_1 = x * (2.0 / (z_m * y))
	tmp = 0
	if y <= -1.1e+72:
		tmp = t_1
	elif y <= 1e+39:
		tmp = x * (-2.0 / (z_m * t))
	else:
		tmp = t_1
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	t_1 = Float64(x * Float64(2.0 / Float64(z_m * y)))
	tmp = 0.0
	if (y <= -1.1e+72)
		tmp = t_1;
	elseif (y <= 1e+39)
		tmp = Float64(x * Float64(-2.0 / Float64(z_m * t)));
	else
		tmp = t_1;
	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)
	t_1 = x * (2.0 / (z_m * y));
	tmp = 0.0;
	if (y <= -1.1e+72)
		tmp = t_1;
	elseif (y <= 1e+39)
		tmp = x * (-2.0 / (z_m * t));
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[y, -1.1e+72], t$95$1, If[LessEqual[y, 1e+39], N[(x * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.1e72 or 9.9999999999999994e38 < y

   1. Initial program 87.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 88.8

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

   4. Applied egg-rr 88.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 78.4%

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

   if -1.1e72 < y < 9.9999999999999994e38

   1. Initial program 94.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. distribute-rgt-out-- N/A

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

      6. *-lowering-*.f64 N/A

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

      7. --lowering--.f64 95.0

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

   4. Applied egg-rr 95.0%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 76.7

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

   7. Simplified 76.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+72}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;y \leq 10^{+39}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.3% accurate, 1.2× speedup

Math

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

FPCore

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

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 * (x * (2.0d0 / (z_m * (y - t))))
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 * (x * (2.0 / (z_m * (y - t))));
}

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

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(x * Float64(2.0 / Float64(z_m * Float64(y - t)))))
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 * (x * (2.0 / (z_m * (y - t))));
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[(x * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.9%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. distribute-rgt-out-- N/A

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

   6. *-lowering-*.f64 N/A

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

   7. --lowering--.f64 92.7

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

4. Applied egg-rr 92.7%

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

5. Final simplification 92.7%

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

Alternative 12: 52.7% accurate, 1.4× speedup

Math

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

FPCore

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

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 * (x * ((-2.0d0) / (z_m * t)))
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 * (x * (-2.0 / (z_m * t)));
}

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 * (x * (-2.0 / (z_m * t)))

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(x * Float64(-2.0 / Float64(z_m * t))))
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 * (x * (-2.0 / (z_m * t)));
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[(x * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.9%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. distribute-rgt-out-- N/A

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

   6. *-lowering-*.f64 N/A

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

   7. --lowering--.f64 92.7

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

4. Applied egg-rr 92.7%

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

5. Taylor expanded in y around 0

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

   1. /-lowering-/.f64 N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 57.4

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

7. Simplified 57.4%

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

8. Final simplification 57.4%

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

Developer Target 1: 97.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024196 
(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))))