Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.8% → 96.0%

Time: 14.4s

Alternatives: 16

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.8% 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: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 4 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{x\_m \cdot 2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\frac{y - t}{2}}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 4e+92)
    (/ (/ (* x_m 2.0) z) (- y t))
    (/ (/ x_m (/ (- y t) 2.0)) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+92) {
		tmp = ((x_m * 2.0) / z) / (y - t);
	} else {
		tmp = (x_m / ((y - t) / 2.0)) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 4d+92) then
        tmp = ((x_m * 2.0d0) / z) / (y - t)
    else
        tmp = (x_m / ((y - t) / 2.0d0)) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+92) {
		tmp = ((x_m * 2.0) / z) / (y - t);
	} else {
		tmp = (x_m / ((y - t) / 2.0)) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 4e+92:
		tmp = ((x_m * 2.0) / z) / (y - t)
	else:
		tmp = (x_m / ((y - t) / 2.0)) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 4e+92)
		tmp = Float64(Float64(Float64(x_m * 2.0) / z) / Float64(y - t));
	else
		tmp = Float64(Float64(x_m / Float64(Float64(y - t) / 2.0)) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 4e+92)
		tmp = ((x_m * 2.0) / z) / (y - t);
	else
		tmp = (x_m / ((y - t) / 2.0)) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 4e+92], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(N[(y - t), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.0000000000000002e92

   1. Initial program 91.9%

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

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

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

      2. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot 2}{z}\right), \color{blue}{\left(y - t\right)}\right) \]

      4. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot 2}{1}}{z}\right), \left(y - t\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{x \cdot 2}{\mathsf{neg}\left(-1\right)}}{z}\right), \left(y - t\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot 2}{\mathsf{neg}\left(-1\right)}\right), z\right), \left(\color{blue}{y} - t\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{x \cdot 2}{1}\right), z\right), \left(y - t\right)\right) \]

      8. /-rgt-identity N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot 2\right), z\right), \left(y - t\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), z\right), \left(y - t\right)\right) \]

      10. --lowering--.f64 94.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), z\right), \mathsf{\_.f64}\left(y, \color{blue}{t}\right)\right) \]

   3. Simplified 94.9%

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

   if 4.0000000000000002e92 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 86.5%

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

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

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. associate-/r* N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{\frac{1}{\frac{{y}^{3} - {t}^{3}}{y \cdot y + \left(t \cdot t + y \cdot t\right)}}}{\frac{z}{x \cdot 2}} \]

      7. clear-num N/A

         \[\leadsto \frac{\frac{y \cdot y + \left(t \cdot t + y \cdot t\right)}{{y}^{3} - {t}^{3}}}{\frac{\color{blue}{z}}{x \cdot 2}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y + \left(t \cdot t + y \cdot t\right)}{{y}^{3} - {t}^{3}}\right), \color{blue}{\left(\frac{z}{x \cdot 2}\right)}\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{{y}^{3} - {t}^{3}}{y \cdot y + \left(t \cdot t + y \cdot t\right)}}\right), \left(\frac{\color{blue}{z}}{x \cdot 2}\right)\right) \]

      10. flip3-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{y - t}\right), \left(\frac{z}{x \cdot 2}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(y - t\right)\right), \left(\frac{\color{blue}{z}}{x \cdot 2}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{z}{x \cdot 2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \color{blue}{\left(x \cdot 2\right)}\right)\right) \]

      14. *-lowering-*.f64 85.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   4. Applied egg-rr 85.7%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{y - t} \cdot x\right), \color{blue}{z}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{2}{y - t}\right), z\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y - t}{2}}\right), z\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{y - t}{2}}\right), z\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y - t}{2}\right)\right), z\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(y - t\right), 2\right)\right), z\right) \]

      13. --lowering--.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, t\right), 2\right)\right), z\right) \]

   6. Applied egg-rr 99.6%

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

Alternative 2: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{\frac{x\_m \cdot 2}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{-2}{t}}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (/ (* x_m 2.0) y) z)))
   (*
    x_s
    (if (<= y -4.2e-53)
      t_1
      (if (<= y 1.9e-15) (/ (/ -2.0 t) (/ z x_m)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = ((x_m * 2.0) / y) / z;
	double tmp;
	if (y <= -4.2e-53) {
		tmp = t_1;
	} else if (y <= 1.9e-15) {
		tmp = (-2.0 / t) / (z / x_m);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((x_m * 2.0d0) / y) / z
    if (y <= (-4.2d-53)) then
        tmp = t_1
    else if (y <= 1.9d-15) then
        tmp = ((-2.0d0) / t) / (z / x_m)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = ((x_m * 2.0) / y) / z;
	double tmp;
	if (y <= -4.2e-53) {
		tmp = t_1;
	} else if (y <= 1.9e-15) {
		tmp = (-2.0 / t) / (z / x_m);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = ((x_m * 2.0) / y) / z
	tmp = 0
	if y <= -4.2e-53:
		tmp = t_1
	elif y <= 1.9e-15:
		tmp = (-2.0 / t) / (z / x_m)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(Float64(x_m * 2.0) / y) / z)
	tmp = 0.0
	if (y <= -4.2e-53)
		tmp = t_1;
	elseif (y <= 1.9e-15)
		tmp = Float64(Float64(-2.0 / t) / Float64(z / x_m));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = ((x_m * 2.0) / y) / z;
	tmp = 0.0;
	if (y <= -4.2e-53)
		tmp = t_1;
	elseif (y <= 1.9e-15)
		tmp = (-2.0 / t) / (z / x_m);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x$95$m * 2.0), $MachinePrecision] / y), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -4.2e-53], t$95$1, If[LessEqual[y, 1.9e-15], N[(N[(-2.0 / t), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.19999999999999955e-53 or 1.9000000000000001e-15 < y

   1. Initial program 90.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \color{blue}{\left(y \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 80.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 80.2%

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

      1. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x \cdot 2}{y}\right), \color{blue}{z}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot 2\right), y\right), z\right) \]

      4. *-lowering-*.f64 83.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), y\right), z\right) \]

   7. Applied egg-rr 83.3%

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

   if -4.19999999999999955e-53 < y < 1.9000000000000001e-15

   1. Initial program 91.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot \left(y - t\right)}\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(y - t\right) \cdot z}\right), x\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), z\right), x\right) \]

      9. --lowering--.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), z\right), x\right) \]

   4. Applied egg-rr 91.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-2}{t \cdot z}\right)}, x\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(t \cdot z\right)\right), x\right) \]

      2. *-lowering-*.f64 81.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(t, z\right)\right), x\right) \]

   7. Simplified 81.5%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{t}\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, t\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      7. /-lowering-/.f64 84.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, t\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 84.1%

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

Alternative 3: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-54}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{-2}{t}}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -8.2e-54)
    (* x_m (/ (/ 2.0 y) z))
    (if (<= y 1.22e-14) (/ (/ -2.0 t) (/ z x_m)) (/ (* x_m 2.0) (* z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -8.2e-54) {
		tmp = x_m * ((2.0 / y) / z);
	} else if (y <= 1.22e-14) {
		tmp = (-2.0 / t) / (z / x_m);
	} else {
		tmp = (x_m * 2.0) / (z * y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-8.2d-54)) then
        tmp = x_m * ((2.0d0 / y) / z)
    else if (y <= 1.22d-14) then
        tmp = ((-2.0d0) / t) / (z / x_m)
    else
        tmp = (x_m * 2.0d0) / (z * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -8.2e-54) {
		tmp = x_m * ((2.0 / y) / z);
	} else if (y <= 1.22e-14) {
		tmp = (-2.0 / t) / (z / x_m);
	} else {
		tmp = (x_m * 2.0) / (z * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -8.2e-54:
		tmp = x_m * ((2.0 / y) / z)
	elif y <= 1.22e-14:
		tmp = (-2.0 / t) / (z / x_m)
	else:
		tmp = (x_m * 2.0) / (z * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -8.2e-54)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z));
	elseif (y <= 1.22e-14)
		tmp = Float64(Float64(-2.0 / t) / Float64(z / x_m));
	else
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -8.2e-54)
		tmp = x_m * ((2.0 / y) / z);
	elseif (y <= 1.22e-14)
		tmp = (-2.0 / t) / (z / x_m);
	else
		tmp = (x_m * 2.0) / (z * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -8.2e-54], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.22e-14], N[(N[(-2.0 / t), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -8.2000000000000001e-54

   1. Initial program 94.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot \left(y - t\right)}\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(y - t\right) \cdot z}\right), x\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), z\right), x\right) \]

      9. --lowering--.f64 94.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), z\right), x\right) \]

   4. Applied egg-rr 94.8%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{y}\right), z\right), x\right) \]
   6. Step-by-step derivation

   7. Simplified 78.8%

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

   if -8.2000000000000001e-54 < y < 1.21999999999999994e-14

   1. Initial program 91.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot \left(y - t\right)}\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(y - t\right) \cdot z}\right), x\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), z\right), x\right) \]

      9. --lowering--.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), z\right), x\right) \]

   4. Applied egg-rr 91.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-2}{t \cdot z}\right)}, x\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(t \cdot z\right)\right), x\right) \]

      2. *-lowering-*.f64 81.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(t, z\right)\right), x\right) \]

   7. Simplified 81.5%

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

      1. associate-*l/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{t}\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, t\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      7. /-lowering-/.f64 84.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, t\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   9. Applied egg-rr 84.1%

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

   if 1.21999999999999994e-14 < y

   1. Initial program 85.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \color{blue}{\left(y \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 82.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 82.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{-2}{t}}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-54}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 1.34 \cdot 10^{-14}:\\ \;\;\;\;\frac{x\_m \cdot \frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -9.2e-54)
    (* x_m (/ (/ 2.0 y) z))
    (if (<= y 1.34e-14) (/ (* x_m (/ -2.0 z)) t) (/ (* x_m 2.0) (* z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -9.2e-54) {
		tmp = x_m * ((2.0 / y) / z);
	} else if (y <= 1.34e-14) {
		tmp = (x_m * (-2.0 / z)) / t;
	} else {
		tmp = (x_m * 2.0) / (z * y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-9.2d-54)) then
        tmp = x_m * ((2.0d0 / y) / z)
    else if (y <= 1.34d-14) then
        tmp = (x_m * ((-2.0d0) / z)) / t
    else
        tmp = (x_m * 2.0d0) / (z * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -9.2e-54) {
		tmp = x_m * ((2.0 / y) / z);
	} else if (y <= 1.34e-14) {
		tmp = (x_m * (-2.0 / z)) / t;
	} else {
		tmp = (x_m * 2.0) / (z * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -9.2e-54:
		tmp = x_m * ((2.0 / y) / z)
	elif y <= 1.34e-14:
		tmp = (x_m * (-2.0 / z)) / t
	else:
		tmp = (x_m * 2.0) / (z * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -9.2e-54)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z));
	elseif (y <= 1.34e-14)
		tmp = Float64(Float64(x_m * Float64(-2.0 / z)) / t);
	else
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -9.2e-54)
		tmp = x_m * ((2.0 / y) / z);
	elseif (y <= 1.34e-14)
		tmp = (x_m * (-2.0 / z)) / t;
	else
		tmp = (x_m * 2.0) / (z * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -9.2e-54], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.34e-14], N[(N[(x$95$m * N[(-2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -9.1999999999999996e-54

   1. Initial program 94.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot \left(y - t\right)}\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(y - t\right) \cdot z}\right), x\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), z\right), x\right) \]

      9. --lowering--.f64 94.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), z\right), x\right) \]

   4. Applied egg-rr 94.8%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{y}\right), z\right), x\right) \]
   6. Step-by-step derivation

   7. Simplified 78.8%

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

   if -9.1999999999999996e-54 < y < 1.34e-14

   1. Initial program 91.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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-/l/ N/A

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

      2. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{x}{z}\right), \color{blue}{t}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2 \cdot x}{z}\right), t\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot x\right), z\right), t\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot -2\right), z\right), t\right) \]

      7. *-lowering-*.f64 83.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, -2\right), z\right), t\right) \]

   5. Simplified 83.7%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{-2}{z}\right), t\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{z} \cdot x\right), t\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-2}{z}\right), x\right), t\right) \]

      4. /-lowering-/.f64 83.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, z\right), x\right), t\right) \]

   7. Applied egg-rr 83.6%

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

   if 1.34e-14 < y

   1. Initial program 85.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \color{blue}{\left(y \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 82.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 82.9%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{-54}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 1.34 \cdot 10^{-14}:\\ \;\;\;\;\frac{x \cdot \frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-53}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{-2}{\frac{t}{\frac{x\_m}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z \cdot y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -2.6e-53)
    (* x_m (/ (/ 2.0 y) z))
    (if (<= y 2.1e-14) (/ -2.0 (/ t (/ x_m z))) (/ (* x_m 2.0) (* z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -2.6e-53) {
		tmp = x_m * ((2.0 / y) / z);
	} else if (y <= 2.1e-14) {
		tmp = -2.0 / (t / (x_m / z));
	} else {
		tmp = (x_m * 2.0) / (z * y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-2.6d-53)) then
        tmp = x_m * ((2.0d0 / y) / z)
    else if (y <= 2.1d-14) then
        tmp = (-2.0d0) / (t / (x_m / z))
    else
        tmp = (x_m * 2.0d0) / (z * y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -2.6e-53) {
		tmp = x_m * ((2.0 / y) / z);
	} else if (y <= 2.1e-14) {
		tmp = -2.0 / (t / (x_m / z));
	} else {
		tmp = (x_m * 2.0) / (z * y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -2.6e-53:
		tmp = x_m * ((2.0 / y) / z)
	elif y <= 2.1e-14:
		tmp = -2.0 / (t / (x_m / z))
	else:
		tmp = (x_m * 2.0) / (z * y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -2.6e-53)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z));
	elseif (y <= 2.1e-14)
		tmp = Float64(-2.0 / Float64(t / Float64(x_m / z)));
	else
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -2.6e-53)
		tmp = x_m * ((2.0 / y) / z);
	elseif (y <= 2.1e-14)
		tmp = -2.0 / (t / (x_m / z));
	else
		tmp = (x_m * 2.0) / (z * y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -2.6e-53], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e-14], N[(-2.0 / N[(t / N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.59999999999999996e-53

   1. Initial program 94.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot \left(y - t\right)}\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(y - t\right) \cdot z}\right), x\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), z\right), x\right) \]

      9. --lowering--.f64 94.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), z\right), x\right) \]

   4. Applied egg-rr 94.8%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{y}\right), z\right), x\right) \]
   6. Step-by-step derivation

   7. Simplified 78.8%

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

   if -2.59999999999999996e-53 < y < 2.0999999999999999e-14

   1. Initial program 91.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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-/l/ N/A

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

      2. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{x}{z}\right), \color{blue}{t}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2 \cdot x}{z}\right), t\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot x\right), z\right), t\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot -2\right), z\right), t\right) \]

      7. *-lowering-*.f64 83.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, -2\right), z\right), t\right) \]

   5. Simplified 83.7%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{-2}{z}\right), t\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{z} \cdot x\right), t\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-2}{z}\right), x\right), t\right) \]

      4. /-lowering-/.f64 83.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, z\right), x\right), t\right) \]

   7. Applied egg-rr 83.6%

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

      1. associate-/r/ N/A

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

      2. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left(\frac{z}{x} \cdot t\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(t \cdot \color{blue}{\frac{z}{x}}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(t \cdot \frac{1}{\color{blue}{\frac{x}{z}}}\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(\frac{t}{\color{blue}{\frac{x}{z}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{/.f64}\left(t, \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

      8. /-lowering-/.f64 82.5%

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   9. Applied egg-rr 82.5%

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

   if 2.0999999999999999e-14 < y

   1. Initial program 85.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \color{blue}{\left(y \cdot z\right)}\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 82.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   5. Simplified 82.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{-2}{\frac{t}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{\frac{2}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-2}{\frac{t}{\frac{x\_m}{z}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ (/ 2.0 y) z))))
   (*
    x_s
    (if (<= y -3.9e-53)
      t_1
      (if (<= y 2.4e-15) (/ -2.0 (/ t (/ x_m z))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((2.0 / y) / z);
	double tmp;
	if (y <= -3.9e-53) {
		tmp = t_1;
	} else if (y <= 2.4e-15) {
		tmp = -2.0 / (t / (x_m / z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * ((2.0d0 / y) / z)
    if (y <= (-3.9d-53)) then
        tmp = t_1
    else if (y <= 2.4d-15) then
        tmp = (-2.0d0) / (t / (x_m / z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((2.0 / y) / z);
	double tmp;
	if (y <= -3.9e-53) {
		tmp = t_1;
	} else if (y <= 2.4e-15) {
		tmp = -2.0 / (t / (x_m / z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * ((2.0 / y) / z)
	tmp = 0
	if y <= -3.9e-53:
		tmp = t_1
	elif y <= 2.4e-15:
		tmp = -2.0 / (t / (x_m / z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(Float64(2.0 / y) / z))
	tmp = 0.0
	if (y <= -3.9e-53)
		tmp = t_1;
	elseif (y <= 2.4e-15)
		tmp = Float64(-2.0 / Float64(t / Float64(x_m / z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * ((2.0 / y) / z);
	tmp = 0.0;
	if (y <= -3.9e-53)
		tmp = t_1;
	elseif (y <= 2.4e-15)
		tmp = -2.0 / (t / (x_m / z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -3.9e-53], t$95$1, If[LessEqual[y, 2.4e-15], N[(-2.0 / N[(t / N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9000000000000002e-53 or 2.39999999999999995e-15 < y

   1. Initial program 90.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot \left(y - t\right)}\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(y - t\right) \cdot z}\right), x\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), z\right), x\right) \]

      9. --lowering--.f64 94.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), z\right), x\right) \]

   4. Applied egg-rr 94.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{y}\right), z\right), x\right) \]
   6. Step-by-step derivation

   7. Simplified 80.6%

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

   if -3.9000000000000002e-53 < y < 2.39999999999999995e-15

   1. Initial program 91.6%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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-/l/ N/A

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

      2. associate-*r/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(-2 \cdot \frac{x}{z}\right), \color{blue}{t}\right) \]

      4. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2 \cdot x}{z}\right), t\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot x\right), z\right), t\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(x \cdot -2\right), z\right), t\right) \]

      7. *-lowering-*.f64 83.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(x, -2\right), z\right), t\right) \]

   5. Simplified 83.7%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{-2}{z}\right), t\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{-2}{z} \cdot x\right), t\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{-2}{z}\right), x\right), t\right) \]

      4. /-lowering-/.f64 83.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, z\right), x\right), t\right) \]

   7. Applied egg-rr 83.6%

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

      1. associate-/r/ N/A

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

      2. associate-/r* N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(-2, \color{blue}{\left(\frac{z}{x} \cdot t\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(t \cdot \color{blue}{\frac{z}{x}}\right)\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(t \cdot \frac{1}{\color{blue}{\frac{x}{z}}}\right)\right) \]

      6. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(-2, \left(\frac{t}{\color{blue}{\frac{x}{z}}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{/.f64}\left(t, \color{blue}{\left(\frac{x}{z}\right)}\right)\right) \]

      8. /-lowering-/.f64 82.5%

         \[\leadsto \mathsf{/.f64}\left(-2, \mathsf{/.f64}\left(t, \mathsf{/.f64}\left(x, \color{blue}{z}\right)\right)\right) \]

   9. Applied egg-rr 82.5%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-2}{\frac{t}{\frac{x}{z}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{\frac{2}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ (/ 2.0 y) z))))
   (*
    x_s
    (if (<= y -4.1e-53)
      t_1
      (if (<= y 2.1e-15) (* x_m (/ (/ -2.0 t) z)) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((2.0 / y) / z);
	double tmp;
	if (y <= -4.1e-53) {
		tmp = t_1;
	} else if (y <= 2.1e-15) {
		tmp = x_m * ((-2.0 / t) / z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * ((2.0d0 / y) / z)
    if (y <= (-4.1d-53)) then
        tmp = t_1
    else if (y <= 2.1d-15) then
        tmp = x_m * (((-2.0d0) / t) / z)
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * ((2.0 / y) / z);
	double tmp;
	if (y <= -4.1e-53) {
		tmp = t_1;
	} else if (y <= 2.1e-15) {
		tmp = x_m * ((-2.0 / t) / z);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * ((2.0 / y) / z)
	tmp = 0
	if y <= -4.1e-53:
		tmp = t_1
	elif y <= 2.1e-15:
		tmp = x_m * ((-2.0 / t) / z)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(Float64(2.0 / y) / z))
	tmp = 0.0
	if (y <= -4.1e-53)
		tmp = t_1;
	elseif (y <= 2.1e-15)
		tmp = Float64(x_m * Float64(Float64(-2.0 / t) / z));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * ((2.0 / y) / z);
	tmp = 0.0;
	if (y <= -4.1e-53)
		tmp = t_1;
	elseif (y <= 2.1e-15)
		tmp = x_m * ((-2.0 / t) / z);
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -4.1e-53], t$95$1, If[LessEqual[y, 2.1e-15], N[(x$95$m * N[(N[(-2.0 / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.1000000000000001e-53 or 2.09999999999999981e-15 < y

   1. Initial program 90.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot \left(y - t\right)}\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(y - t\right) \cdot z}\right), x\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), z\right), x\right) \]

      9. --lowering--.f64 94.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), z\right), x\right) \]

   4. Applied egg-rr 94.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \color{blue}{y}\right), z\right), x\right) \]
   6. Step-by-step derivation

   7. Simplified 80.6%

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

   if -4.1000000000000001e-53 < y < 2.09999999999999981e-15

   1. Initial program 91.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot \left(y - t\right)}\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(y - t\right) \cdot z}\right), x\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), z\right), x\right) \]

      9. --lowering--.f64 91.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), z\right), x\right) \]

   4. Applied egg-rr 91.1%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{-2}{t}\right)}, z\right), x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 81.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, t\right), z\right), x\right) \]

   7. Simplified 81.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{-53}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 4 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{2}{y - t}}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\frac{y - t}{2}}}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 4e+92)
    (/ (/ 2.0 (- y t)) (/ z x_m))
    (/ (/ x_m (/ (- y t) 2.0)) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+92) {
		tmp = (2.0 / (y - t)) / (z / x_m);
	} else {
		tmp = (x_m / ((y - t) / 2.0)) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 4d+92) then
        tmp = (2.0d0 / (y - t)) / (z / x_m)
    else
        tmp = (x_m / ((y - t) / 2.0d0)) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+92) {
		tmp = (2.0 / (y - t)) / (z / x_m);
	} else {
		tmp = (x_m / ((y - t) / 2.0)) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 4e+92:
		tmp = (2.0 / (y - t)) / (z / x_m)
	else:
		tmp = (x_m / ((y - t) / 2.0)) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 4e+92)
		tmp = Float64(Float64(2.0 / Float64(y - t)) / Float64(z / x_m));
	else
		tmp = Float64(Float64(x_m / Float64(Float64(y - t) / 2.0)) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 4e+92)
		tmp = (2.0 / (y - t)) / (z / x_m);
	else
		tmp = (x_m / ((y - t) / 2.0)) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 4e+92], N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(N[(y - t), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.0000000000000002e92

   1. Initial program 91.9%

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

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

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. associate-/r* N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{\frac{1}{\frac{{y}^{3} - {t}^{3}}{y \cdot y + \left(t \cdot t + y \cdot t\right)}}}{\frac{z}{x \cdot 2}} \]

      7. clear-num N/A

         \[\leadsto \frac{\frac{y \cdot y + \left(t \cdot t + y \cdot t\right)}{{y}^{3} - {t}^{3}}}{\frac{\color{blue}{z}}{x \cdot 2}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y + \left(t \cdot t + y \cdot t\right)}{{y}^{3} - {t}^{3}}\right), \color{blue}{\left(\frac{z}{x \cdot 2}\right)}\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{{y}^{3} - {t}^{3}}{y \cdot y + \left(t \cdot t + y \cdot t\right)}}\right), \left(\frac{\color{blue}{z}}{x \cdot 2}\right)\right) \]

      10. flip3-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{y - t}\right), \left(\frac{z}{x \cdot 2}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(y - t\right)\right), \left(\frac{\color{blue}{z}}{x \cdot 2}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{z}{x \cdot 2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \color{blue}{\left(x \cdot 2\right)}\right)\right) \]

      14. *-lowering-*.f64 95.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   4. Applied egg-rr 95.4%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{z}{x}\right)\right) \]

      9. /-lowering-/.f64 95.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   6. Applied egg-rr 95.4%

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

   if 4.0000000000000002e92 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 86.5%

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

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

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. associate-/r* N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{\frac{1}{\frac{{y}^{3} - {t}^{3}}{y \cdot y + \left(t \cdot t + y \cdot t\right)}}}{\frac{z}{x \cdot 2}} \]

      7. clear-num N/A

         \[\leadsto \frac{\frac{y \cdot y + \left(t \cdot t + y \cdot t\right)}{{y}^{3} - {t}^{3}}}{\frac{\color{blue}{z}}{x \cdot 2}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y + \left(t \cdot t + y \cdot t\right)}{{y}^{3} - {t}^{3}}\right), \color{blue}{\left(\frac{z}{x \cdot 2}\right)}\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{{y}^{3} - {t}^{3}}{y \cdot y + \left(t \cdot t + y \cdot t\right)}}\right), \left(\frac{\color{blue}{z}}{x \cdot 2}\right)\right) \]

      10. flip3-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{y - t}\right), \left(\frac{z}{x \cdot 2}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(y - t\right)\right), \left(\frac{\color{blue}{z}}{x \cdot 2}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{z}{x \cdot 2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \color{blue}{\left(x \cdot 2\right)}\right)\right) \]

      14. *-lowering-*.f64 85.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   4. Applied egg-rr 85.7%

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

      1. associate-/r/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. associate-*l/ N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{y - t} \cdot x\right), \color{blue}{z}\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{2}{y - t}\right), z\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y - t}{2}}\right), z\right) \]

      10. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{y - t}{2}}\right), z\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y - t}{2}\right)\right), z\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(y - t\right), 2\right)\right), z\right) \]

      13. --lowering--.f64 99.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, t\right), 2\right)\right), z\right) \]

   6. Applied egg-rr 99.6%

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

Alternative 9: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 4 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{2}{y - t}}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - t}}{\frac{z}{2}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 4e+92)
    (/ (/ 2.0 (- y t)) (/ z x_m))
    (/ (/ x_m (- y t)) (/ z 2.0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+92) {
		tmp = (2.0 / (y - t)) / (z / x_m);
	} else {
		tmp = (x_m / (y - t)) / (z / 2.0);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 4d+92) then
        tmp = (2.0d0 / (y - t)) / (z / x_m)
    else
        tmp = (x_m / (y - t)) / (z / 2.0d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+92) {
		tmp = (2.0 / (y - t)) / (z / x_m);
	} else {
		tmp = (x_m / (y - t)) / (z / 2.0);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 4e+92:
		tmp = (2.0 / (y - t)) / (z / x_m)
	else:
		tmp = (x_m / (y - t)) / (z / 2.0)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 4e+92)
		tmp = Float64(Float64(2.0 / Float64(y - t)) / Float64(z / x_m));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) / Float64(z / 2.0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 4e+92)
		tmp = (2.0 / (y - t)) / (z / x_m);
	else
		tmp = (x_m / (y - t)) / (z / 2.0);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 4e+92], N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.0000000000000002e92

   1. Initial program 91.9%

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

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

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. associate-/r* N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{\frac{1}{\frac{{y}^{3} - {t}^{3}}{y \cdot y + \left(t \cdot t + y \cdot t\right)}}}{\frac{z}{x \cdot 2}} \]

      7. clear-num N/A

         \[\leadsto \frac{\frac{y \cdot y + \left(t \cdot t + y \cdot t\right)}{{y}^{3} - {t}^{3}}}{\frac{\color{blue}{z}}{x \cdot 2}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y + \left(t \cdot t + y \cdot t\right)}{{y}^{3} - {t}^{3}}\right), \color{blue}{\left(\frac{z}{x \cdot 2}\right)}\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{{y}^{3} - {t}^{3}}{y \cdot y + \left(t \cdot t + y \cdot t\right)}}\right), \left(\frac{\color{blue}{z}}{x \cdot 2}\right)\right) \]

      10. flip3-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{y - t}\right), \left(\frac{z}{x \cdot 2}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(y - t\right)\right), \left(\frac{\color{blue}{z}}{x \cdot 2}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{z}{x \cdot 2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \color{blue}{\left(x \cdot 2\right)}\right)\right) \]

      14. *-lowering-*.f64 95.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   4. Applied egg-rr 95.4%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{z}{x}\right)\right) \]

      9. /-lowering-/.f64 95.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   6. Applied egg-rr 95.4%

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

   if 4.0000000000000002e92 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 86.5%

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

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{y - t}\right), \color{blue}{\left(\frac{z}{2}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(y - t\right)\right), \left(\frac{\color{blue}{z}}{2}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{z}{2}\right)\right) \]

      9. /-lowering-/.f64 99.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \color{blue}{2}\right)\right) \]

   4. Applied egg-rr 99.6%

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

Alternative 10: 95.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 4 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{2}{y - t}}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 4e+92)
    (/ (/ 2.0 (- y t)) (/ z x_m))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+92) {
		tmp = (2.0 / (y - t)) / (z / x_m);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 4d+92) then
        tmp = (2.0d0 / (y - t)) / (z / x_m)
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+92) {
		tmp = (2.0 / (y - t)) / (z / x_m);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 4e+92:
		tmp = (2.0 / (y - t)) / (z / x_m)
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 4e+92)
		tmp = Float64(Float64(2.0 / Float64(y - t)) / Float64(z / x_m));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 4e+92)
		tmp = (2.0 / (y - t)) / (z / x_m);
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 4e+92], N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.0000000000000002e92

   1. Initial program 91.9%

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

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

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

      4. associate-/l/ N/A

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

      5. associate-/r* N/A

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

      6. flip3-- N/A

         \[\leadsto \frac{\frac{1}{\frac{{y}^{3} - {t}^{3}}{y \cdot y + \left(t \cdot t + y \cdot t\right)}}}{\frac{z}{x \cdot 2}} \]

      7. clear-num N/A

         \[\leadsto \frac{\frac{y \cdot y + \left(t \cdot t + y \cdot t\right)}{{y}^{3} - {t}^{3}}}{\frac{\color{blue}{z}}{x \cdot 2}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y \cdot y + \left(t \cdot t + y \cdot t\right)}{{y}^{3} - {t}^{3}}\right), \color{blue}{\left(\frac{z}{x \cdot 2}\right)}\right) \]

      9. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{{y}^{3} - {t}^{3}}{y \cdot y + \left(t \cdot t + y \cdot t\right)}}\right), \left(\frac{\color{blue}{z}}{x \cdot 2}\right)\right) \]

      10. flip3-- N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{y - t}\right), \left(\frac{z}{x \cdot 2}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \left(y - t\right)\right), \left(\frac{\color{blue}{z}}{x \cdot 2}\right)\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{z}{x \cdot 2}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \color{blue}{\left(x \cdot 2\right)}\right)\right) \]

      14. *-lowering-*.f64 95.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \mathsf{*.f64}\left(x, \color{blue}{2}\right)\right)\right) \]

   4. Applied egg-rr 95.4%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

      3. associate-*l/ N/A

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

      4. associate-/r/ N/A

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

      5. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), \color{blue}{\left(\frac{z}{x}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), \left(\frac{\color{blue}{z}}{x}\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{z}{x}\right)\right) \]

      9. /-lowering-/.f64 95.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   6. Applied egg-rr 95.4%

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

   if 4.0000000000000002e92 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 86.5%

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

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y - t}\right), \color{blue}{\left(\frac{2}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y - t\right)\right), \left(\frac{\color{blue}{2}}{z}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{2}{z}\right)\right) \]

      7. /-lowering-/.f64 99.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(2, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 99.7%

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

Alternative 11: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 4.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z 4.3e+15)
    (/ (* x_m 2.0) (* z (- y t)))
    (/ (/ 2.0 z) (/ (- y t) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 4.3e+15) {
		tmp = (x_m * 2.0) / (z * (y - t));
	} else {
		tmp = (2.0 / z) / ((y - t) / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 4.3d+15) then
        tmp = (x_m * 2.0d0) / (z * (y - t))
    else
        tmp = (2.0d0 / z) / ((y - t) / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 4.3e+15) {
		tmp = (x_m * 2.0) / (z * (y - t));
	} else {
		tmp = (2.0 / z) / ((y - t) / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 4.3e+15:
		tmp = (x_m * 2.0) / (z * (y - t))
	else:
		tmp = (2.0 / z) / ((y - t) / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 4.3e+15)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * Float64(y - t)));
	else
		tmp = Float64(Float64(2.0 / z) / Float64(Float64(y - t) / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 4.3e+15)
		tmp = (x_m * 2.0) / (z * (y - t));
	else
		tmp = (2.0 / z) / ((y - t) / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 4.3e+15], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.3e15

   1. Initial program 93.1%

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(z \cdot \color{blue}{\left(y - t\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\left(y - t\right) \cdot \color{blue}{z}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(\left(y - t\right), \color{blue}{z}\right)\right) \]

      4. --lowering--.f64 93.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, t\right), z\right)\right) \]

   4. Applied egg-rr 93.6%

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

   if 4.3e15 < z

   1. Initial program 85.4%

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

      1. clear-num N/A

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

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

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{z}\right), \color{blue}{\left(\frac{y - t}{x}\right)}\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, z\right), \left(\frac{\color{blue}{y - t}}{x}\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, z\right), \mathsf{/.f64}\left(\left(y - t\right), \color{blue}{x}\right)\right) \]

      10. --lowering--.f64 96.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, z\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, t\right), x\right)\right) \]

   4. Applied egg-rr 96.3%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+16}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z 8e+16)
    (/ (* x_m 2.0) (* z (- y t)))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 8e+16) {
		tmp = (x_m * 2.0) / (z * (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 8d+16) then
        tmp = (x_m * 2.0d0) / (z * (y - t))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 8e+16) {
		tmp = (x_m * 2.0) / (z * (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 8e+16:
		tmp = (x_m * 2.0) / (z * (y - t))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 8e+16)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 8e+16)
		tmp = (x_m * 2.0) / (z * (y - t));
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 8e+16], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8e16

   1. Initial program 93.1%

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

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(z \cdot \color{blue}{\left(y - t\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \left(\left(y - t\right) \cdot \color{blue}{z}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(\left(y - t\right), \color{blue}{z}\right)\right) \]

      4. --lowering--.f64 93.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, 2\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, t\right), z\right)\right) \]

   4. Applied egg-rr 93.6%

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

   if 8e16 < z

   1. Initial program 85.4%

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

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y - t}\right), \color{blue}{\left(\frac{2}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y - t\right)\right), \left(\frac{\color{blue}{2}}{z}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{2}{z}\right)\right) \]

      7. /-lowering-/.f64 95.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(2, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 95.4%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+16}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+16}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z 4e+16)
    (* x_m (/ (/ 2.0 (- y t)) z))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 4e+16) {
		tmp = x_m * ((2.0 / (y - t)) / z);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 4d+16) then
        tmp = x_m * ((2.0d0 / (y - t)) / z)
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 4e+16) {
		tmp = x_m * ((2.0 / (y - t)) / z);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 4e+16:
		tmp = x_m * ((2.0 / (y - t)) / z)
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 4e+16)
		tmp = Float64(x_m * Float64(Float64(2.0 / Float64(y - t)) / z));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 4e+16)
		tmp = x_m * ((2.0 / (y - t)) / z);
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 4e+16], N[(x$95$m * N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4e16

   1. Initial program 93.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot \left(y - t\right)}\right), x\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(y - t\right) \cdot z}\right), x\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), z\right), x\right) \]

      9. --lowering--.f64 93.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), z\right), x\right) \]

   4. Applied egg-rr 93.2%

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

   if 4e16 < z

   1. Initial program 85.4%

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

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y - t}\right), \color{blue}{\left(\frac{2}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y - t\right)\right), \left(\frac{\color{blue}{2}}{z}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{2}{z}\right)\right) \]

      7. /-lowering-/.f64 95.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(2, \color{blue}{z}\right)\right) \]

   4. Applied egg-rr 95.4%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 92.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

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

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{y - t}\right), \color{blue}{\left(\frac{2}{z}\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(y - t\right)\right), \left(\frac{\color{blue}{2}}{z}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, t\right)\right), \left(\frac{2}{z}\right)\right) \]

   7. /-lowering-/.f64 93.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(y, t\right)\right), \mathsf{/.f64}\left(2, \color{blue}{z}\right)\right) \]

4. Applied egg-rr 93.2%

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

Alternative 15: 52.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot \left(y - t\right)}\right), x\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(y - t\right) \cdot z}\right), x\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), z\right), x\right) \]

   9. --lowering--.f64 92.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), z\right), x\right) \]

4. Applied egg-rr 92.5%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{-2}{t}\right)}, z\right), x\right) \]
6. Step-by-step derivation

   1. /-lowering-/.f64 57.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(-2, t\right), z\right), x\right) \]

7. Simplified 57.1%

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

8. Final simplification 57.1%

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

Alternative 16: 52.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.1%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{y \cdot z - t \cdot z}\right), \color{blue}{x}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{z \cdot \left(y - t\right)}\right), x\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(y - t\right) \cdot z}\right), x\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2}{y - t}}{z}\right), x\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{y - t}\right), z\right), x\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(y - t\right)\right), z\right), x\right) \]

   9. --lowering--.f64 92.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{\_.f64}\left(y, t\right)\right), z\right), x\right) \]

4. Applied egg-rr 92.5%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{-2}{t \cdot z}\right)}, x\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \left(t \cdot z\right)\right), x\right) \]

   2. *-lowering-*.f64 57.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(-2, \mathsf{*.f64}\left(t, z\right)\right), x\right) \]

7. Simplified 57.1%

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

8. Final simplification 57.1%

   \[\leadsto x \cdot \frac{-2}{z \cdot t} \]
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 2024158 
(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))))