Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.7% → 96.6%

Time: 13.0s

Alternatives: 13

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.7% 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: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 0.005:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - t}}{\frac{z}{2}}\\ \end{array} \end{array} \]

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) 0.005)
    (* x_m (/ (/ 2.0 (- y t)) z))
    (/ (/ 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) <= 0.005) {
		tmp = x_m * ((2.0 / (y - t)) / z);
	} 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) <= 0.005d0) then
        tmp = x_m * ((2.0d0 / (y - t)) / z)
    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) <= 0.005) {
		tmp = x_m * ((2.0 / (y - t)) / z);
	} 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) <= 0.005:
		tmp = x_m * ((2.0 / (y - t)) / z)
	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) <= 0.005)
		tmp = Float64(x_m * Float64(Float64(2.0 / Float64(y - t)) / z));
	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) <= 0.005)
		tmp = x_m * ((2.0 / (y - t)) / z);
	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], 0.005], 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[(z / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 0.0050000000000000001

   1. Initial program 92.4%

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

      1. associate-/l* N/A

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

      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 95.3%

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

   4. Applied egg-rr 95.3%

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

   if 0.0050000000000000001 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 87.3%

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

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

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

      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 96.8%

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

   4. Applied egg-rr 96.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 0.005:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{\frac{z}{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y}{x\_m}}\\ \mathbf{elif}\;y \leq 112000:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z}\\ \end{array} \end{array} \]

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 (<= y -7.2e+16)
    (/ (/ 2.0 z) (/ y x_m))
    (if (<= y 112000.0) (* x_m (/ (/ -2.0 z) t)) (/ (* x_m 2.0) (* y 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 (y <= -7.2e+16) {
		tmp = (2.0 / z) / (y / x_m);
	} else if (y <= 112000.0) {
		tmp = x_m * ((-2.0 / z) / t);
	} else {
		tmp = (x_m * 2.0) / (y * 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 (y <= (-7.2d+16)) then
        tmp = (2.0d0 / z) / (y / x_m)
    else if (y <= 112000.0d0) then
        tmp = x_m * (((-2.0d0) / z) / t)
    else
        tmp = (x_m * 2.0d0) / (y * 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 (y <= -7.2e+16) {
		tmp = (2.0 / z) / (y / x_m);
	} else if (y <= 112000.0) {
		tmp = x_m * ((-2.0 / z) / t);
	} else {
		tmp = (x_m * 2.0) / (y * 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 y <= -7.2e+16:
		tmp = (2.0 / z) / (y / x_m)
	elif y <= 112000.0:
		tmp = x_m * ((-2.0 / z) / t)
	else:
		tmp = (x_m * 2.0) / (y * 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 (y <= -7.2e+16)
		tmp = Float64(Float64(2.0 / z) / Float64(y / x_m));
	elseif (y <= 112000.0)
		tmp = Float64(x_m * Float64(Float64(-2.0 / z) / t));
	else
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * 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 (y <= -7.2e+16)
		tmp = (2.0 / z) / (y / x_m);
	elseif (y <= 112000.0)
		tmp = x_m * ((-2.0 / z) / t);
	else
		tmp = (x_m * 2.0) / (y * 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[y, -7.2e+16], N[(N[(2.0 / z), $MachinePrecision] / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 112000.0], N[(x$95$m * N[(N[(-2.0 / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -7.2e16

   1. Initial program 89.4%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 74.1%

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

   5. Simplified 74.1%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

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

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

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

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

      7. /-lowering-/.f64 76.2%

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

   7. Applied egg-rr 76.2%

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

   if -7.2e16 < y < 112000

   1. Initial program 92.5%

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

      1. associate-/l* N/A

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

      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 94.2%

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

   4. Applied egg-rr 94.2%

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

   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 78.9%

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

   7. Simplified 78.9%

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

      1. associate-/l/ N/A

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

      2. metadata-eval N/A

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

      3. distribute-neg-frac N/A

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

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

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 79.2%

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

   9. Applied egg-rr 79.2%

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

   if 112000 < y

   1. Initial program 89.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 77.2%

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

   5. Simplified 77.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 112000:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -460000000000:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 3000:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z}\\ \end{array} \end{array} \]

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 (<= y -460000000000.0)
    (* x_m (/ (/ 2.0 y) z))
    (if (<= y 3000.0) (* x_m (/ (/ -2.0 z) t)) (/ (* x_m 2.0) (* y 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 (y <= -460000000000.0) {
		tmp = x_m * ((2.0 / y) / z);
	} else if (y <= 3000.0) {
		tmp = x_m * ((-2.0 / z) / t);
	} else {
		tmp = (x_m * 2.0) / (y * 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 (y <= (-460000000000.0d0)) then
        tmp = x_m * ((2.0d0 / y) / z)
    else if (y <= 3000.0d0) then
        tmp = x_m * (((-2.0d0) / z) / t)
    else
        tmp = (x_m * 2.0d0) / (y * 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 (y <= -460000000000.0) {
		tmp = x_m * ((2.0 / y) / z);
	} else if (y <= 3000.0) {
		tmp = x_m * ((-2.0 / z) / t);
	} else {
		tmp = (x_m * 2.0) / (y * 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 y <= -460000000000.0:
		tmp = x_m * ((2.0 / y) / z)
	elif y <= 3000.0:
		tmp = x_m * ((-2.0 / z) / t)
	else:
		tmp = (x_m * 2.0) / (y * 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 (y <= -460000000000.0)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z));
	elseif (y <= 3000.0)
		tmp = Float64(x_m * Float64(Float64(-2.0 / z) / t));
	else
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * 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 (y <= -460000000000.0)
		tmp = x_m * ((2.0 / y) / z);
	elseif (y <= 3000.0)
		tmp = x_m * ((-2.0 / z) / t);
	else
		tmp = (x_m * 2.0) / (y * 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[y, -460000000000.0], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3000.0], N[(x$95$m * N[(N[(-2.0 / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -4.6e11

   1. Initial program 89.4%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 92.2%

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

   4. Applied egg-rr 92.2%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 74.2%

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

   7. Simplified 74.2%

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

   if -4.6e11 < y < 3e3

   1. Initial program 92.5%

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

      1. associate-/l* N/A

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

      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 94.2%

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

   4. Applied egg-rr 94.2%

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

   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 78.9%

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

   7. Simplified 78.9%

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

      1. associate-/l/ N/A

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

      2. metadata-eval N/A

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

      3. distribute-neg-frac N/A

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

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

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 79.2%

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

   9. Applied egg-rr 79.2%

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

   if 3e3 < y

   1. Initial program 89.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 77.2%

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

   5. Simplified 77.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -460000000000:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 3000:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{\frac{2}{y}}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 132000000000:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -1.15e+14)
      t_1
      (if (<= y 132000000000.0) (* x_m (/ (/ -2.0 z) t)) 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 <= -1.15e+14) {
		tmp = t_1;
	} else if (y <= 132000000000.0) {
		tmp = x_m * ((-2.0 / z) / t);
	} 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 <= (-1.15d+14)) then
        tmp = t_1
    else if (y <= 132000000000.0d0) then
        tmp = x_m * (((-2.0d0) / z) / t)
    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 <= -1.15e+14) {
		tmp = t_1;
	} else if (y <= 132000000000.0) {
		tmp = x_m * ((-2.0 / z) / t);
	} 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 <= -1.15e+14:
		tmp = t_1
	elif y <= 132000000000.0:
		tmp = x_m * ((-2.0 / z) / t)
	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 <= -1.15e+14)
		tmp = t_1;
	elseif (y <= 132000000000.0)
		tmp = Float64(x_m * Float64(Float64(-2.0 / z) / t));
	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 <= -1.15e+14)
		tmp = t_1;
	elseif (y <= 132000000000.0)
		tmp = x_m * ((-2.0 / z) / t);
	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, -1.15e+14], t$95$1, If[LessEqual[y, 132000000000.0], N[(x$95$m * N[(N[(-2.0 / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.15e14 or 1.32e11 < y

   1. Initial program 89.5%

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

      1. associate-/l* N/A

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

      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 92.8%

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

   4. Applied egg-rr 92.8%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 75.3%

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

   7. Simplified 75.3%

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

   if -1.15e14 < y < 1.32e11

   1. Initial program 92.5%

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

      1. associate-/l* N/A

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

      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 94.2%

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

   4. Applied egg-rr 94.2%

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

   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 78.9%

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

   7. Simplified 78.9%

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

      1. associate-/l/ N/A

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

      2. metadata-eval N/A

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

      3. distribute-neg-frac N/A

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

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

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 79.2%

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

   9. Applied egg-rr 79.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 132000000000:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{z} \cdot \frac{2}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 49000:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 z) (/ 2.0 y))))
   (*
    x_s
    (if (<= y -4.7e+72)
      t_1
      (if (<= y 49000.0) (* x_m (/ (/ -2.0 z) t)) 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 / z) * (2.0 / y);
	double tmp;
	if (y <= -4.7e+72) {
		tmp = t_1;
	} else if (y <= 49000.0) {
		tmp = x_m * ((-2.0 / z) / t);
	} 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 / z) * (2.0d0 / y)
    if (y <= (-4.7d+72)) then
        tmp = t_1
    else if (y <= 49000.0d0) then
        tmp = x_m * (((-2.0d0) / z) / t)
    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 / z) * (2.0 / y);
	double tmp;
	if (y <= -4.7e+72) {
		tmp = t_1;
	} else if (y <= 49000.0) {
		tmp = x_m * ((-2.0 / z) / t);
	} 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 / z) * (2.0 / y)
	tmp = 0
	if y <= -4.7e+72:
		tmp = t_1
	elif y <= 49000.0:
		tmp = x_m * ((-2.0 / z) / t)
	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(x_m / z) * Float64(2.0 / y))
	tmp = 0.0
	if (y <= -4.7e+72)
		tmp = t_1;
	elseif (y <= 49000.0)
		tmp = Float64(x_m * Float64(Float64(-2.0 / z) / t));
	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 / z) * (2.0 / y);
	tmp = 0.0;
	if (y <= -4.7e+72)
		tmp = t_1;
	elseif (y <= 49000.0)
		tmp = x_m * ((-2.0 / z) / t);
	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[(x$95$m / z), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -4.7e+72], t$95$1, If[LessEqual[y, 49000.0], N[(x$95$m * N[(N[(-2.0 / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.70000000000000034e72 or 49000 < y

   1. Initial program 87.9%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 77.0%

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

   5. Simplified 77.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. /-lowering-/.f64 71.8%

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

   7. Applied egg-rr 71.8%

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

   if -4.70000000000000034e72 < y < 49000

   1. Initial program 93.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 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 76.8%

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

   7. Simplified 76.8%

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

      1. associate-/l/ N/A

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

      2. metadata-eval N/A

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

      3. distribute-neg-frac N/A

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

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

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

      5. distribute-neg-frac N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 77.1%

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

   9. Applied egg-rr 77.1%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+72}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 49000:\\ \;\;\;\;x \cdot \frac{\frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{z} \cdot \frac{2}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1050000000000:\\ \;\;\;\;x\_m \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 z) (/ 2.0 y))))
   (*
    x_s
    (if (<= y -1.95e+74)
      t_1
      (if (<= y 1050000000000.0) (* 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 / z) * (2.0 / y);
	double tmp;
	if (y <= -1.95e+74) {
		tmp = t_1;
	} else if (y <= 1050000000000.0) {
		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 / z) * (2.0d0 / y)
    if (y <= (-1.95d+74)) then
        tmp = t_1
    else if (y <= 1050000000000.0d0) 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 / z) * (2.0 / y);
	double tmp;
	if (y <= -1.95e+74) {
		tmp = t_1;
	} else if (y <= 1050000000000.0) {
		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 / z) * (2.0 / y)
	tmp = 0
	if y <= -1.95e+74:
		tmp = t_1
	elif y <= 1050000000000.0:
		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(Float64(x_m / z) * Float64(2.0 / y))
	tmp = 0.0
	if (y <= -1.95e+74)
		tmp = t_1;
	elseif (y <= 1050000000000.0)
		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 / z) * (2.0 / y);
	tmp = 0.0;
	if (y <= -1.95e+74)
		tmp = t_1;
	elseif (y <= 1050000000000.0)
		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[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.95e+74], t$95$1, If[LessEqual[y, 1050000000000.0], 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 < -1.95000000000000004e74 or 1.05e12 < y

   1. Initial program 87.9%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 77.0%

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

   5. Simplified 77.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. /-lowering-/.f64 71.8%

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

   7. Applied egg-rr 71.8%

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

   if -1.95000000000000004e74 < y < 1.05e12

   1. Initial program 93.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 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 77.0%

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

   7. Simplified 77.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+74}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 1050000000000:\\ \;\;\;\;x \cdot \frac{\frac{-2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m}{z} \cdot \frac{2}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 470000000:\\ \;\;\;\;x\_m \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 z) (/ 2.0 y))))
   (*
    x_s
    (if (<= y -5.7e+72)
      t_1
      (if (<= y 470000000.0) (* 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 / z) * (2.0 / y);
	double tmp;
	if (y <= -5.7e+72) {
		tmp = t_1;
	} else if (y <= 470000000.0) {
		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 / z) * (2.0d0 / y)
    if (y <= (-5.7d+72)) then
        tmp = t_1
    else if (y <= 470000000.0d0) 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 / z) * (2.0 / y);
	double tmp;
	if (y <= -5.7e+72) {
		tmp = t_1;
	} else if (y <= 470000000.0) {
		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 / z) * (2.0 / y)
	tmp = 0
	if y <= -5.7e+72:
		tmp = t_1
	elif y <= 470000000.0:
		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(Float64(x_m / z) * Float64(2.0 / y))
	tmp = 0.0
	if (y <= -5.7e+72)
		tmp = t_1;
	elseif (y <= 470000000.0)
		tmp = Float64(x_m * Float64(-2.0 / Float64(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 / z) * (2.0 / y);
	tmp = 0.0;
	if (y <= -5.7e+72)
		tmp = t_1;
	elseif (y <= 470000000.0)
		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[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -5.7e+72], t$95$1, If[LessEqual[y, 470000000.0], N[(x$95$m * N[(-2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.6999999999999997e72 or 4.7e8 < y

   1. Initial program 87.9%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 77.0%

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

   5. Simplified 77.0%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

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

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

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

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

      5. /-lowering-/.f64 71.8%

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

   7. Applied egg-rr 71.8%

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

   if -5.6999999999999997e72 < y < 4.7e8

   1. Initial program 93.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 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 76.8%

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

   7. Simplified 76.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+72}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 470000000:\\ \;\;\;\;x \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 2 \cdot 10^{-21}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{z}{\frac{x\_m}{y - t}}}\\ \end{array} \end{array} \]

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) 2e-21)
    (* x_m (/ (/ 2.0 (- y t)) z))
    (/ 2.0 (/ z (/ x_m (- y 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) {
	double tmp;
	if ((x_m * 2.0) <= 2e-21) {
		tmp = x_m * ((2.0 / (y - t)) / z);
	} else {
		tmp = 2.0 / (z / (x_m / (y - t)));
	}
	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) <= 2d-21) then
        tmp = x_m * ((2.0d0 / (y - t)) / z)
    else
        tmp = 2.0d0 / (z / (x_m / (y - t)))
    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) <= 2e-21) {
		tmp = x_m * ((2.0 / (y - t)) / z);
	} else {
		tmp = 2.0 / (z / (x_m / (y - t)));
	}
	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) <= 2e-21:
		tmp = x_m * ((2.0 / (y - t)) / z)
	else:
		tmp = 2.0 / (z / (x_m / (y - t)))
	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) <= 2e-21)
		tmp = Float64(x_m * Float64(Float64(2.0 / Float64(y - t)) / z));
	else
		tmp = Float64(2.0 / Float64(z / Float64(x_m / Float64(y - t))));
	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) <= 2e-21)
		tmp = x_m * ((2.0 / (y - t)) / z);
	else
		tmp = 2.0 / (z / (x_m / (y - t)));
	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], 2e-21], N[(x$95$m * N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(z / N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 1.99999999999999982e-21

   1. Initial program 92.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 95.2%

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

   4. Applied egg-rr 95.2%

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

   if 1.99999999999999982e-21 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 88.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 89.5%

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

   4. Applied egg-rr 89.5%

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

      1. associate-*l/ N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. --lowering--.f64 97.0%

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

   6. Applied egg-rr 97.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 2 \cdot 10^{-21}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{z}{\frac{x}{y - t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{2}{y - t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+18}:\\ \;\;\;\;x\_m \cdot \frac{t\_1}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 (/ 2.0 (- y t))))
   (* x_s (if (<= z 8e+18) (* x_m (/ t_1 z)) (* t_1 (/ x_m 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 t_1 = 2.0 / (y - t);
	double tmp;
	if (z <= 8e+18) {
		tmp = x_m * (t_1 / z);
	} else {
		tmp = t_1 * (x_m / 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) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (y - t)
    if (z <= 8d+18) then
        tmp = x_m * (t_1 / z)
    else
        tmp = t_1 * (x_m / 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 t_1 = 2.0 / (y - t);
	double tmp;
	if (z <= 8e+18) {
		tmp = x_m * (t_1 / z);
	} else {
		tmp = t_1 * (x_m / 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):
	t_1 = 2.0 / (y - t)
	tmp = 0
	if z <= 8e+18:
		tmp = x_m * (t_1 / z)
	else:
		tmp = t_1 * (x_m / 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)
	t_1 = Float64(2.0 / Float64(y - t))
	tmp = 0.0
	if (z <= 8e+18)
		tmp = Float64(x_m * Float64(t_1 / z));
	else
		tmp = Float64(t_1 * Float64(x_m / 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)
	t_1 = 2.0 / (y - t);
	tmp = 0.0;
	if (z <= 8e+18)
		tmp = x_m * (t_1 / z);
	else
		tmp = t_1 * (x_m / 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_] := Block[{t$95$1 = N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, 8e+18], N[(x$95$m * N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 8e18

   1. Initial program 92.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 94.9%

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

   4. Applied egg-rr 94.9%

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

   if 8e18 < z

   1. Initial program 87.8%

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

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

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

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

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

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

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

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

      6. --lowering--.f64 95.3%

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

   4. Applied egg-rr 95.3%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+43}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x\_m}{z}\\ \end{array} \end{array} \]

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 2.3e+43)
    (* x_m (/ (/ 2.0 z) (- y t)))
    (* (/ 2.0 (- y t)) (/ x_m 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 <= 2.3e+43) {
		tmp = x_m * ((2.0 / z) / (y - t));
	} else {
		tmp = (2.0 / (y - t)) * (x_m / 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 <= 2.3d+43) then
        tmp = x_m * ((2.0d0 / z) / (y - t))
    else
        tmp = (2.0d0 / (y - t)) * (x_m / 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 <= 2.3e+43) {
		tmp = x_m * ((2.0 / z) / (y - t));
	} else {
		tmp = (2.0 / (y - t)) * (x_m / 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 <= 2.3e+43:
		tmp = x_m * ((2.0 / z) / (y - t))
	else:
		tmp = (2.0 / (y - t)) * (x_m / 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 <= 2.3e+43)
		tmp = Float64(x_m * Float64(Float64(2.0 / z) / Float64(y - t)));
	else
		tmp = Float64(Float64(2.0 / Float64(y - t)) * Float64(x_m / 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 <= 2.3e+43)
		tmp = x_m * ((2.0 / z) / (y - t));
	else
		tmp = (2.0 / (y - t)) * (x_m / 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, 2.3e+43], N[(x$95$m * N[(N[(2.0 / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.3000000000000002e43

   1. Initial program 92.4%

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

      1. associate-/l* N/A

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

      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 95.0%

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

   4. Applied egg-rr 95.0%

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

      1. associate-/l/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. --lowering--.f64 95.0%

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

   6. Applied egg-rr 95.0%

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

   if 2.3000000000000002e43 < z

   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. times-frac N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 94.9%

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

   4. Applied egg-rr 94.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+43}:\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-46}:\\ \;\;\;\;x\_m \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{-2}{t}\\ \end{array} \end{array} \]

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 3.6e-46) (* x_m (/ -2.0 (* t z))) (* (/ x_m z) (/ -2.0 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) {
	double tmp;
	if (z <= 3.6e-46) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = (x_m / z) * (-2.0 / t);
	}
	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 <= 3.6d-46) then
        tmp = x_m * ((-2.0d0) / (t * z))
    else
        tmp = (x_m / z) * ((-2.0d0) / t)
    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 <= 3.6e-46) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = (x_m / z) * (-2.0 / t);
	}
	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 <= 3.6e-46:
		tmp = x_m * (-2.0 / (t * z))
	else:
		tmp = (x_m / z) * (-2.0 / t)
	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 <= 3.6e-46)
		tmp = Float64(x_m * Float64(-2.0 / Float64(t * z)));
	else
		tmp = Float64(Float64(x_m / z) * Float64(-2.0 / t));
	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 <= 3.6e-46)
		tmp = x_m * (-2.0 / (t * z));
	else
		tmp = (x_m / z) * (-2.0 / t);
	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, 3.6e-46], N[(x$95$m * N[(-2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.6e-46

   1. Initial program 91.5%

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

      1. associate-/l* N/A

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

      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 94.5%

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

   4. Applied egg-rr 94.5%

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

   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 55.9%

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

   7. Simplified 55.9%

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

   if 3.6e-46 < z

   1. Initial program 89.9%

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

      1. associate-/l* N/A

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

      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 91.2%

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

   4. Applied egg-rr 91.2%

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

   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 53.0%

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

   7. Simplified 53.0%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

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

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

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

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

      7. /-lowering-/.f64 61.2%

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

   9. Applied egg-rr 61.2%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-46}:\\ \;\;\;\;x \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.9% accurate, 1.2× speedup

Math

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

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 (* (/ 2.0 (- y t)) (/ x_m 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 * ((2.0 / (y - t)) * (x_m / 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 * ((2.0d0 / (y - t)) * (x_m / 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 * ((2.0 / (y - t)) * (x_m / 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 * ((2.0 / (y - t)) * (x_m / 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(2.0 / Float64(y - t)) * Float64(x_m / 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 * ((2.0 / (y - t)) * (x_m / 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[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

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

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

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

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

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

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

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

   6. --lowering--.f64 89.8%

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

4. Applied egg-rr 89.8%

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

5. Final simplification 89.8%

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

Alternative 13: 52.1% accurate, 1.6× speedup

Math

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

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(-2.0 / Float64(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[(-2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 91.0%

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

   1. associate-/l* N/A

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

   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 93.5%

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

4. Applied egg-rr 93.5%

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

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 55.0%

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

7. Simplified 55.0%

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

8. Final simplification 55.0%

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

Developer Target 1: 96.9% 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 2024152 
(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))))