Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.9% → 96.9%

Time: 11.3s

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: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 4 \cdot 10^{-56}:\\ \;\;\;\;\frac{x\_m \cdot 2}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 4e-56)
    (/ (* x_m 2.0) (* (- y t) z))
    (/ (/ 2.0 z) (/ (- y t) x_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e-56) {
		tmp = (x_m * 2.0) / ((y - t) * z);
	} else {
		tmp = (2.0 / z) / ((y - t) / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 4d-56) then
        tmp = (x_m * 2.0d0) / ((y - t) * z)
    else
        tmp = (2.0d0 / z) / ((y - t) / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e-56) {
		tmp = (x_m * 2.0) / ((y - t) * z);
	} else {
		tmp = (2.0 / z) / ((y - t) / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 4e-56:
		tmp = (x_m * 2.0) / ((y - t) * z)
	else:
		tmp = (2.0 / z) / ((y - t) / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 4e-56)
		tmp = Float64(Float64(x_m * 2.0) / Float64(Float64(y - t) * z));
	else
		tmp = Float64(Float64(2.0 / z) / Float64(Float64(y - t) / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 4e-56)
		tmp = (x_m * 2.0) / ((y - t) * z);
	else
		tmp = (2.0 / z) / ((y - t) / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 4e-56], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.0000000000000002e-56

   1. Initial program 92.1%

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

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

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

      2. *-commutative N/A

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

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

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

      4. --lowering--.f64 94.3%

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

   4. Applied egg-rr 94.3%

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

   if 4.0000000000000002e-56 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 84.5%

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

      1. clear-num N/A

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

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

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. associate-/r* N/A

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

      6. clear-num N/A

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

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

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

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

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

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

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

      10. --lowering--.f64 99.7%

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

   4. Applied egg-rr 99.7%

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

Alternative 2: 74.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 0.00025:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{\frac{z}{2}}\\ \end{array} \end{array} \]

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 (<= y -5.6e-7)
    (/ (* x_m 2.0) (* y z))
    (if (<= y 0.00025) (/ (/ (* x_m -2.0) z) t) (/ (/ x_m y) (/ 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 (y <= -5.6e-7) {
		tmp = (x_m * 2.0) / (y * z);
	} else if (y <= 0.00025) {
		tmp = ((x_m * -2.0) / z) / t;
	} else {
		tmp = (x_m / y) / (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 (y <= (-5.6d-7)) then
        tmp = (x_m * 2.0d0) / (y * z)
    else if (y <= 0.00025d0) then
        tmp = ((x_m * (-2.0d0)) / z) / t
    else
        tmp = (x_m / y) / (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 (y <= -5.6e-7) {
		tmp = (x_m * 2.0) / (y * z);
	} else if (y <= 0.00025) {
		tmp = ((x_m * -2.0) / z) / t;
	} else {
		tmp = (x_m / y) / (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 y <= -5.6e-7:
		tmp = (x_m * 2.0) / (y * z)
	elif y <= 0.00025:
		tmp = ((x_m * -2.0) / z) / t
	else:
		tmp = (x_m / y) / (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 (y <= -5.6e-7)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z));
	elseif (y <= 0.00025)
		tmp = Float64(Float64(Float64(x_m * -2.0) / z) / t);
	else
		tmp = Float64(Float64(x_m / y) / 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 (y <= -5.6e-7)
		tmp = (x_m * 2.0) / (y * z);
	elseif (y <= 0.00025)
		tmp = ((x_m * -2.0) / z) / t;
	else
		tmp = (x_m / y) / (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[y, -5.6e-7], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00025], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / z), $MachinePrecision] / t), $MachinePrecision], N[(N[(x$95$m / y), $MachinePrecision] / N[(z / 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -5.60000000000000038e-7

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 77.4%

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

   5. Simplified 77.4%

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

   if -5.60000000000000038e-7 < y < 2.5000000000000001e-4

   1. Initial program 91.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l/ N/A

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

      2. associate-*r/ N/A

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

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

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

      4. associate-*r/ N/A

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 75.2%

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

   5. Simplified 75.2%

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

   if 2.5000000000000001e-4 < y

   1. Initial program 86.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 78.7%

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

   5. Simplified 78.7%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 83.5%

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

   7. Applied egg-rr 83.5%

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

Alternative 3: 74.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{\frac{z}{2}}\\ \end{array} \end{array} \]

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 (<= y -1.45e-6)
    (/ (* x_m 2.0) (* y z))
    (if (<= y 9e-5) (* (/ x_m z) (/ -2.0 t)) (/ (/ x_m y) (/ 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 (y <= -1.45e-6) {
		tmp = (x_m * 2.0) / (y * z);
	} else if (y <= 9e-5) {
		tmp = (x_m / z) * (-2.0 / t);
	} else {
		tmp = (x_m / y) / (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 (y <= (-1.45d-6)) then
        tmp = (x_m * 2.0d0) / (y * z)
    else if (y <= 9d-5) then
        tmp = (x_m / z) * ((-2.0d0) / t)
    else
        tmp = (x_m / y) / (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 (y <= -1.45e-6) {
		tmp = (x_m * 2.0) / (y * z);
	} else if (y <= 9e-5) {
		tmp = (x_m / z) * (-2.0 / t);
	} else {
		tmp = (x_m / y) / (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 y <= -1.45e-6:
		tmp = (x_m * 2.0) / (y * z)
	elif y <= 9e-5:
		tmp = (x_m / z) * (-2.0 / t)
	else:
		tmp = (x_m / y) / (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 (y <= -1.45e-6)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z));
	elseif (y <= 9e-5)
		tmp = Float64(Float64(x_m / z) * Float64(-2.0 / t));
	else
		tmp = Float64(Float64(x_m / y) / 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 (y <= -1.45e-6)
		tmp = (x_m * 2.0) / (y * z);
	elseif (y <= 9e-5)
		tmp = (x_m / z) * (-2.0 / t);
	else
		tmp = (x_m / y) / (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[y, -1.45e-6], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9e-5], N[(N[(x$95$m / z), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / y), $MachinePrecision] / N[(z / 2.0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.4500000000000001e-6

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 77.4%

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

   5. Simplified 77.4%

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

   if -1.4500000000000001e-6 < y < 9.00000000000000057e-5

   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 90.5%

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

   4. Applied egg-rr 90.5%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 74.3%

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

   7. Simplified 74.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. /-lowering-/.f64 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 9.00000000000000057e-5 < y

   1. Initial program 86.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 78.7%

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

   5. Simplified 78.7%

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

      1. times-frac N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. /-lowering-/.f64 83.5%

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

   7. Applied egg-rr 83.5%

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

Alternative 4: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 0.000115:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -5.2e-6)
    (/ (* x_m 2.0) (* y z))
    (if (<= y 0.000115) (* (/ x_m z) (/ -2.0 t)) (/ (/ 2.0 z) (/ y x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -5.2e-6) {
		tmp = (x_m * 2.0) / (y * z);
	} else if (y <= 0.000115) {
		tmp = (x_m / z) * (-2.0 / t);
	} else {
		tmp = (2.0 / z) / (y / x_m);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-5.2d-6)) then
        tmp = (x_m * 2.0d0) / (y * z)
    else if (y <= 0.000115d0) then
        tmp = (x_m / z) * ((-2.0d0) / t)
    else
        tmp = (2.0d0 / z) / (y / x_m)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -5.2e-6) {
		tmp = (x_m * 2.0) / (y * z);
	} else if (y <= 0.000115) {
		tmp = (x_m / z) * (-2.0 / t);
	} else {
		tmp = (2.0 / z) / (y / x_m);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -5.2e-6:
		tmp = (x_m * 2.0) / (y * z)
	elif y <= 0.000115:
		tmp = (x_m / z) * (-2.0 / t)
	else:
		tmp = (2.0 / z) / (y / x_m)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -5.2e-6)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z));
	elseif (y <= 0.000115)
		tmp = Float64(Float64(x_m / z) * Float64(-2.0 / t));
	else
		tmp = Float64(Float64(2.0 / z) / Float64(y / x_m));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -5.2e-6)
		tmp = (x_m * 2.0) / (y * z);
	elseif (y <= 0.000115)
		tmp = (x_m / z) * (-2.0 / t);
	else
		tmp = (2.0 / z) / (y / x_m);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -5.2e-6], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.000115], N[(N[(x$95$m / z), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -5.20000000000000019e-6

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 77.4%

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

   5. Simplified 77.4%

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

   if -5.20000000000000019e-6 < y < 1.15e-4

   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 90.5%

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

   4. Applied egg-rr 90.5%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 74.3%

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

   7. Simplified 74.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. /-lowering-/.f64 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 1.15e-4 < y

   1. Initial program 86.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 78.7%

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

   5. Simplified 78.7%

      \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z}} \]
   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 82.1%

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

   7. Applied egg-rr 82.1%

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

Alternative 5: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 0.00052:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -1.5e-6)
    (/ (* x_m 2.0) (* y z))
    (if (<= y 0.00052) (* (/ x_m z) (/ -2.0 t)) (* (/ x_m z) (/ 2.0 y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -1.5e-6) {
		tmp = (x_m * 2.0) / (y * z);
	} else if (y <= 0.00052) {
		tmp = (x_m / z) * (-2.0 / t);
	} else {
		tmp = (x_m / z) * (2.0 / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.5d-6)) then
        tmp = (x_m * 2.0d0) / (y * z)
    else if (y <= 0.00052d0) then
        tmp = (x_m / z) * ((-2.0d0) / t)
    else
        tmp = (x_m / z) * (2.0d0 / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -1.5e-6) {
		tmp = (x_m * 2.0) / (y * z);
	} else if (y <= 0.00052) {
		tmp = (x_m / z) * (-2.0 / t);
	} else {
		tmp = (x_m / z) * (2.0 / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -1.5e-6:
		tmp = (x_m * 2.0) / (y * z)
	elif y <= 0.00052:
		tmp = (x_m / z) * (-2.0 / t)
	else:
		tmp = (x_m / z) * (2.0 / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -1.5e-6)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z));
	elseif (y <= 0.00052)
		tmp = Float64(Float64(x_m / z) * Float64(-2.0 / t));
	else
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -1.5e-6)
		tmp = (x_m * 2.0) / (y * z);
	elseif (y <= 0.00052)
		tmp = (x_m / z) * (-2.0 / t);
	else
		tmp = (x_m / z) * (2.0 / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -1.5e-6], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00052], N[(N[(x$95$m / z), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.5e-6

   1. Initial program 90.8%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 77.4%

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

   5. Simplified 77.4%

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

   if -1.5e-6 < y < 5.19999999999999954e-4

   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 90.5%

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

   4. Applied egg-rr 90.5%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 74.3%

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

   7. Simplified 74.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. /-lowering-/.f64 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 5.19999999999999954e-4 < y

   1. Initial program 86.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 78.7%

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

   5. Simplified 78.7%

      \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z}} \]
   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 81.0%

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

   7. Applied egg-rr 81.0%

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

Alternative 6: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;x\_m \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 0.000215:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -3.4e-6)
    (* x_m (/ 2.0 (* y z)))
    (if (<= y 0.000215) (* (/ x_m z) (/ -2.0 t)) (* (/ x_m z) (/ 2.0 y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -3.4e-6) {
		tmp = x_m * (2.0 / (y * z));
	} else if (y <= 0.000215) {
		tmp = (x_m / z) * (-2.0 / t);
	} else {
		tmp = (x_m / z) * (2.0 / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-3.4d-6)) then
        tmp = x_m * (2.0d0 / (y * z))
    else if (y <= 0.000215d0) then
        tmp = (x_m / z) * ((-2.0d0) / t)
    else
        tmp = (x_m / z) * (2.0d0 / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -3.4e-6) {
		tmp = x_m * (2.0 / (y * z));
	} else if (y <= 0.000215) {
		tmp = (x_m / z) * (-2.0 / t);
	} else {
		tmp = (x_m / z) * (2.0 / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -3.4e-6:
		tmp = x_m * (2.0 / (y * z))
	elif y <= 0.000215:
		tmp = (x_m / z) * (-2.0 / t)
	else:
		tmp = (x_m / z) * (2.0 / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -3.4e-6)
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z)));
	elseif (y <= 0.000215)
		tmp = Float64(Float64(x_m / z) * Float64(-2.0 / t));
	else
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -3.4e-6)
		tmp = x_m * (2.0 / (y * z));
	elseif (y <= 0.000215)
		tmp = (x_m / z) * (-2.0 / t);
	else
		tmp = (x_m / z) * (2.0 / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -3.4e-6], N[(x$95$m * N[(2.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.000215], N[(N[(x$95$m / z), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.40000000000000006e-6

   1. Initial program 90.8%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 90.9%

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

   4. Applied egg-rr 90.9%

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

   5. Taylor expanded in y around inf

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 77.3%

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

   7. Simplified 77.3%

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

   if -3.40000000000000006e-6 < y < 2.14999999999999995e-4

   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 90.5%

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

   4. Applied egg-rr 90.5%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 74.3%

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

   7. Simplified 74.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. /-lowering-/.f64 75.2%

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

   9. Applied egg-rr 75.2%

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

   if 2.14999999999999995e-4 < y

   1. Initial program 86.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 78.7%

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

   5. Simplified 78.7%

      \[\leadsto \frac{x \cdot 2}{\color{blue}{y \cdot z}} \]
   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 81.0%

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

   7. Applied egg-rr 81.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 0.000215:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{2}{y \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.0003:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 -1.8e-7) t_1 (if (<= y 0.0003) (* (/ x_m z) (/ -2.0 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 <= -1.8e-7) {
		tmp = t_1;
	} else if (y <= 0.0003) {
		tmp = (x_m / z) * (-2.0 / 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 <= (-1.8d-7)) then
        tmp = t_1
    else if (y <= 0.0003d0) then
        tmp = (x_m / z) * ((-2.0d0) / 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 <= -1.8e-7) {
		tmp = t_1;
	} else if (y <= 0.0003) {
		tmp = (x_m / z) * (-2.0 / 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 <= -1.8e-7:
		tmp = t_1
	elif y <= 0.0003:
		tmp = (x_m / z) * (-2.0 / 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(2.0 / Float64(y * z)))
	tmp = 0.0
	if (y <= -1.8e-7)
		tmp = t_1;
	elseif (y <= 0.0003)
		tmp = Float64(Float64(x_m / z) * Float64(-2.0 / 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 <= -1.8e-7)
		tmp = t_1;
	elseif (y <= 0.0003)
		tmp = (x_m / z) * (-2.0 / 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[(2.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.8e-7], t$95$1, If[LessEqual[y, 0.0003], N[(N[(x$95$m / z), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.79999999999999997e-7 or 2.99999999999999974e-4 < y

   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 91.4%

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

   4. Applied egg-rr 91.4%

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

   5. Taylor expanded in y around inf

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 78.0%

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

   7. Simplified 78.0%

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

   if -1.79999999999999997e-7 < y < 2.99999999999999974e-4

   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 90.5%

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

   4. Applied egg-rr 90.5%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 74.3%

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

   7. Simplified 74.3%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. /-lowering-/.f64 75.2%

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

   9. Applied egg-rr 75.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 0.0003:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{2}{y \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.000108:\\ \;\;\;\;x\_m \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

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 (/ 2.0 (* y z)))))
   (*
    x_s
    (if (<= y -2.2e-7)
      t_1
      (if (<= y 0.000108) (* x_m (/ -2.0 (* t z))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (2.0 / (y * z));
	double tmp;
	if (y <= -2.2e-7) {
		tmp = t_1;
	} else if (y <= 0.000108) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * (2.0d0 / (y * z))
    if (y <= (-2.2d-7)) then
        tmp = t_1
    else if (y <= 0.000108d0) then
        tmp = x_m * ((-2.0d0) / (t * z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (2.0 / (y * z));
	double tmp;
	if (y <= -2.2e-7) {
		tmp = t_1;
	} else if (y <= 0.000108) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * (2.0 / (y * z))
	tmp = 0
	if y <= -2.2e-7:
		tmp = t_1
	elif y <= 0.000108:
		tmp = x_m * (-2.0 / (t * z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(2.0 / Float64(y * z)))
	tmp = 0.0
	if (y <= -2.2e-7)
		tmp = t_1;
	elseif (y <= 0.000108)
		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 * (2.0 / (y * z));
	tmp = 0.0;
	if (y <= -2.2e-7)
		tmp = t_1;
	elseif (y <= 0.000108)
		tmp = x_m * (-2.0 / (t * z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(2.0 / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -2.2e-7], t$95$1, If[LessEqual[y, 0.000108], 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 < -2.2000000000000001e-7 or 1.08e-4 < y

   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 91.4%

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

   4. Applied egg-rr 91.4%

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

   5. Taylor expanded in y around inf

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

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

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 78.0%

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

   7. Simplified 78.0%

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

   if -2.2000000000000001e-7 < y < 1.08e-4

   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 90.5%

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

   4. Applied egg-rr 90.5%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 74.3%

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

   7. Simplified 74.3%

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

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

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

      2. associate-/l/ N/A

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 74.3%

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

   9. Applied egg-rr 74.3%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 0.000108:\\ \;\;\;\;x \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.9% accurate, 0.7× speedup

Math

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

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-53)
    (/ (* 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-53) {
		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-53) 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-53) {
		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-53:
		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-53)
		tmp = Float64(Float64(x_m * 2.0) / Float64(Float64(y - t) * z));
	else
		tmp = Float64(Float64(2.0 / 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-53)
		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-53], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 2.00000000000000006e-53

   1. Initial program 92.1%

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

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

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

      2. *-commutative N/A

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

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

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

      4. --lowering--.f64 94.3%

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

   4. Applied egg-rr 94.3%

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

   if 2.00000000000000006e-53 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 84.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      7. /-lowering-/.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.6% accurate, 0.7× speedup

Math

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

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-53)
    (/ 2.0 (/ (* (- y t) z) x_m))
    (* (/ 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-53) {
		tmp = 2.0 / (((y - t) * z) / x_m);
	} 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-53) then
        tmp = 2.0d0 / (((y - t) * z) / x_m)
    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-53) {
		tmp = 2.0 / (((y - t) * z) / x_m);
	} 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-53:
		tmp = 2.0 / (((y - t) * z) / x_m)
	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-53)
		tmp = Float64(2.0 / Float64(Float64(Float64(y - t) * z) / x_m));
	else
		tmp = Float64(Float64(2.0 / 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-53)
		tmp = 2.0 / (((y - t) * z) / x_m);
	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-53], N[(2.0 / N[(N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 2.00000000000000006e-53

   1. Initial program 92.1%

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

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 87.4%

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

   4. Applied egg-rr 87.4%

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

      1. clear-num N/A

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

      2. frac-times N/A

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

      3. metadata-eval N/A

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

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

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

      5. associate-*l/ N/A

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

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

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

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

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

      8. --lowering--.f64 94.1%

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

   6. Applied egg-rr 94.1%

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

   if 2.00000000000000006e-53 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 84.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

      7. /-lowering-/.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\frac{\left(y - t\right) \cdot z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y - t}\\ \end{array} \end{array} \]

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-65)
    (* (/ x_m z) (/ 2.0 (- y t)))
    (* (/ 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-65) {
		tmp = (x_m / z) * (2.0 / (y - t));
	} 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-65) then
        tmp = (x_m / z) * (2.0d0 / (y - t))
    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-65) {
		tmp = (x_m / z) * (2.0 / (y - t));
	} 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-65:
		tmp = (x_m / z) * (2.0 / (y - t))
	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-65)
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(2.0 / 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-65)
		tmp = (x_m / z) * (2.0 / (y - t));
	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-65], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 1.99999999999999985e-65

   1. Initial program 92.1%

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

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

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

      2. 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 90.3%

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

   4. Applied egg-rr 90.3%

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

   if 1.99999999999999985e-65 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 84.6%

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

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

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

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

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

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

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

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

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

      7. /-lowering-/.f64 99.5%

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

   4. Applied egg-rr 99.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 2 \cdot 10^{-65}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.8% accurate, 1.2× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.6%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 90.0%

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

4. Applied egg-rr 90.0%

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

Alternative 13: 53.3% 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 89.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

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

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

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

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

   9. --lowering--.f64 90.9%

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

4. Applied egg-rr 90.9%

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

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 49.2%

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

7. Simplified 49.2%

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

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

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

   2. associate-/l/ N/A

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

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

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 49.2%

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

9. Applied egg-rr 49.2%

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

10. Final simplification 49.2%

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

Developer Target 1: 97.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024155 
(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))))