Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.1% → 96.9%

Time: 9.6s

Alternatives: 8

Speedup: 1.3×

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: 89.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.8× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 5e-6)
    (/ (+ x_m x_m) (* (- y t) z))
    (/ (* 2.0 (/ x_m (- y 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) {
	double tmp;
	if (x_m <= 5e-6) {
		tmp = (x_m + x_m) / ((y - t) * z);
	} else {
		tmp = (2.0 * (x_m / (y - t))) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x_m <= 5d-6) then
        tmp = (x_m + x_m) / ((y - t) * z)
    else
        tmp = (2.0d0 * (x_m / (y - t))) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (x_m <= 5e-6) {
		tmp = (x_m + x_m) / ((y - t) * z);
	} else {
		tmp = (2.0 * (x_m / (y - t))) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if x_m <= 5e-6:
		tmp = (x_m + x_m) / ((y - t) * z)
	else:
		tmp = (2.0 * (x_m / (y - t))) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (x_m <= 5e-6)
		tmp = Float64(Float64(x_m + x_m) / Float64(Float64(y - t) * z));
	else
		tmp = Float64(Float64(2.0 * Float64(x_m / Float64(y - t))) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (x_m <= 5e-6)
		tmp = (x_m + x_m) / ((y - t) * z);
	else
		tmp = (2.0 * (x_m / (y - t))) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[x$95$m, 5e-6], N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000041e-6

   1. Initial program 89.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 92.7

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

   4. Applied rewrites 92.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 92.7

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

   6. Applied rewrites 92.7%

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

   if 5.00000000000000041e-6 < x

   1. Initial program 75.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 98.1

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

   4. Applied rewrites 98.1%

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

Alternative 2: 60.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot 2}{y \cdot z - t \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-311} \lor \neg \left(t\_1 \leq 5 \cdot 10^{-324}\right):\\ \;\;\;\;\frac{x\_m + x\_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \end{array} \]

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) (* t z)))))
   (*
    x_s
    (if (or (<= t_1 -1e-311) (not (<= t_1 5e-324)))
      (/ (+ x_m x_m) (* y z))
      0.0))))

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 t_1 = (x_m * 2.0) / ((y * z) - (t * z));
	double tmp;
	if ((t_1 <= -1e-311) || !(t_1 <= 5e-324)) {
		tmp = (x_m + x_m) / (y * z);
	} else {
		tmp = 0.0;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (x_m * 2.0d0) / ((y * z) - (t * z))
    if ((t_1 <= (-1d-311)) .or. (.not. (t_1 <= 5d-324))) then
        tmp = (x_m + x_m) / (y * z)
    else
        tmp = 0.0d0
    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 t_1 = (x_m * 2.0) / ((y * z) - (t * z));
	double tmp;
	if ((t_1 <= -1e-311) || !(t_1 <= 5e-324)) {
		tmp = (x_m + x_m) / (y * z);
	} else {
		tmp = 0.0;
	}
	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):
	t_1 = (x_m * 2.0) / ((y * z) - (t * z))
	tmp = 0
	if (t_1 <= -1e-311) or not (t_1 <= 5e-324):
		tmp = (x_m + x_m) / (y * z)
	else:
		tmp = 0.0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
	tmp = 0.0
	if ((t_1 <= -1e-311) || !(t_1 <= 5e-324))
		tmp = Float64(Float64(x_m + x_m) / Float64(y * z));
	else
		tmp = 0.0;
	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)
	t_1 = (x_m * 2.0) / ((y * z) - (t * z));
	tmp = 0.0;
	if ((t_1 <= -1e-311) || ~((t_1 <= 5e-324)))
		tmp = (x_m + x_m) / (y * z);
	else
		tmp = 0.0;
	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_] := Block[{t$95$1 = N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[Or[LessEqual[t$95$1, -1e-311], N[Not[LessEqual[t$95$1, 5e-324]], $MachinePrecision]], N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision], 0.0]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -9.99999999999948e-312 or 4.94066e-324 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 90.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot 2}{\color{blue}{-1 \cdot \left(t \cdot z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 53.7

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 54.2

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

   8. Applied rewrites 54.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 54.2

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

   10. Applied rewrites 54.2%

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

   if -9.99999999999948e-312 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < 4.94066e-324

   1. Initial program 74.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. flip-+ N/A

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

      5. +-inverses N/A

         \[\leadsto \frac{\frac{\frac{x}{y - t} \cdot \frac{x}{y - t} - \frac{x}{y - t} \cdot \frac{x}{y - t}}{\color{blue}{0}}}{z} \]

      6. +-inverses N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. +-inverses N/A

         \[\leadsto \frac{\color{blue}{0}}{\left(\frac{x}{y - t} \cdot \frac{x}{y - t} - \frac{x}{y - t} \cdot \frac{x}{y - t}\right) \cdot z} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{0}{\color{blue}{\left(\frac{x}{y - t} \cdot \frac{x}{y - t} - \frac{x}{y - t} \cdot \frac{x}{y - t}\right) \cdot z}} \]

      11. +-inverses 0.0

          \[\leadsto \frac{0}{\color{blue}{0} \cdot z} \]

   6. Applied rewrites 0.0%

      \[\leadsto \color{blue}{\frac{0}{0 \cdot z}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{0}{0 \cdot z}} \]

      2. div0 74.9

         \[\leadsto \color{blue}{0} \]

   8. Applied rewrites 74.9%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - t \cdot z} \leq -1 \cdot 10^{-311} \lor \neg \left(\frac{x \cdot 2}{y \cdot z - t \cdot z} \leq 5 \cdot 10^{-324}\right):\\ \;\;\;\;\frac{x + x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{x\_m + x\_m}{\left(y - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y - t}\\ \end{array} \end{array} \]

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 (<= z 5.2e+54)
    (/ (+ x_m x_m) (* (- y t) z))
    (* (/ 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) {
	double tmp;
	if (z <= 5.2e+54) {
		tmp = (x_m + x_m) / ((y - t) * z);
	} else {
		tmp = (x_m / z) * (2.0 / (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 (z <= 5.2d+54) then
        tmp = (x_m + x_m) / ((y - t) * z)
    else
        tmp = (x_m / z) * (2.0d0 / (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 (z <= 5.2e+54) {
		tmp = (x_m + x_m) / ((y - t) * z);
	} else {
		tmp = (x_m / z) * (2.0 / (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 z <= 5.2e+54:
		tmp = (x_m + x_m) / ((y - t) * z)
	else:
		tmp = (x_m / z) * (2.0 / (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 (z <= 5.2e+54)
		tmp = Float64(Float64(x_m + x_m) / Float64(Float64(y - t) * z));
	else
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / 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 (z <= 5.2e+54)
		tmp = (x_m + x_m) / ((y - t) * z);
	else
		tmp = (x_m / z) * (2.0 / (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[z, 5.2e+54], N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.20000000000000013e54

   1. Initial program 88.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 91.3

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

   6. Applied rewrites 91.3%

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

   if 5.20000000000000013e54 < z

   1. Initial program 75.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 97.6

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

   4. Applied rewrites 97.6%

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

Alternative 4: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+17} \lor \neg \left(t \leq 2.3 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x\_m}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot y} \cdot 2\\ \end{array} \end{array} \]

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 (or (<= t -7.5e+17) (not (<= t 2.3e-47)))
    (* (/ x_m (* t z)) -2.0)
    (* (/ x_m (* z y)) 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 ((t <= -7.5e+17) || !(t <= 2.3e-47)) {
		tmp = (x_m / (t * z)) * -2.0;
	} else {
		tmp = (x_m / (z * y)) * 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 ((t <= (-7.5d+17)) .or. (.not. (t <= 2.3d-47))) then
        tmp = (x_m / (t * z)) * (-2.0d0)
    else
        tmp = (x_m / (z * y)) * 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 ((t <= -7.5e+17) || !(t <= 2.3e-47)) {
		tmp = (x_m / (t * z)) * -2.0;
	} else {
		tmp = (x_m / (z * y)) * 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 (t <= -7.5e+17) or not (t <= 2.3e-47):
		tmp = (x_m / (t * z)) * -2.0
	else:
		tmp = (x_m / (z * y)) * 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 ((t <= -7.5e+17) || !(t <= 2.3e-47))
		tmp = Float64(Float64(x_m / Float64(t * z)) * -2.0);
	else
		tmp = Float64(Float64(x_m / Float64(z * y)) * 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 ((t <= -7.5e+17) || ~((t <= 2.3e-47)))
		tmp = (x_m / (t * z)) * -2.0;
	else
		tmp = (x_m / (z * y)) * 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[Or[LessEqual[t, -7.5e+17], N[Not[LessEqual[t, 2.3e-47]], $MachinePrecision]], N[(N[(x$95$m / N[(t * z), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(x$95$m / N[(z * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -7.5e17 or 2.29999999999999982e-47 < t

   1. Initial program 81.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

   if -7.5e17 < t < 2.29999999999999982e-47

   1. Initial program 90.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+17} \lor \neg \left(t \leq 2.3 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot y} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+17} \lor \neg \left(t \leq 2.3 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x\_m}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m + x\_m}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= t -7.5e+17) (not (<= t 2.3e-47)))
    (* (/ x_m (* t z)) -2.0)
    (/ (+ x_m x_m) (* y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t <= -7.5e+17) || !(t <= 2.3e-47)) {
		tmp = (x_m / (t * z)) * -2.0;
	} else {
		tmp = (x_m + x_m) / (y * z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-7.5d+17)) .or. (.not. (t <= 2.3d-47))) then
        tmp = (x_m / (t * z)) * (-2.0d0)
    else
        tmp = (x_m + x_m) / (y * z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t <= -7.5e+17) || !(t <= 2.3e-47)) {
		tmp = (x_m / (t * z)) * -2.0;
	} else {
		tmp = (x_m + x_m) / (y * z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (t <= -7.5e+17) or not (t <= 2.3e-47):
		tmp = (x_m / (t * z)) * -2.0
	else:
		tmp = (x_m + x_m) / (y * z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((t <= -7.5e+17) || !(t <= 2.3e-47))
		tmp = Float64(Float64(x_m / Float64(t * z)) * -2.0);
	else
		tmp = Float64(Float64(x_m + x_m) / Float64(y * z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((t <= -7.5e+17) || ~((t <= 2.3e-47)))
		tmp = (x_m / (t * z)) * -2.0;
	else
		tmp = (x_m + x_m) / (y * z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[t, -7.5e+17], N[Not[LessEqual[t, 2.3e-47]], $MachinePrecision]], N[(N[(x$95$m / N[(t * z), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -7.5e17 or 2.29999999999999982e-47 < t

   1. Initial program 81.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

   if -7.5e17 < t < 2.29999999999999982e-47

   1. Initial program 90.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot 2}{\color{blue}{-1 \cdot \left(t \cdot z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 27.3

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

   5. Applied rewrites 27.3%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 75.6

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

   8. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 75.6

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

   10. Applied rewrites 75.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+17} \lor \neg \left(t \leq 2.3 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+17} \lor \neg \left(t \leq 2.3 \cdot 10^{-47}\right):\\ \;\;\;\;x\_m \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m + x\_m}{y \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= t -7.5e+17) (not (<= t 2.3e-47)))
    (* x_m (/ -2.0 (* t z)))
    (/ (+ x_m x_m) (* y z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t <= -7.5e+17) || !(t <= 2.3e-47)) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = (x_m + x_m) / (y * z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-7.5d+17)) .or. (.not. (t <= 2.3d-47))) then
        tmp = x_m * ((-2.0d0) / (t * z))
    else
        tmp = (x_m + x_m) / (y * z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t <= -7.5e+17) || !(t <= 2.3e-47)) {
		tmp = x_m * (-2.0 / (t * z));
	} else {
		tmp = (x_m + x_m) / (y * z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (t <= -7.5e+17) or not (t <= 2.3e-47):
		tmp = x_m * (-2.0 / (t * z))
	else:
		tmp = (x_m + x_m) / (y * z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((t <= -7.5e+17) || !(t <= 2.3e-47))
		tmp = Float64(x_m * Float64(-2.0 / Float64(t * z)));
	else
		tmp = Float64(Float64(x_m + x_m) / Float64(y * z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((t <= -7.5e+17) || ~((t <= 2.3e-47)))
		tmp = x_m * (-2.0 / (t * z));
	else
		tmp = (x_m + x_m) / (y * z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[t, -7.5e+17], N[Not[LessEqual[t, 2.3e-47]], $MachinePrecision]], N[(x$95$m * N[(-2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m + x$95$m), $MachinePrecision] / N[(y * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -7.5e17 or 2.29999999999999982e-47 < t

   1. Initial program 81.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 73.0%

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

   if -7.5e17 < t < 2.29999999999999982e-47

   1. Initial program 90.9%

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

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot 2}{\color{blue}{-1 \cdot \left(t \cdot z\right)}} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 27.3

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

   5. Applied rewrites 27.3%

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

   6. Taylor expanded in y around inf

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

      1. lower-*.f64 75.6

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

   8. Applied rewrites 75.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 75.6

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

   10. Applied rewrites 75.6%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+17} \lor \neg \left(t \leq 2.3 \cdot 10^{-47}\right):\\ \;\;\;\;x \cdot \frac{-2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 1.3× speedup

Math

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

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

Derivation

1. Initial program 85.8%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 89.1

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

4. Applied rewrites 89.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f64 89.1

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

6. Applied rewrites 89.1%

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

Alternative 8: 26.7% accurate, 30.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 0 \end{array} \]

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 0.0))

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) {
	return x_s * 0.0;
}

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 * 0.0d0
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 * 0.0;
}

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 * 0.0

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	return Float64(x_s * 0.0)
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 * 0.0;
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 * 0.0), $MachinePrecision]

Derivation

1. Initial program 85.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift--.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. times-frac N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 92.8

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

4. Applied rewrites 92.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. count-2-rev N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

      \[\leadsto \frac{\frac{\frac{x}{y - t} \cdot \frac{x}{y - t} - \frac{x}{y - t} \cdot \frac{x}{y - t}}{\color{blue}{0}}}{z} \]

   6. +-inverses N/A

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

   7. associate-/l/ N/A

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

   8. lower-/.f64 N/A

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

   9. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0}}{\left(\frac{x}{y - t} \cdot \frac{x}{y - t} - \frac{x}{y - t} \cdot \frac{x}{y - t}\right) \cdot z} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{0}{\color{blue}{\left(\frac{x}{y - t} \cdot \frac{x}{y - t} - \frac{x}{y - t} \cdot \frac{x}{y - t}\right) \cdot z}} \]

   11. +-inverses 0.0

       \[\leadsto \frac{0}{\color{blue}{0} \cdot z} \]

6. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\frac{0}{0 \cdot z}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{0}{0 \cdot z}} \]

   2. div0 29.6

      \[\leadsto \color{blue}{0} \]

8. Applied rewrites 29.6%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Developer Target 1: 97.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024337 
(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))))