Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.6% → 94.7%

Time: 10.9s

Alternatives: 8

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: 89.6% 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: 94.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot 2}{y \cdot z\_m - z\_m \cdot t} \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\frac{x\_m}{z\_m \cdot \left(\left(y - t\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot 2}{z\_m}}{y - t}\\ \end{array}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s z_s x_m y z_m t)
 :precision binary64
 (*
  x_s
  (*
   z_s
   (if (<= (/ (* x_m 2.0) (- (* y z_m) (* z_m t))) -5e-278)
     (/ x_m (* z_m (* (- y t) 0.5)))
     (/ (/ (* x_m 2.0) z_m) (- y t))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -5e-278) {
		tmp = x_m / (z_m * ((y - t) * 0.5));
	} else {
		tmp = ((x_m * 2.0) / z_m) / (y - t);
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, z_s, x_m, y, z_m, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x_m * 2.0d0) / ((y * z_m) - (z_m * t))) <= (-5d-278)) then
        tmp = x_m / (z_m * ((y - t) * 0.5d0))
    else
        tmp = ((x_m * 2.0d0) / z_m) / (y - t)
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -5e-278) {
		tmp = x_m / (z_m * ((y - t) * 0.5));
	} else {
		tmp = ((x_m * 2.0) / z_m) / (y - t);
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, z_s, x_m, y, z_m, t):
	tmp = 0
	if ((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -5e-278:
		tmp = x_m / (z_m * ((y - t) * 0.5))
	else:
		tmp = ((x_m * 2.0) / z_m) / (y - t)
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, z_s, x_m, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(x_m * 2.0) / Float64(Float64(y * z_m) - Float64(z_m * t))) <= -5e-278)
		tmp = Float64(x_m / Float64(z_m * Float64(Float64(y - t) * 0.5)));
	else
		tmp = Float64(Float64(Float64(x_m * 2.0) / z_m) / Float64(y - t));
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -5e-278)
		tmp = x_m / (z_m * ((y - t) * 0.5));
	else
		tmp = ((x_m * 2.0) / z_m) / (y - t);
	end
	tmp_2 = x_s * (z_s * tmp);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, z$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(x$95$s * N[(z$95$s * If[LessEqual[N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-278], N[(x$95$m / N[(z$95$m * N[(N[(y - t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -4.99999999999999985e-278

   1. Initial program 96.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 96.7

          \[\leadsto \frac{x}{z \cdot \left(\left(y - t\right) \cdot \color{blue}{0.5}\right)} \]

   4. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{x}{z \cdot \left(\left(y - t\right) \cdot 0.5\right)}} \]

   if -4.99999999999999985e-278 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 82.2%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 93.0

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

   4. Applied rewrites 93.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -5 \cdot 10^{-278}:\\ \;\;\;\;\frac{x}{z \cdot \left(\left(y - t\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot 2}{y \cdot z\_m - z\_m \cdot t} \leq -1 \cdot 10^{-266}:\\ \;\;\;\;\frac{x\_m}{z\_m \cdot \left(\left(y - t\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \end{array}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s z_s x_m y z_m t)
 :precision binary64
 (*
  x_s
  (*
   z_s
   (if (<= (/ (* x_m 2.0) (- (* y z_m) (* z_m t))) -1e-266)
     (/ x_m (* z_m (* (- y t) 0.5)))
     (* (/ x_m z_m) (/ 2.0 (- y t)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-266) {
		tmp = x_m / (z_m * ((y - t) * 0.5));
	} else {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, z_s, x_m, y, z_m, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x_m * 2.0d0) / ((y * z_m) - (z_m * t))) <= (-1d-266)) then
        tmp = x_m / (z_m * ((y - t) * 0.5d0))
    else
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-266) {
		tmp = x_m / (z_m * ((y - t) * 0.5));
	} else {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, z_s, x_m, y, z_m, t):
	tmp = 0
	if ((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-266:
		tmp = x_m / (z_m * ((y - t) * 0.5))
	else:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, z_s, x_m, y, z_m, t)
	tmp = 0.0
	if (Float64(Float64(x_m * 2.0) / Float64(Float64(y * z_m) - Float64(z_m * t))) <= -1e-266)
		tmp = Float64(x_m / Float64(z_m * Float64(Float64(y - t) * 0.5)));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -1e-266)
		tmp = x_m / (z_m * ((y - t) * 0.5));
	else
		tmp = (x_m / z_m) * (2.0 / (y - t));
	end
	tmp_2 = x_s * (z_s * tmp);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, z$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(x$95$s * N[(z$95$s * If[LessEqual[N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-266], N[(x$95$m / N[(z$95$m * N[(N[(y - t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -9.9999999999999998e-267

   1. Initial program 96.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 96.7

          \[\leadsto \frac{x}{z \cdot \left(\left(y - t\right) \cdot \color{blue}{0.5}\right)} \]

   4. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{x}{z \cdot \left(\left(y - t\right) \cdot 0.5\right)}} \]

   if -9.9999999999999998e-267 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 82.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 93.5

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

   4. Applied rewrites 93.5%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -1 \cdot 10^{-266}:\\ \;\;\;\;\frac{x}{z \cdot \left(\left(y - t\right) \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{x\_m \cdot -2}{z\_m \cdot t}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \frac{x\_m}{y \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{-2}{z\_m \cdot t}\\ \end{array}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s z_s x_m y z_m t)
 :precision binary64
 (*
  x_s
  (*
   z_s
   (if (<= t -5.6e-75)
     (/ (* x_m -2.0) (* z_m t))
     (if (<= t 3.8e-134)
       (* 2.0 (/ x_m (* y z_m)))
       (* x_m (/ -2.0 (* z_m t))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -5.6e-75) {
		tmp = (x_m * -2.0) / (z_m * t);
	} else if (t <= 3.8e-134) {
		tmp = 2.0 * (x_m / (y * z_m));
	} else {
		tmp = x_m * (-2.0 / (z_m * t));
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, z_s, x_m, y, z_m, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-5.6d-75)) then
        tmp = (x_m * (-2.0d0)) / (z_m * t)
    else if (t <= 3.8d-134) then
        tmp = 2.0d0 * (x_m / (y * z_m))
    else
        tmp = x_m * ((-2.0d0) / (z_m * t))
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -5.6e-75) {
		tmp = (x_m * -2.0) / (z_m * t);
	} else if (t <= 3.8e-134) {
		tmp = 2.0 * (x_m / (y * z_m));
	} else {
		tmp = x_m * (-2.0 / (z_m * t));
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, z_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -5.6e-75:
		tmp = (x_m * -2.0) / (z_m * t)
	elif t <= 3.8e-134:
		tmp = 2.0 * (x_m / (y * z_m))
	else:
		tmp = x_m * (-2.0 / (z_m * t))
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, z_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -5.6e-75)
		tmp = Float64(Float64(x_m * -2.0) / Float64(z_m * t));
	elseif (t <= 3.8e-134)
		tmp = Float64(2.0 * Float64(x_m / Float64(y * z_m)));
	else
		tmp = Float64(x_m * Float64(-2.0 / Float64(z_m * t)));
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -5.6e-75)
		tmp = (x_m * -2.0) / (z_m * t);
	elseif (t <= 3.8e-134)
		tmp = 2.0 * (x_m / (y * z_m));
	else
		tmp = x_m * (-2.0 / (z_m * t));
	end
	tmp_2 = x_s * (z_s * tmp);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, z$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(x$95$s * N[(z$95$s * If[LessEqual[t, -5.6e-75], N[(N[(x$95$m * -2.0), $MachinePrecision] / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e-134], N[(2.0 * N[(x$95$m / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -5.59999999999999996e-75

   1. Initial program 77.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if -5.59999999999999996e-75 < t < 3.80000000000000003e-134

   1. Initial program 93.2%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 77.7

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

   5. Applied rewrites 77.7%

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

   if 3.80000000000000003e-134 < t

   1. Initial program 89.9%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.0

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 93.4

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

   4. Applied rewrites 93.4%

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

   5. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 75.9

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

   7. Applied rewrites 75.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot -2}{z\_m \cdot t}\\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \frac{x\_m}{y \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s z_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ (* x_m -2.0) (* z_m t))))
   (*
    x_s
    (*
     z_s
     (if (<= t -5.6e-75)
       t_1
       (if (<= t 3.8e-134) (* 2.0 (/ x_m (* y z_m))) t_1))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m * -2.0) / (z_m * t);
	double tmp;
	if (t <= -5.6e-75) {
		tmp = t_1;
	} else if (t <= 3.8e-134) {
		tmp = 2.0 * (x_m / (y * z_m));
	} else {
		tmp = t_1;
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, z_s, x_m, y, z_m, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * (-2.0d0)) / (z_m * t)
    if (t <= (-5.6d-75)) then
        tmp = t_1
    else if (t <= 3.8d-134) then
        tmp = 2.0d0 * (x_m / (y * z_m))
    else
        tmp = t_1
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double t_1 = (x_m * -2.0) / (z_m * t);
	double tmp;
	if (t <= -5.6e-75) {
		tmp = t_1;
	} else if (t <= 3.8e-134) {
		tmp = 2.0 * (x_m / (y * z_m));
	} else {
		tmp = t_1;
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, z_s, x_m, y, z_m, t):
	t_1 = (x_m * -2.0) / (z_m * t)
	tmp = 0
	if t <= -5.6e-75:
		tmp = t_1
	elif t <= 3.8e-134:
		tmp = 2.0 * (x_m / (y * z_m))
	else:
		tmp = t_1
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, z_s, x_m, y, z_m, t)
	t_1 = Float64(Float64(x_m * -2.0) / Float64(z_m * t))
	tmp = 0.0
	if (t <= -5.6e-75)
		tmp = t_1;
	elseif (t <= 3.8e-134)
		tmp = Float64(2.0 * Float64(x_m / Float64(y * z_m)));
	else
		tmp = t_1;
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t)
	t_1 = (x_m * -2.0) / (z_m * t);
	tmp = 0.0;
	if (t <= -5.6e-75)
		tmp = t_1;
	elseif (t <= 3.8e-134)
		tmp = 2.0 * (x_m / (y * z_m));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * (z_s * tmp);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, z$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[(N[(x$95$m * -2.0), $MachinePrecision] / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(z$95$s * If[LessEqual[t, -5.6e-75], t$95$1, If[LessEqual[t, 3.8e-134], N[(2.0 * N[(x$95$m / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.59999999999999996e-75 or 3.80000000000000003e-134 < t

   1. Initial program 84.4%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 76.1

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

   5. Applied rewrites 76.1%

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

   if -5.59999999999999996e-75 < t < 3.80000000000000003e-134

   1. Initial program 93.2%

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

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 77.7

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

   5. Applied rewrites 77.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{-75}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot -2}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+100}:\\ \;\;\;\;\frac{x\_m}{t} \cdot \frac{-2}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m \cdot \left(\left(y - t\right) \cdot 0.5\right)}\\ \end{array}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s z_s x_m y z_m t)
 :precision binary64
 (*
  x_s
  (*
   z_s
   (if (<= t -7e+100)
     (* (/ x_m t) (/ -2.0 z_m))
     (/ x_m (* z_m (* (- y t) 0.5)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -7e+100) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else {
		tmp = x_m / (z_m * ((y - t) * 0.5));
	}
	return x_s * (z_s * tmp);
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, z_s, x_m, y, z_m, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-7d+100)) then
        tmp = (x_m / t) * ((-2.0d0) / z_m)
    else
        tmp = x_m / (z_m * ((y - t) * 0.5d0))
    end if
    code = x_s * (z_s * tmp)
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -7e+100) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else {
		tmp = x_m / (z_m * ((y - t) * 0.5));
	}
	return x_s * (z_s * tmp);
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, z_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -7e+100:
		tmp = (x_m / t) * (-2.0 / z_m)
	else:
		tmp = x_m / (z_m * ((y - t) * 0.5))
	return x_s * (z_s * tmp)

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, z_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -7e+100)
		tmp = Float64(Float64(x_m / t) * Float64(-2.0 / z_m));
	else
		tmp = Float64(x_m / Float64(z_m * Float64(Float64(y - t) * 0.5)));
	end
	return Float64(x_s * Float64(z_s * tmp))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, z_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -7e+100)
		tmp = (x_m / t) * (-2.0 / z_m);
	else
		tmp = x_m / (z_m * ((y - t) * 0.5));
	end
	tmp_2 = x_s * (z_s * tmp);
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, z$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(x$95$s * N[(z$95$s * If[LessEqual[t, -7e+100], N[(N[(x$95$m / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(z$95$m * N[(N[(y - t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -6.99999999999999953e100

   1. Initial program 67.9%

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

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

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

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 92.6

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 78.2

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

   7. Applied rewrites 78.2%

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

   9. Applied rewrites 93.7%

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

   if -6.99999999999999953e100 < t

   1. Initial program 91.3%

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

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lower--.f64 N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval 92.9

          \[\leadsto \frac{x}{z \cdot \left(\left(y - t\right) \cdot \color{blue}{0.5}\right)} \]

   4. Applied rewrites 92.9%

      \[\leadsto \color{blue}{\frac{x}{z \cdot \left(\left(y - t\right) \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 92.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(z\_s \cdot \frac{x\_m}{z\_m \cdot \left(\left(y - t\right) \cdot 0.5\right)}\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s z_s x_m y z_m t)
 :precision binary64
 (* x_s (* z_s (/ x_m (* z_m (* (- y t) 0.5))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	return x_s * (z_s * (x_m / (z_m * ((y - t) * 0.5))));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, z_s, x_m, y, z_m, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    code = x_s * (z_s * (x_m / (z_m * ((y - t) * 0.5d0))))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double z_s, double x_m, double y, double z_m, double t) {
	return x_s * (z_s * (x_m / (z_m * ((y - t) * 0.5))));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, z_s, x_m, y, z_m, t):
	return x_s * (z_s * (x_m / (z_m * ((y - t) * 0.5))))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, z_s, x_m, y, z_m, t)
	return Float64(x_s * Float64(z_s * Float64(x_m / Float64(z_m * Float64(Float64(y - t) * 0.5)))))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, z_s, x_m, y, z_m, t)
	tmp = x_s * (z_s * (x_m / (z_m * ((y - t) * 0.5))));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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_, z$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(x$95$s * N[(z$95$s * N[(x$95$m / N[(z$95$m * N[(N[(y - t), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.7%

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

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. associate-/l* N/A

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

   12. lower-*.f64 N/A

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

   13. div-inv N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

   18. lower--.f64 N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval 90.6

       \[\leadsto \frac{x}{z \cdot \left(\left(y - t\right) \cdot \color{blue}{0.5}\right)} \]

4. Applied rewrites 90.6%

   \[\leadsto \color{blue}{\frac{x}{z \cdot \left(\left(y - t\right) \cdot 0.5\right)}} \]
5. Add Preprocessing

Alternative 7: 91.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 88.0

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

   7. lift--.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 90.5

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

4. Applied rewrites 90.5%

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

5. Final simplification 90.5%

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

Alternative 8: 53.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 47.4

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

5. Applied rewrites 47.4%

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

6. Final simplification 47.4%

   \[\leadsto 2 \cdot \frac{x}{y \cdot z} \]
7. Add Preprocessing

Developer Target 1: 97.2% 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 2024238 
(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))))