Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.5% → 96.1%

Time: 7.5s

Alternatives: 9

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: 89.5% 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: 96.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.4 \cdot 10^{-95}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z\_m} \cdot x}{y - t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2.4e-95)
    (/ (* x 2.0) (* (- y t) z_m))
    (/ (* (/ 2.0 z_m) x) (- y t)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.4e-95) {
		tmp = (x * 2.0) / ((y - t) * z_m);
	} else {
		tmp = ((2.0 / z_m) * x) / (y - t);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 2.4d-95) then
        tmp = (x * 2.0d0) / ((y - t) * z_m)
    else
        tmp = ((2.0d0 / z_m) * x) / (y - t)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.4e-95) {
		tmp = (x * 2.0) / ((y - t) * z_m);
	} else {
		tmp = ((2.0 / z_m) * x) / (y - t);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 2.4e-95:
		tmp = (x * 2.0) / ((y - t) * z_m)
	else:
		tmp = ((2.0 / z_m) * x) / (y - t)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2.4e-95)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(Float64(2.0 / z_m) * x) / Float64(y - t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 2.4e-95)
		tmp = (x * 2.0) / ((y - t) * z_m);
	else
		tmp = ((2.0 / z_m) * x) / (y - t);
	end
	tmp_2 = 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]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 2.4e-95], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / z$95$m), $MachinePrecision] * x), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.4e-95

   1. Initial program 88.8%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 91.3

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

   4. Applied rewrites 91.3%

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

   if 2.4e-95 < z

   1. Initial program 91.1%

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

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 98.4

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

   4. Applied rewrites 98.4%

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

Alternative 2: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -9.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z\_m \cdot y} \cdot 2\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-35}:\\ \;\;\;\;\frac{x \cdot 2}{\left(-t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m \cdot y} \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y -9.1e-7)
    (* (/ x (* z_m y)) 2.0)
    (if (<= y 9e-35) (/ (* x 2.0) (* (- t) z_m)) (* (/ 2.0 (* z_m y)) x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -9.1e-7) {
		tmp = (x / (z_m * y)) * 2.0;
	} else if (y <= 9e-35) {
		tmp = (x * 2.0) / (-t * z_m);
	} else {
		tmp = (2.0 / (z_m * y)) * x;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-9.1d-7)) then
        tmp = (x / (z_m * y)) * 2.0d0
    else if (y <= 9d-35) then
        tmp = (x * 2.0d0) / (-t * z_m)
    else
        tmp = (2.0d0 / (z_m * y)) * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -9.1e-7) {
		tmp = (x / (z_m * y)) * 2.0;
	} else if (y <= 9e-35) {
		tmp = (x * 2.0) / (-t * z_m);
	} else {
		tmp = (2.0 / (z_m * y)) * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if y <= -9.1e-7:
		tmp = (x / (z_m * y)) * 2.0
	elif y <= 9e-35:
		tmp = (x * 2.0) / (-t * z_m)
	else:
		tmp = (2.0 / (z_m * y)) * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (y <= -9.1e-7)
		tmp = Float64(Float64(x / Float64(z_m * y)) * 2.0);
	elseif (y <= 9e-35)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(-t) * z_m));
	else
		tmp = Float64(Float64(2.0 / Float64(z_m * y)) * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (y <= -9.1e-7)
		tmp = (x / (z_m * y)) * 2.0;
	elseif (y <= 9e-35)
		tmp = (x * 2.0) / (-t * z_m);
	else
		tmp = (2.0 / (z_m * y)) * x;
	end
	tmp_2 = 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]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[y, -9.1e-7], N[(N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[y, 9e-35], N[(N[(x * 2.0), $MachinePrecision] / N[((-t) * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -9.0999999999999997e-7

   1. Initial program 90.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 82.5

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

   5. Applied rewrites 82.5%

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

   if -9.0999999999999997e-7 < y < 9.0000000000000002e-35

   1. Initial program 90.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 78.2

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

   5. Applied rewrites 78.2%

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

   if 9.0000000000000002e-35 < y

   1. Initial program 88.3%

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

      1. 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. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 88.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.2

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

   6. Applied rewrites 88.2%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.2

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

   9. Applied rewrites 69.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z \cdot y} \cdot 2\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-35}:\\ \;\;\;\;\frac{x \cdot 2}{\left(-t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{x \cdot 2}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + x}{y - t}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 5.6e-59)
    (/ (* x 2.0) (* (- y t) z_m))
    (/ (/ (+ x x) (- y t)) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 5.6e-59) {
		tmp = (x * 2.0) / ((y - t) * z_m);
	} else {
		tmp = ((x + x) / (y - t)) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 5.6d-59) then
        tmp = (x * 2.0d0) / ((y - t) * z_m)
    else
        tmp = ((x + x) / (y - t)) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 5.6e-59) {
		tmp = (x * 2.0) / ((y - t) * z_m);
	} else {
		tmp = ((x + x) / (y - t)) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 5.6e-59:
		tmp = (x * 2.0) / ((y - t) * z_m)
	else:
		tmp = ((x + x) / (y - t)) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 5.6e-59)
		tmp = Float64(Float64(x * 2.0) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(Float64(x + x) / Float64(y - t)) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 5.6e-59)
		tmp = (x * 2.0) / ((y - t) * z_m);
	else
		tmp = ((x + x) / (y - t)) / z_m;
	end
	tmp_2 = 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]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 5.6e-59], N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + x), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.59999999999999961e-59

   1. Initial program 89.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 91.4

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

   4. Applied rewrites 91.4%

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

   if 5.59999999999999961e-59 < z

   1. Initial program 90.8%

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

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

      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. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower--.f64 97.6

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

   4. Applied rewrites 97.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 97.6

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

   6. Applied rewrites 97.6%

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

Alternative 4: 74.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -9.1 \cdot 10^{-7} \lor \neg \left(y \leq 9 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x}{z\_m \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot z\_m} \cdot -2\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= y -9.1e-7) (not (<= y 9e-35)))
    (* (/ x (* z_m y)) 2.0)
    (* (/ x (* t z_m)) -2.0))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((y <= -9.1e-7) || !(y <= 9e-35)) {
		tmp = (x / (z_m * y)) * 2.0;
	} else {
		tmp = (x / (t * z_m)) * -2.0;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-9.1d-7)) .or. (.not. (y <= 9d-35))) then
        tmp = (x / (z_m * y)) * 2.0d0
    else
        tmp = (x / (t * z_m)) * (-2.0d0)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((y <= -9.1e-7) || !(y <= 9e-35)) {
		tmp = (x / (z_m * y)) * 2.0;
	} else {
		tmp = (x / (t * z_m)) * -2.0;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (y <= -9.1e-7) or not (y <= 9e-35):
		tmp = (x / (z_m * y)) * 2.0
	else:
		tmp = (x / (t * z_m)) * -2.0
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if ((y <= -9.1e-7) || !(y <= 9e-35))
		tmp = Float64(Float64(x / Float64(z_m * y)) * 2.0);
	else
		tmp = Float64(Float64(x / Float64(t * z_m)) * -2.0);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((y <= -9.1e-7) || ~((y <= 9e-35)))
		tmp = (x / (z_m * y)) * 2.0;
	else
		tmp = (x / (t * z_m)) * -2.0;
	end
	tmp_2 = 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]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[Or[LessEqual[y, -9.1e-7], N[Not[LessEqual[y, 9e-35]], $MachinePrecision]], N[(N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -9.0999999999999997e-7 or 9.0000000000000002e-35 < y

   1. Initial program 89.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-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 76.7

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

   5. Applied rewrites 76.7%

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

   if -9.0999999999999997e-7 < y < 9.0000000000000002e-35

   1. Initial program 90.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-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 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.1 \cdot 10^{-7} \lor \neg \left(y \leq 9 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{x}{z \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -9.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z\_m \cdot y} \cdot 2\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{t \cdot z\_m} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m \cdot y} \cdot x\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y -9.1e-7)
    (* (/ x (* z_m y)) 2.0)
    (if (<= y 9e-35) (* (/ x (* t z_m)) -2.0) (* (/ 2.0 (* z_m y)) x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -9.1e-7) {
		tmp = (x / (z_m * y)) * 2.0;
	} else if (y <= 9e-35) {
		tmp = (x / (t * z_m)) * -2.0;
	} else {
		tmp = (2.0 / (z_m * y)) * x;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-9.1d-7)) then
        tmp = (x / (z_m * y)) * 2.0d0
    else if (y <= 9d-35) then
        tmp = (x / (t * z_m)) * (-2.0d0)
    else
        tmp = (2.0d0 / (z_m * y)) * x
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -9.1e-7) {
		tmp = (x / (z_m * y)) * 2.0;
	} else if (y <= 9e-35) {
		tmp = (x / (t * z_m)) * -2.0;
	} else {
		tmp = (2.0 / (z_m * y)) * x;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if y <= -9.1e-7:
		tmp = (x / (z_m * y)) * 2.0
	elif y <= 9e-35:
		tmp = (x / (t * z_m)) * -2.0
	else:
		tmp = (2.0 / (z_m * y)) * x
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (y <= -9.1e-7)
		tmp = Float64(Float64(x / Float64(z_m * y)) * 2.0);
	elseif (y <= 9e-35)
		tmp = Float64(Float64(x / Float64(t * z_m)) * -2.0);
	else
		tmp = Float64(Float64(2.0 / Float64(z_m * y)) * x);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (y <= -9.1e-7)
		tmp = (x / (z_m * y)) * 2.0;
	elseif (y <= 9e-35)
		tmp = (x / (t * z_m)) * -2.0;
	else
		tmp = (2.0 / (z_m * y)) * x;
	end
	tmp_2 = 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]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[y, -9.1e-7], N[(N[(x / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], If[LessEqual[y, 9e-35], N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -9.0999999999999997e-7

   1. Initial program 90.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 82.5

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

   5. Applied rewrites 82.5%

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

   if -9.0999999999999997e-7 < y < 9.0000000000000002e-35

   1. Initial program 90.1%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. *-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 78.1

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

   5. Applied rewrites 78.1%

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

   if 9.0000000000000002e-35 < y

   1. Initial program 88.3%

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

      1. 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. lift--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

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

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-/.f64 88.2

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.2

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

   6. Applied rewrites 88.2%

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

   7. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 69.2

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

   9. Applied rewrites 69.2%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{x}{z \cdot y} \cdot 2\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-35}:\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t)
	tmp = z_s * ((x * 2.0) / ((y - t) * 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]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.7%

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

   1. lift--.f64 N/A

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

   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 91.2

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

4. Applied rewrites 91.2%

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

Alternative 7: 91.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t)
	tmp = z_s * ((2.0 / (z_m * (y - t))) * x);
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]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * N[(N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.7%

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

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

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

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 91.3

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

4. Applied rewrites 91.3%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lower-/.f64 90.7

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 90.7

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

6. Applied rewrites 90.7%

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

Alternative 8: 53.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t)
	tmp = z_s * ((x / (t * z_m)) * -2.0);
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]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * N[(N[(x / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.7%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 53.9

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

5. Applied rewrites 53.9%

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

Alternative 9: 21.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m, t)
	tmp = z_s * ((x / (z_m * t)) * 2.0);
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]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * N[(N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.7%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 53.9

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

5. Applied rewrites 53.9%

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

7. Applied rewrites 20.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 20.4%

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

Developer Target 1: 96.8% 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 2024326 
(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))))