Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.5% → 95.6%

Time: 7.5s

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: 89.5% 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: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot 2}{y \cdot z\_m - t \cdot z\_m} \leq -4 \cdot 10^{-320}:\\ \;\;\;\;\frac{x\_m \cdot 2}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z\_m} \cdot x\_m}{y - t}\\ \end{array}\right) \end{array} \]

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) (* t z_m))) -4e-320)
     (/ (* x_m 2.0) (* (- y t) z_m))
     (/ (* (/ 2.0 z_m) x_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) - (t * z_m))) <= -4e-320) {
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	} else {
		tmp = ((2.0 / z_m) * x_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) - (t * z_m))) <= (-4d-320)) then
        tmp = (x_m * 2.0d0) / ((y - t) * z_m)
    else
        tmp = ((2.0d0 / z_m) * x_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) - (t * z_m))) <= -4e-320) {
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	} else {
		tmp = ((2.0 / z_m) * x_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) - (t * z_m))) <= -4e-320:
		tmp = (x_m * 2.0) / ((y - t) * z_m)
	else:
		tmp = ((2.0 / z_m) * x_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(t * z_m))) <= -4e-320)
		tmp = Float64(Float64(x_m * 2.0) / Float64(Float64(y - t) * z_m));
	else
		tmp = Float64(Float64(Float64(2.0 / z_m) * x_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) - (t * z_m))) <= -4e-320)
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	else
		tmp = ((2.0 / z_m) * x_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[(t * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-320], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / z$95$m), $MachinePrecision] * x$95$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))) < -3.99996e-320

   1. Initial program 98.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 98.5

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

   4. Applied rewrites 98.5%

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

   if -3.99996e-320 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 83.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. times-frac N/A

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

      9. associate-*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 96.9

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

   4. Applied rewrites 96.9%

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

Alternative 2: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot 2}{y \cdot z\_m - t \cdot z\_m} \leq -4 \cdot 10^{-320}:\\ \;\;\;\;\frac{x\_m \cdot 2}{\left(y - t\right) \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \end{array}\right) \end{array} \]

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) (* t z_m))) -4e-320)
     (/ (* x_m 2.0) (* (- y t) z_m))
     (* (/ 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) - (t * z_m))) <= -4e-320) {
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	} 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) - (t * z_m))) <= (-4d-320)) then
        tmp = (x_m * 2.0d0) / ((y - t) * z_m)
    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) - (t * z_m))) <= -4e-320) {
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	} 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) - (t * z_m))) <= -4e-320:
		tmp = (x_m * 2.0) / ((y - t) * z_m)
	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(t * z_m))) <= -4e-320)
		tmp = Float64(Float64(x_m * 2.0) / Float64(Float64(y - t) * z_m));
	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) - (t * z_m))) <= -4e-320)
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	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[(t * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-320], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $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))) < -3.99996e-320

   1. Initial program 98.5%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 98.5

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

   4. Applied rewrites 98.5%

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

   if -3.99996e-320 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 83.9%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. 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 96.9

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

   4. Applied rewrites 96.9%

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

Alternative 3: 91.6% accurate, 0.6× speedup

Math

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

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 (<= (- (* y z_m) (* t z_m)) (- INFINITY))
     (/ -2.0 (* (/ t x_m) z_m))
     (/ (* x_m 2.0) (* (- y t) 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) {
	double tmp;
	if (((y * z_m) - (t * z_m)) <= -((double) INFINITY)) {
		tmp = -2.0 / ((t / x_m) * z_m);
	} else {
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	}
	return x_s * (z_s * tmp);
}

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 (((y * z_m) - (t * z_m)) <= -Double.POSITIVE_INFINITY) {
		tmp = -2.0 / ((t / x_m) * z_m);
	} else {
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	}
	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 ((y * z_m) - (t * z_m)) <= -math.inf:
		tmp = -2.0 / ((t / x_m) * z_m)
	else:
		tmp = (x_m * 2.0) / ((y - t) * z_m)
	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(y * z_m) - Float64(t * z_m)) <= Float64(-Inf))
		tmp = Float64(-2.0 / Float64(Float64(t / x_m) * z_m));
	else
		tmp = Float64(Float64(x_m * 2.0) / Float64(Float64(y - t) * z_m));
	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 (((y * z_m) - (t * z_m)) <= -Inf)
		tmp = -2.0 / ((t / x_m) * z_m);
	else
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	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[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(-2.0 / N[(N[(t / x$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -inf.0

   1. Initial program 60.8%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 37.0%

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

   9. Applied rewrites 53.9%

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

   if -inf.0 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 96.0

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

   4. Applied rewrites 96.0%

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

Alternative 4: 91.6% accurate, 0.6× speedup

Math

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

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 (<= (- (* y z_m) (* t z_m)) (- INFINITY))
     (* (/ x_m t) (/ -2.0 z_m))
     (/ (* x_m 2.0) (* (- y t) 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) {
	double tmp;
	if (((y * z_m) - (t * z_m)) <= -((double) INFINITY)) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else {
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	}
	return x_s * (z_s * tmp);
}

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 (((y * z_m) - (t * z_m)) <= -Double.POSITIVE_INFINITY) {
		tmp = (x_m / t) * (-2.0 / z_m);
	} else {
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	}
	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 ((y * z_m) - (t * z_m)) <= -math.inf:
		tmp = (x_m / t) * (-2.0 / z_m)
	else:
		tmp = (x_m * 2.0) / ((y - t) * z_m)
	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(y * z_m) - Float64(t * z_m)) <= Float64(-Inf))
		tmp = Float64(Float64(x_m / t) * Float64(-2.0 / z_m));
	else
		tmp = Float64(Float64(x_m * 2.0) / Float64(Float64(y - t) * z_m));
	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 (((y * z_m) - (t * z_m)) <= -Inf)
		tmp = (x_m / t) * (-2.0 / z_m);
	else
		tmp = (x_m * 2.0) / ((y - t) * z_m);
	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[(y * z$95$m), $MachinePrecision] - N[(t * z$95$m), $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(N[(x$95$m / t), $MachinePrecision] * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 y z) (*.f64 t z)) < -inf.0

   1. Initial program 60.8%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 37.0

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

   5. Applied rewrites 37.0%

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

   7. Applied rewrites 53.8%

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

   if -inf.0 < (-.f64 (*.f64 y z) (*.f64 t z))

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 96.0

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

   4. Applied rewrites 96.0%

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

Alternative 5: 73.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+59} \lor \neg \left(t \leq 5.2 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{x\_m}{t \cdot z\_m} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m \cdot y} \cdot 2\\ \end{array}\right) \end{array} \]

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 (or (<= t -7e+59) (not (<= t 5.2e-59)))
     (* (/ x_m (* t z_m)) -2.0)
     (* (/ x_m (* z_m y)) 2.0)))))

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+59) || !(t <= 5.2e-59)) {
		tmp = (x_m / (t * z_m)) * -2.0;
	} else {
		tmp = (x_m / (z_m * y)) * 2.0;
	}
	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+59)) .or. (.not. (t <= 5.2d-59))) then
        tmp = (x_m / (t * z_m)) * (-2.0d0)
    else
        tmp = (x_m / (z_m * y)) * 2.0d0
    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+59) || !(t <= 5.2e-59)) {
		tmp = (x_m / (t * z_m)) * -2.0;
	} else {
		tmp = (x_m / (z_m * y)) * 2.0;
	}
	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+59) or not (t <= 5.2e-59):
		tmp = (x_m / (t * z_m)) * -2.0
	else:
		tmp = (x_m / (z_m * y)) * 2.0
	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+59) || !(t <= 5.2e-59))
		tmp = Float64(Float64(x_m / Float64(t * z_m)) * -2.0);
	else
		tmp = Float64(Float64(x_m / Float64(z_m * y)) * 2.0);
	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+59) || ~((t <= 5.2e-59)))
		tmp = (x_m / (t * z_m)) * -2.0;
	else
		tmp = (x_m / (z_m * y)) * 2.0;
	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[Or[LessEqual[t, -7e+59], N[Not[LessEqual[t, 5.2e-59]], $MachinePrecision]], N[(N[(x$95$m / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], N[(N[(x$95$m / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -7e59 or 5.19999999999999996e-59 < t

   1. Initial program 87.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 78.6

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

   5. Applied rewrites 78.6%

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

   if -7e59 < t < 5.19999999999999996e-59

   1. Initial program 89.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   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 78.4

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

   5. Applied rewrites 78.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+59} \lor \neg \left(t \leq 5.2 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{x}{t \cdot z} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot y} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+59}:\\ \;\;\;\;\frac{x\_m}{t \cdot z\_m} \cdot -2\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{x\_m}{z\_m \cdot y} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot x\_m}{t \cdot z\_m}\\ \end{array}\right) \end{array} \]

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+59)
     (* (/ x_m (* t z_m)) -2.0)
     (if (<= t 5.2e-59)
       (* (/ x_m (* z_m y)) 2.0)
       (/ (* -2.0 x_m) (* t 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) {
	double tmp;
	if (t <= -7e+59) {
		tmp = (x_m / (t * z_m)) * -2.0;
	} else if (t <= 5.2e-59) {
		tmp = (x_m / (z_m * y)) * 2.0;
	} else {
		tmp = (-2.0 * x_m) / (t * z_m);
	}
	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+59)) then
        tmp = (x_m / (t * z_m)) * (-2.0d0)
    else if (t <= 5.2d-59) then
        tmp = (x_m / (z_m * y)) * 2.0d0
    else
        tmp = ((-2.0d0) * x_m) / (t * z_m)
    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+59) {
		tmp = (x_m / (t * z_m)) * -2.0;
	} else if (t <= 5.2e-59) {
		tmp = (x_m / (z_m * y)) * 2.0;
	} else {
		tmp = (-2.0 * x_m) / (t * z_m);
	}
	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+59:
		tmp = (x_m / (t * z_m)) * -2.0
	elif t <= 5.2e-59:
		tmp = (x_m / (z_m * y)) * 2.0
	else:
		tmp = (-2.0 * x_m) / (t * z_m)
	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+59)
		tmp = Float64(Float64(x_m / Float64(t * z_m)) * -2.0);
	elseif (t <= 5.2e-59)
		tmp = Float64(Float64(x_m / Float64(z_m * y)) * 2.0);
	else
		tmp = Float64(Float64(-2.0 * x_m) / Float64(t * z_m));
	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+59)
		tmp = (x_m / (t * z_m)) * -2.0;
	elseif (t <= 5.2e-59)
		tmp = (x_m / (z_m * y)) * 2.0;
	else
		tmp = (-2.0 * x_m) / (t * z_m);
	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+59], N[(N[(x$95$m / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[t, 5.2e-59], N[(N[(x$95$m / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(-2.0 * x$95$m), $MachinePrecision] / N[(t * z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -7e59

   1. Initial program 91.0%

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

   3. Taylor expanded in y around 0

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

      1. *-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 84.4

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

   5. Applied rewrites 84.4%

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

   if -7e59 < t < 5.19999999999999996e-59

   1. Initial program 89.7%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{2 \cdot \frac{x}{y \cdot z}} \]
   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 78.4

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

   5. Applied rewrites 78.4%

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

   if 5.19999999999999996e-59 < t

   1. Initial program 86.0%

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

   3. Taylor expanded in y around 0

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

      1. *-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 75.6

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

   5. Applied rewrites 75.6%

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

   7. Applied rewrites 75.6%

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

Alternative 7: 91.6% accurate, 1.2× speedup

Math

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

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

Derivation

1. Initial program 88.7%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 92.3

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

4. Applied rewrites 92.3%

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

Alternative 8: 53.1% accurate, 1.4× speedup

Math

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

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 (* t z_m)) -2.0))))

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 / (t * z_m)) * -2.0));
}

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 / (t * z_m)) * (-2.0d0)))
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 / (t * z_m)) * -2.0));
}

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 / (t * z_m)) * -2.0))

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

Derivation

1. Initial program 88.7%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 52.8

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

5. Applied rewrites 52.8%

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

Developer Target 1: 97.3% 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 2024315 
(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))))