Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.8% → 94.6%

Time: 10.0s

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: 89.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot 2}{y \cdot z\_m - z\_m \cdot t} \leq 0:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z\_m}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (/ (* x_m 2.0) (- (* y z_m) (* z_m t))) 0.0)
     (* 2.0 (/ (/ x_m z_m) (- y t)))
     (/ (* x_m 2.0) (* z_m (- y t)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= 0.0) {
		tmp = 2.0 * ((x_m / z_m) / (y - t));
	} else {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x_m * 2.0d0) / ((y * z_m) - (z_m * t))) <= 0.0d0) then
        tmp = 2.0d0 * ((x_m / z_m) / (y - t))
    else
        tmp = (x_m * 2.0d0) / (z_m * (y - t))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= 0.0) {
		tmp = 2.0 * ((x_m / z_m) / (y - t));
	} else {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if ((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= 0.0:
		tmp = 2.0 * ((x_m / z_m) / (y - t))
	else:
		tmp = (x_m * 2.0) / (z_m * (y - t))
	return z_s * (x_s * tmp)

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z$95$m * 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))) < -0.0

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in x around 0 91.5%

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

      1. associate-/r* 92.7%

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

   7. Simplified 92.7%

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

   if -0.0 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 93.4%

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

      1. distribute-rgt-out-- 97.7%

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

   3. Simplified 97.7%

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

4. Final simplification 94.6%

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

Alternative 2: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-31} \lor \neg \left(t \leq 6.2 \cdot 10^{-17}\right):\\ \;\;\;\;x\_m \cdot \frac{-2}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{2}{y \cdot z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (or (<= t -1.06e-31) (not (<= t 6.2e-17)))
     (* x_m (/ -2.0 (* z_m t)))
     (* x_m (/ 2.0 (* y z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((t <= -1.06e-31) || !(t <= 6.2e-17)) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else {
		tmp = x_m * (2.0 / (y * z_m));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_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 <= (-1.06d-31)) .or. (.not. (t <= 6.2d-17))) then
        tmp = x_m * ((-2.0d0) / (z_m * t))
    else
        tmp = x_m * (2.0d0 / (y * z_m))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((t <= -1.06e-31) || !(t <= 6.2e-17)) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else {
		tmp = x_m * (2.0 / (y * z_m));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (t <= -1.06e-31) or not (t <= 6.2e-17):
		tmp = x_m * (-2.0 / (z_m * t))
	else:
		tmp = x_m * (2.0 / (y * z_m))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if ((t <= -1.06e-31) || !(t <= 6.2e-17))
		tmp = Float64(x_m * Float64(-2.0 / Float64(z_m * t)));
	else
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z_m)));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((t <= -1.06e-31) || ~((t <= 6.2e-17)))
		tmp = x_m * (-2.0 / (z_m * t));
	else
		tmp = x_m * (2.0 / (y * z_m));
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[Or[LessEqual[t, -1.06e-31], N[Not[LessEqual[t, 6.2e-17]], $MachinePrecision]], N[(x$95$m * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(2.0 / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -1.06e-31 or 6.1999999999999997e-17 < t

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in y around 0 79.8%

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

      1. associate-*r/ 79.8%

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

      2. *-commutative 79.8%

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

      3. *-commutative 79.8%

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

      4. associate-/l* 79.8%

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

   7. Simplified 79.8%

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

   if -1.06e-31 < t < 6.1999999999999997e-17

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 95.2%

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

   3. Simplified 95.2%

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

      1. distribute-rgt-out-- 92.9%

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

      2. associate-/l* 91.3%

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

      3. *-commutative 91.3%

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

      4. distribute-rgt-out-- 93.7%

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

   6. Applied egg-rr 93.7%

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

   7. Taylor expanded in y around inf 80.1%

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

      1. *-commutative 80.1%

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

   9. Simplified 80.1%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{-31} \lor \neg \left(t \leq 6.2 \cdot 10^{-17}\right):\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-30} \lor \neg \left(t \leq 2.8 \cdot 10^{-16}\right):\\ \;\;\;\;x\_m \cdot \frac{-2}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (or (<= t -1.65e-30) (not (<= t 2.8e-16)))
     (* x_m (/ -2.0 (* z_m t)))
     (* (/ x_m z_m) (/ 2.0 y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((t <= -1.65e-30) || !(t <= 2.8e-16)) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else {
		tmp = (x_m / z_m) * (2.0 / y);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_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 <= (-1.65d-30)) .or. (.not. (t <= 2.8d-16))) then
        tmp = x_m * ((-2.0d0) / (z_m * t))
    else
        tmp = (x_m / z_m) * (2.0d0 / y)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((t <= -1.65e-30) || !(t <= 2.8e-16)) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else {
		tmp = (x_m / z_m) * (2.0 / y);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (t <= -1.65e-30) or not (t <= 2.8e-16):
		tmp = x_m * (-2.0 / (z_m * t))
	else:
		tmp = (x_m / z_m) * (2.0 / y)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if ((t <= -1.65e-30) || !(t <= 2.8e-16))
		tmp = Float64(x_m * Float64(-2.0 / Float64(z_m * t)));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / y));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((t <= -1.65e-30) || ~((t <= 2.8e-16)))
		tmp = x_m * (-2.0 / (z_m * t));
	else
		tmp = (x_m / z_m) * (2.0 / y);
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[Or[LessEqual[t, -1.65e-30], N[Not[LessEqual[t, 2.8e-16]], $MachinePrecision]], N[(x$95$m * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -1.6500000000000001e-30 or 2.8000000000000001e-16 < t

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in y around 0 79.8%

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

      1. associate-*r/ 79.8%

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

      2. *-commutative 79.8%

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

      3. *-commutative 79.8%

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

      4. associate-/l* 79.8%

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

   7. Simplified 79.8%

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

   if -1.6500000000000001e-30 < t < 2.8000000000000001e-16

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in y around inf 81.6%

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

      1. *-commutative 80.1%

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

   7. Simplified 81.6%

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

      1. times-frac 81.5%

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

   9. Applied egg-rr 81.5%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{-30} \lor \neg \left(t \leq 2.8 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-30} \lor \neg \left(t \leq 1.65 \cdot 10^{-19}\right):\\ \;\;\;\;x\_m \cdot \frac{-2}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (or (<= t -3.5e-30) (not (<= t 1.65e-19)))
     (* x_m (/ -2.0 (* z_m t)))
     (/ (* x_m 2.0) (* y z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((t <= -3.5e-30) || !(t <= 1.65e-19)) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else {
		tmp = (x_m * 2.0) / (y * z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_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 <= (-3.5d-30)) .or. (.not. (t <= 1.65d-19))) then
        tmp = x_m * ((-2.0d0) / (z_m * t))
    else
        tmp = (x_m * 2.0d0) / (y * z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((t <= -3.5e-30) || !(t <= 1.65e-19)) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else {
		tmp = (x_m * 2.0) / (y * z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (t <= -3.5e-30) or not (t <= 1.65e-19):
		tmp = x_m * (-2.0 / (z_m * t))
	else:
		tmp = (x_m * 2.0) / (y * z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if ((t <= -3.5e-30) || !(t <= 1.65e-19))
		tmp = Float64(x_m * Float64(-2.0 / Float64(z_m * t)));
	else
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((t <= -3.5e-30) || ~((t <= 1.65e-19)))
		tmp = x_m * (-2.0 / (z_m * t));
	else
		tmp = (x_m * 2.0) / (y * z_m);
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[Or[LessEqual[t, -3.5e-30], N[Not[LessEqual[t, 1.65e-19]], $MachinePrecision]], N[(x$95$m * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -3.5000000000000003e-30 or 1.6499999999999999e-19 < t

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in y around 0 79.8%

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

      1. associate-*r/ 79.8%

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

      2. *-commutative 79.8%

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

      3. *-commutative 79.8%

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

      4. associate-/l* 79.8%

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

   7. Simplified 79.8%

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

   if -3.5000000000000003e-30 < t < 1.6499999999999999e-19

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in y around inf 81.6%

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

      1. *-commutative 80.1%

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

   7. Simplified 81.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-30} \lor \neg \left(t \leq 1.65 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-29} \lor \neg \left(t \leq 8.8 \cdot 10^{-19}\right):\\ \;\;\;\;x\_m \cdot \frac{-2}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z\_m}}{y \cdot 0.5}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (or (<= t -7.5e-29) (not (<= t 8.8e-19)))
     (* x_m (/ -2.0 (* z_m t)))
     (/ (/ x_m z_m) (* y 0.5))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((t <= -7.5e-29) || !(t <= 8.8e-19)) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else {
		tmp = (x_m / z_m) / (y * 0.5);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_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 <= (-7.5d-29)) .or. (.not. (t <= 8.8d-19))) then
        tmp = x_m * ((-2.0d0) / (z_m * t))
    else
        tmp = (x_m / z_m) / (y * 0.5d0)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((t <= -7.5e-29) || !(t <= 8.8e-19)) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else {
		tmp = (x_m / z_m) / (y * 0.5);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (t <= -7.5e-29) or not (t <= 8.8e-19):
		tmp = x_m * (-2.0 / (z_m * t))
	else:
		tmp = (x_m / z_m) / (y * 0.5)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if ((t <= -7.5e-29) || !(t <= 8.8e-19))
		tmp = Float64(x_m * Float64(-2.0 / Float64(z_m * t)));
	else
		tmp = Float64(Float64(x_m / z_m) / Float64(y * 0.5));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((t <= -7.5e-29) || ~((t <= 8.8e-19)))
		tmp = x_m * (-2.0 / (z_m * t));
	else
		tmp = (x_m / z_m) / (y * 0.5);
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[Or[LessEqual[t, -7.5e-29], N[Not[LessEqual[t, 8.8e-19]], $MachinePrecision]], N[(x$95$m * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] / N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -7.50000000000000006e-29 or 8.7999999999999994e-19 < t

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in y around 0 79.8%

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

      1. associate-*r/ 79.8%

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

      2. *-commutative 79.8%

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

      3. *-commutative 79.8%

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

      4. associate-/l* 79.8%

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

   7. Simplified 79.8%

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

   if -7.50000000000000006e-29 < t < 8.7999999999999994e-19

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in y around inf 81.6%

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

      1. *-commutative 80.1%

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

   7. Simplified 81.6%

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

      1. times-frac 81.5%

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

   9. Applied egg-rr 81.5%

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

       1. clear-num 81.5%

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

       2. un-div-inv 81.7%

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

       3. div-inv 81.7%

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

       4. metadata-eval 81.7%

          \[\leadsto \frac{\frac{x}{z}}{y \cdot \color{blue}{0.5}} \]

   11. Applied egg-rr 81.7%

       \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-29} \lor \neg \left(t \leq 8.8 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{y \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-30}:\\ \;\;\;\;x\_m \cdot \frac{-2}{z\_m \cdot t}\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{x\_m}{z\_m}}{y \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{z\_m}}{t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -3.5e-30)
     (* x_m (/ -2.0 (* z_m t)))
     (if (<= t 9.4e-17)
       (/ (/ x_m z_m) (* y 0.5))
       (/ (/ (* x_m -2.0) z_m) t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -3.5e-30) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else if (t <= 9.4e-17) {
		tmp = (x_m / z_m) / (y * 0.5);
	} else {
		tmp = ((x_m * -2.0) / z_m) / t;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_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 <= (-3.5d-30)) then
        tmp = x_m * ((-2.0d0) / (z_m * t))
    else if (t <= 9.4d-17) then
        tmp = (x_m / z_m) / (y * 0.5d0)
    else
        tmp = ((x_m * (-2.0d0)) / z_m) / t
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (t <= -3.5e-30) {
		tmp = x_m * (-2.0 / (z_m * t));
	} else if (t <= 9.4e-17) {
		tmp = (x_m / z_m) / (y * 0.5);
	} else {
		tmp = ((x_m * -2.0) / z_m) / t;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if t <= -3.5e-30:
		tmp = x_m * (-2.0 / (z_m * t))
	elif t <= 9.4e-17:
		tmp = (x_m / z_m) / (y * 0.5)
	else:
		tmp = ((x_m * -2.0) / z_m) / t
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (t <= -3.5e-30)
		tmp = Float64(x_m * Float64(-2.0 / Float64(z_m * t)));
	elseif (t <= 9.4e-17)
		tmp = Float64(Float64(x_m / z_m) / Float64(y * 0.5));
	else
		tmp = Float64(Float64(Float64(x_m * -2.0) / z_m) / t);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (t <= -3.5e-30)
		tmp = x_m * (-2.0 / (z_m * t));
	elseif (t <= 9.4e-17)
		tmp = (x_m / z_m) / (y * 0.5);
	else
		tmp = ((x_m * -2.0) / z_m) / t;
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[t, -3.5e-30], N[(x$95$m * N[(-2.0 / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.4e-17], N[(N[(x$95$m / z$95$m), $MachinePrecision] / N[(y * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / z$95$m), $MachinePrecision] / t), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -3.5000000000000003e-30

   1. Initial program 93.0%

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

      1. distribute-rgt-out-- 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in y around 0 81.6%

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

      1. associate-*r/ 81.6%

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

      2. *-commutative 81.6%

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

      3. *-commutative 81.6%

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

      4. associate-/l* 81.7%

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

   7. Simplified 81.7%

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

   if -3.5000000000000003e-30 < t < 9.3999999999999999e-17

   1. Initial program 92.9%

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

      1. distribute-rgt-out-- 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in y around inf 81.6%

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

      1. *-commutative 80.1%

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

   7. Simplified 81.6%

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

      1. times-frac 81.5%

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

   9. Applied egg-rr 81.5%

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

       1. clear-num 81.5%

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

       2. un-div-inv 81.7%

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

       3. div-inv 81.7%

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

       4. metadata-eval 81.7%

          \[\leadsto \frac{\frac{x}{z}}{y \cdot \color{blue}{0.5}} \]

   11. Applied egg-rr 81.7%

       \[\leadsto \color{blue}{\frac{\frac{x}{z}}{y \cdot 0.5}} \]

   if 9.3999999999999999e-17 < t

   1. Initial program 88.2%

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

      1. distribute-rgt-out-- 90.2%

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

   3. Simplified 90.2%

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

      1. distribute-rgt-out-- 88.2%

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

      2. associate-/l* 88.2%

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

      3. *-commutative 88.2%

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

      4. distribute-rgt-out-- 90.2%

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

   6. Applied egg-rr 90.2%

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

   7. Taylor expanded in y around 0 77.5%

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

      1. *-commutative 77.5%

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

      2. associate-*r/ 77.6%

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

      3. metadata-eval 77.6%

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

      4. distribute-rgt-neg-in 77.6%

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

      5. *-commutative 77.6%

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

      6. associate-/r* 82.0%

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

      7. distribute-rgt-neg-in 82.0%

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

      8. metadata-eval 82.0%

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

   9. Applied egg-rr 82.0%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{x}{z}}{y \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.3 \cdot 10^{-109}:\\ \;\;\;\;x\_m \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z\_m}}{y - t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 2.3e-109)
     (* x_m (/ 2.0 (* z_m (- y t))))
     (* 2.0 (/ (/ x_m z_m) (- y t)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.3e-109) {
		tmp = x_m * (2.0 / (z_m * (y - t)));
	} else {
		tmp = 2.0 * ((x_m / z_m) / (y - t));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_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 (z_m <= 2.3d-109) then
        tmp = x_m * (2.0d0 / (z_m * (y - t)))
    else
        tmp = 2.0d0 * ((x_m / z_m) / (y - t))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.3e-109) {
		tmp = x_m * (2.0 / (z_m * (y - t)));
	} else {
		tmp = 2.0 * ((x_m / z_m) / (y - t));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if z_m <= 2.3e-109:
		tmp = x_m * (2.0 / (z_m * (y - t)))
	else:
		tmp = 2.0 * ((x_m / z_m) / (y - t))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2.3e-109)
		tmp = Float64(x_m * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / z_m) / Float64(y - t)));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 2.3e-109)
		tmp = x_m * (2.0 / (z_m * (y - t)));
	else
		tmp = 2.0 * ((x_m / z_m) / (y - t));
	end
	tmp_2 = z_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.3e-109], N[(x$95$m * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.3000000000000001e-109

   1. Initial program 93.2%

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

      1. distribute-rgt-out-- 94.8%

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

   3. Simplified 94.8%

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

      1. distribute-rgt-out-- 93.2%

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

      2. associate-/l* 92.3%

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

      3. *-commutative 92.3%

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

      4. distribute-rgt-out-- 94.0%

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

   6. Applied egg-rr 94.0%

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

   if 2.3000000000000001e-109 < z

   1. Initial program 88.6%

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

      1. distribute-rgt-out-- 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in x around 0 91.4%

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

      1. associate-/r* 99.8%

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

   7. Simplified 99.8%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{-109}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

   1. distribute-rgt-out-- 93.8%

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

3. Simplified 93.8%

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

5. Taylor expanded in x around 0 93.8%

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

   1. associate-/r* 93.3%

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

7. Simplified 93.3%

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

8. Final simplification 93.3%

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

Alternative 9: 53.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.9%

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

   1. distribute-rgt-out-- 93.8%

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

3. Simplified 93.8%

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

5. Taylor expanded in y around 0 54.3%

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

   1. *-commutative 54.3%

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

7. Simplified 54.3%

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

8. Final simplification 54.3%

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

Developer target: 97.4% 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 2024079 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))