Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.5% → 96.7%

Time: 9.0s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{-21}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z\_m}}{y - t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2e-21)
    (/ (* x 2.0) (* z_m (- y t)))
    (* 2.0 (/ (/ x z_m) (- y t))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2e-21) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = 2.0 * ((x / z_m) / (y - t));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 2d-21) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else
        tmp = 2.0d0 * ((x / z_m) / (y - t))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2e-21) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = 2.0 * ((x / z_m) / (y - t));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 2e-21:
		tmp = (x * 2.0) / (z_m * (y - t))
	else:
		tmp = 2.0 * ((x / z_m) / (y - t))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2e-21)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(2.0 * Float64(Float64(x / z_m) / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 2e-21)
		tmp = (x * 2.0) / (z_m * (y - t));
	else
		tmp = 2.0 * ((x / z_m) / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 2e-21], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.99999999999999982e-21

   1. Initial program 94.1%

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

      1. distribute-rgt-out-- 94.6%

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

   3. Simplified 94.6%

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

   if 1.99999999999999982e-21 < z

   1. Initial program 74.9%

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

      1. distribute-rgt-out-- 79.8%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in x around 0 79.8%

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

      1. associate-/r* 98.4%

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

   7. Simplified 98.4%

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

Alternative 2: 73.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-22} \lor \neg \left(y \leq 6 \cdot 10^{+49}\right):\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= y -5.5e-22) (not (<= y 6e+49)))
    (* x (/ 2.0 (* z_m y)))
    (* -2.0 (/ (/ x z_m) t)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((y <= -5.5e-22) || !(y <= 6e+49)) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = -2.0 * ((x / z_m) / t);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-5.5d-22)) .or. (.not. (y <= 6d+49))) then
        tmp = x * (2.0d0 / (z_m * y))
    else
        tmp = (-2.0d0) * ((x / z_m) / t)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if ((y <= -5.5e-22) || !(y <= 6e+49)) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = -2.0 * ((x / z_m) / t);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if (y <= -5.5e-22) or not (y <= 6e+49):
		tmp = x * (2.0 / (z_m * y))
	else:
		tmp = -2.0 * ((x / z_m) / t)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if ((y <= -5.5e-22) || !(y <= 6e+49))
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if ((y <= -5.5e-22) || ~((y <= 6e+49)))
		tmp = x * (2.0 / (z_m * y));
	else
		tmp = -2.0 * ((x / z_m) / t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[Or[LessEqual[y, -5.5e-22], N[Not[LessEqual[y, 6e+49]], $MachinePrecision]], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -5.5000000000000001e-22 or 6.0000000000000005e49 < y

   1. Initial program 88.5%

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

      1. distribute-rgt-out-- 91.0%

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

   3. Simplified 91.0%

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

      1. associate-/l* 90.9%

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

      2. *-commutative 90.9%

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

   6. Applied egg-rr 90.9%

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

   7. Taylor expanded in y around inf 83.5%

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

      1. *-commutative 83.5%

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

   9. Simplified 83.5%

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

   if -5.5000000000000001e-22 < y < 6.0000000000000005e49

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 76.1%

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

      1. associate-/l/ 78.6%

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

   7. Simplified 78.6%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-22} \lor \neg \left(y \leq 6 \cdot 10^{+49}\right):\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.85 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot y}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{t}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z\_m}}{y \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y -3.85e-22)
    (/ (* x 2.0) (* z_m y))
    (if (<= y 6e+49) (/ (/ (* x -2.0) t) z_m) (/ (/ x z_m) (* y 0.5))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -3.85e-22) {
		tmp = (x * 2.0) / (z_m * y);
	} else if (y <= 6e+49) {
		tmp = ((x * -2.0) / t) / z_m;
	} else {
		tmp = (x / z_m) / (y * 0.5);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-3.85d-22)) then
        tmp = (x * 2.0d0) / (z_m * y)
    else if (y <= 6d+49) then
        tmp = ((x * (-2.0d0)) / t) / z_m
    else
        tmp = (x / z_m) / (y * 0.5d0)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -3.85e-22) {
		tmp = (x * 2.0) / (z_m * y);
	} else if (y <= 6e+49) {
		tmp = ((x * -2.0) / t) / z_m;
	} else {
		tmp = (x / z_m) / (y * 0.5);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if y <= -3.85e-22:
		tmp = (x * 2.0) / (z_m * y)
	elif y <= 6e+49:
		tmp = ((x * -2.0) / t) / z_m
	else:
		tmp = (x / z_m) / (y * 0.5)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (y <= -3.85e-22)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * y));
	elseif (y <= 6e+49)
		tmp = Float64(Float64(Float64(x * -2.0) / t) / z_m);
	else
		tmp = Float64(Float64(x / z_m) / Float64(y * 0.5));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (y <= -3.85e-22)
		tmp = (x * 2.0) / (z_m * y);
	elseif (y <= 6e+49)
		tmp = ((x * -2.0) / t) / z_m;
	else
		tmp = (x / z_m) / (y * 0.5);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[y, -3.85e-22], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+49], N[(N[(N[(x * -2.0), $MachinePrecision] / t), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] / N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.8500000000000001e-22

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around inf 85.6%

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

      1. *-commutative 85.6%

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

   7. Simplified 85.6%

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

   if -3.8500000000000001e-22 < y < 6.0000000000000005e49

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 76.1%

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

      1. associate-*r/ 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-/r* 80.5%

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

   7. Simplified 80.5%

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

   if 6.0000000000000005e49 < y

   1. Initial program 82.6%

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

      1. distribute-rgt-out-- 86.4%

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

   3. Simplified 86.4%

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

      1. associate-/l* 86.3%

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

      2. *-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

   7. Taylor expanded in y around inf 80.9%

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

      1. *-commutative 80.9%

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

   9. Simplified 80.9%

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

       1. associate-*l/ 81.1%

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

       2. *-commutative 81.1%

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

       3. frac-times 87.5%

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

       4. clear-num 87.5%

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

       5. un-div-inv 87.6%

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

       6. div-inv 87.6%

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

       7. metadata-eval 87.6%

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

   11. Applied egg-rr 87.6%

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

Alternative 4: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.36 \cdot 10^{-27}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot y}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{x \cdot \frac{-2}{t}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z\_m}}{y \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y -1.36e-27)
    (/ (* x 2.0) (* z_m y))
    (if (<= y 6.2e+49) (/ (* x (/ -2.0 t)) z_m) (/ (/ x z_m) (* y 0.5))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.36e-27) {
		tmp = (x * 2.0) / (z_m * y);
	} else if (y <= 6.2e+49) {
		tmp = (x * (-2.0 / t)) / z_m;
	} else {
		tmp = (x / z_m) / (y * 0.5);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.36d-27)) then
        tmp = (x * 2.0d0) / (z_m * y)
    else if (y <= 6.2d+49) then
        tmp = (x * ((-2.0d0) / t)) / z_m
    else
        tmp = (x / z_m) / (y * 0.5d0)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.36e-27) {
		tmp = (x * 2.0) / (z_m * y);
	} else if (y <= 6.2e+49) {
		tmp = (x * (-2.0 / t)) / z_m;
	} else {
		tmp = (x / z_m) / (y * 0.5);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if y <= -1.36e-27:
		tmp = (x * 2.0) / (z_m * y)
	elif y <= 6.2e+49:
		tmp = (x * (-2.0 / t)) / z_m
	else:
		tmp = (x / z_m) / (y * 0.5)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (y <= -1.36e-27)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * y));
	elseif (y <= 6.2e+49)
		tmp = Float64(Float64(x * Float64(-2.0 / t)) / z_m);
	else
		tmp = Float64(Float64(x / z_m) / Float64(y * 0.5));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (y <= -1.36e-27)
		tmp = (x * 2.0) / (z_m * y);
	elseif (y <= 6.2e+49)
		tmp = (x * (-2.0 / t)) / z_m;
	else
		tmp = (x / z_m) / (y * 0.5);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[y, -1.36e-27], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e+49], N[(N[(x * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] / N[(y * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.36e-27

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around inf 85.6%

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

      1. *-commutative 85.6%

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

   7. Simplified 85.6%

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

   if -1.36e-27 < y < 6.19999999999999985e49

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 76.1%

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

      1. associate-*r/ 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-/r* 80.5%

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

      4. associate-/l* 80.4%

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

   7. Applied egg-rr 80.4%

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

   if 6.19999999999999985e49 < y

   1. Initial program 82.6%

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

      1. distribute-rgt-out-- 86.4%

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

   3. Simplified 86.4%

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

      1. associate-/l* 86.3%

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

      2. *-commutative 86.3%

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

   6. Applied egg-rr 86.3%

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

   7. Taylor expanded in y around inf 80.9%

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

      1. *-commutative 80.9%

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

   9. Simplified 80.9%

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

       1. associate-*l/ 81.1%

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

       2. *-commutative 81.1%

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

       3. frac-times 87.5%

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

       4. clear-num 87.5%

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

       5. un-div-inv 87.6%

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

       6. div-inv 87.6%

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

       7. metadata-eval 87.6%

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

   11. Applied egg-rr 87.6%

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

Alternative 5: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot y}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{x \cdot \frac{-2}{t}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y -5.2e-22)
    (/ (* x 2.0) (* z_m y))
    (if (<= y 7.2e+53) (/ (* x (/ -2.0 t)) z_m) (* (/ x z_m) (/ 2.0 y))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -5.2e-22) {
		tmp = (x * 2.0) / (z_m * y);
	} else if (y <= 7.2e+53) {
		tmp = (x * (-2.0 / t)) / z_m;
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-5.2d-22)) then
        tmp = (x * 2.0d0) / (z_m * y)
    else if (y <= 7.2d+53) then
        tmp = (x * ((-2.0d0) / t)) / z_m
    else
        tmp = (x / z_m) * (2.0d0 / y)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -5.2e-22) {
		tmp = (x * 2.0) / (z_m * y);
	} else if (y <= 7.2e+53) {
		tmp = (x * (-2.0 / t)) / z_m;
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if y <= -5.2e-22:
		tmp = (x * 2.0) / (z_m * y)
	elif y <= 7.2e+53:
		tmp = (x * (-2.0 / t)) / z_m
	else:
		tmp = (x / z_m) * (2.0 / y)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (y <= -5.2e-22)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * y));
	elseif (y <= 7.2e+53)
		tmp = Float64(Float64(x * Float64(-2.0 / t)) / z_m);
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (y <= -5.2e-22)
		tmp = (x * 2.0) / (z_m * y);
	elseif (y <= 7.2e+53)
		tmp = (x * (-2.0 / t)) / z_m;
	else
		tmp = (x / z_m) * (2.0 / y);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[y, -5.2e-22], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+53], N[(N[(x * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -5.2e-22

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around inf 85.6%

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

      1. *-commutative 85.6%

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

   7. Simplified 85.6%

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

   if -5.2e-22 < y < 7.2e53

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 76.1%

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

      1. associate-*r/ 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-/r* 80.5%

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

      4. associate-/l* 80.4%

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

   7. Applied egg-rr 80.4%

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

   if 7.2e53 < y

   1. Initial program 82.6%

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

      1. distribute-rgt-out-- 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0 86.4%

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

      1. associate-/r* 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in y around inf 81.1%

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

      1. associate-*r/ 81.1%

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

      2. times-frac 87.5%

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

      3. *-commutative 87.5%

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

   10. Simplified 87.5%

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

Alternative 6: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot y}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+49}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y -1.9e-23)
    (/ (* x 2.0) (* z_m y))
    (if (<= y 6.5e+49) (* -2.0 (/ (/ x z_m) t)) (* (/ x z_m) (/ 2.0 y))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.9e-23) {
		tmp = (x * 2.0) / (z_m * y);
	} else if (y <= 6.5e+49) {
		tmp = -2.0 * ((x / z_m) / t);
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.9d-23)) then
        tmp = (x * 2.0d0) / (z_m * y)
    else if (y <= 6.5d+49) then
        tmp = (-2.0d0) * ((x / z_m) / t)
    else
        tmp = (x / z_m) * (2.0d0 / y)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.9e-23) {
		tmp = (x * 2.0) / (z_m * y);
	} else if (y <= 6.5e+49) {
		tmp = -2.0 * ((x / z_m) / t);
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if y <= -1.9e-23:
		tmp = (x * 2.0) / (z_m * y)
	elif y <= 6.5e+49:
		tmp = -2.0 * ((x / z_m) / t)
	else:
		tmp = (x / z_m) * (2.0 / y)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (y <= -1.9e-23)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * y));
	elseif (y <= 6.5e+49)
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / t));
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (y <= -1.9e-23)
		tmp = (x * 2.0) / (z_m * y);
	elseif (y <= 6.5e+49)
		tmp = -2.0 * ((x / z_m) / t);
	else
		tmp = (x / z_m) * (2.0 / y);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[y, -1.9e-23], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+49], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.90000000000000006e-23

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around inf 85.6%

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

      1. *-commutative 85.6%

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

   7. Simplified 85.6%

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

   if -1.90000000000000006e-23 < y < 6.5000000000000005e49

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 76.1%

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

      1. associate-/l/ 78.6%

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

   7. Simplified 78.6%

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

   if 6.5000000000000005e49 < y

   1. Initial program 82.6%

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

      1. distribute-rgt-out-- 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0 86.4%

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

      1. associate-/r* 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in y around inf 81.1%

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

      1. associate-*r/ 81.1%

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

      2. times-frac 87.5%

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

      3. *-commutative 87.5%

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

   10. Simplified 87.5%

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

Alternative 7: 73.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+49}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y -1.05e-25)
    (* x (/ 2.0 (* z_m y)))
    (if (<= y 6e+49) (* -2.0 (/ (/ x z_m) t)) (* (/ x z_m) (/ 2.0 y))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.05e-25) {
		tmp = x * (2.0 / (z_m * y));
	} else if (y <= 6e+49) {
		tmp = -2.0 * ((x / z_m) / t);
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.05d-25)) then
        tmp = x * (2.0d0 / (z_m * y))
    else if (y <= 6d+49) then
        tmp = (-2.0d0) * ((x / z_m) / t)
    else
        tmp = (x / z_m) * (2.0d0 / y)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.05e-25) {
		tmp = x * (2.0 / (z_m * y));
	} else if (y <= 6e+49) {
		tmp = -2.0 * ((x / z_m) / t);
	} else {
		tmp = (x / z_m) * (2.0 / y);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if y <= -1.05e-25:
		tmp = x * (2.0 / (z_m * y))
	elif y <= 6e+49:
		tmp = -2.0 * ((x / z_m) / t)
	else:
		tmp = (x / z_m) * (2.0 / y)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (y <= -1.05e-25)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	elseif (y <= 6e+49)
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / t));
	else
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (y <= -1.05e-25)
		tmp = x * (2.0 / (z_m * y));
	elseif (y <= 6e+49)
		tmp = -2.0 * ((x / z_m) / t);
	else
		tmp = (x / z_m) * (2.0 / y);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[y, -1.05e-25], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e+49], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000001e-25

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 94.5%

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

   3. Simplified 94.5%

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

      1. associate-/l* 94.4%

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

      2. *-commutative 94.4%

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

   6. Applied egg-rr 94.4%

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

   7. Taylor expanded in y around inf 85.5%

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

      1. *-commutative 85.5%

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

   9. Simplified 85.5%

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

   if -1.05000000000000001e-25 < y < 6.0000000000000005e49

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 76.1%

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

      1. associate-/l/ 78.6%

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

   7. Simplified 78.6%

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

   if 6.0000000000000005e49 < y

   1. Initial program 82.6%

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

      1. distribute-rgt-out-- 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in x around 0 86.4%

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

      1. associate-/r* 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in y around inf 81.1%

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

      1. associate-*r/ 81.1%

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

      2. times-frac 87.5%

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

      3. *-commutative 87.5%

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

   10. Simplified 87.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+49}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+51}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{2}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= y -9.5e-23)
    (* x (/ 2.0 (* z_m y)))
    (if (<= y 6.5e+51) (* -2.0 (/ (/ x z_m) t)) (* (/ x y) (/ 2.0 z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -9.5e-23) {
		tmp = x * (2.0 / (z_m * y));
	} else if (y <= 6.5e+51) {
		tmp = -2.0 * ((x / z_m) / t);
	} else {
		tmp = (x / y) * (2.0 / z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-9.5d-23)) then
        tmp = x * (2.0d0 / (z_m * y))
    else if (y <= 6.5d+51) then
        tmp = (-2.0d0) * ((x / z_m) / t)
    else
        tmp = (x / y) * (2.0d0 / z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (y <= -9.5e-23) {
		tmp = x * (2.0 / (z_m * y));
	} else if (y <= 6.5e+51) {
		tmp = -2.0 * ((x / z_m) / t);
	} else {
		tmp = (x / y) * (2.0 / z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if y <= -9.5e-23:
		tmp = x * (2.0 / (z_m * y))
	elif y <= 6.5e+51:
		tmp = -2.0 * ((x / z_m) / t)
	else:
		tmp = (x / y) * (2.0 / z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (y <= -9.5e-23)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	elseif (y <= 6.5e+51)
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / t));
	else
		tmp = Float64(Float64(x / y) * Float64(2.0 / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (y <= -9.5e-23)
		tmp = x * (2.0 / (z_m * y));
	elseif (y <= 6.5e+51)
		tmp = -2.0 * ((x / z_m) / t);
	else
		tmp = (x / y) * (2.0 / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[y, -9.5e-23], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+51], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -9.50000000000000058e-23

   1. Initial program 93.1%

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

      1. distribute-rgt-out-- 94.5%

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

   3. Simplified 94.5%

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

      1. associate-/l* 94.4%

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

      2. *-commutative 94.4%

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

   6. Applied egg-rr 94.4%

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

   7. Taylor expanded in y around inf 85.5%

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

      1. *-commutative 85.5%

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

   9. Simplified 85.5%

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

   if -9.50000000000000058e-23 < y < 6.5e51

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

   5. Taylor expanded in y around 0 76.1%

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

      1. associate-/l/ 78.6%

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

   7. Simplified 78.6%

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

   if 6.5e51 < y

   1. Initial program 82.6%

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

      1. distribute-rgt-out-- 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in y around inf 81.1%

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

      1. *-commutative 81.1%

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

   7. Simplified 81.1%

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

      1. *-commutative 81.1%

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

      2. times-frac 84.4%

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

   9. Applied egg-rr 84.4%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+51}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.5 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z\_m}}{y - t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 4.5e-80)
    (* x (/ 2.0 (* z_m (- y t))))
    (* 2.0 (/ (/ x z_m) (- y t))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 4.5e-80) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = 2.0 * ((x / z_m) / (y - t));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 4.5d-80) then
        tmp = x * (2.0d0 / (z_m * (y - t)))
    else
        tmp = 2.0d0 * ((x / z_m) / (y - t))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 4.5e-80) {
		tmp = x * (2.0 / (z_m * (y - t)));
	} else {
		tmp = 2.0 * ((x / z_m) / (y - t));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 4.5e-80:
		tmp = x * (2.0 / (z_m * (y - t)))
	else:
		tmp = 2.0 * ((x / z_m) / (y - t))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 4.5e-80)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(2.0 * Float64(Float64(x / z_m) / Float64(y - t)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 4.5e-80)
		tmp = x * (2.0 / (z_m * (y - t)));
	else
		tmp = 2.0 * ((x / z_m) / (y - t));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 4.5e-80], N[(x * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.5000000000000003e-80

   1. Initial program 93.7%

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

      1. distribute-rgt-out-- 94.3%

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

   3. Simplified 94.3%

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

      1. associate-/l* 94.1%

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

      2. *-commutative 94.1%

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

   6. Applied egg-rr 94.1%

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

   if 4.5000000000000003e-80 < z

   1. Initial program 78.5%

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

      1. distribute-rgt-out-- 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in x around 0 82.7%

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

      1. associate-/r* 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{-80}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.3 \cdot 10^{-25}:\\ \;\;\;\;-2 \cdot \frac{x}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z\_m}}{t}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2.3e-25) (* -2.0 (/ x (* z_m t))) (* -2.0 (/ (/ x z_m) t)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.3e-25) {
		tmp = -2.0 * (x / (z_m * t));
	} else {
		tmp = -2.0 * ((x / z_m) / t);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 2.3d-25) then
        tmp = (-2.0d0) * (x / (z_m * t))
    else
        tmp = (-2.0d0) * ((x / z_m) / t)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.3e-25) {
		tmp = -2.0 * (x / (z_m * t));
	} else {
		tmp = -2.0 * ((x / z_m) / t);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if z_m <= 2.3e-25:
		tmp = -2.0 * (x / (z_m * t))
	else:
		tmp = -2.0 * ((x / z_m) / t)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2.3e-25)
		tmp = Float64(-2.0 * Float64(x / Float64(z_m * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x / z_m) / t));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 2.3e-25)
		tmp = -2.0 * (x / (z_m * t));
	else
		tmp = -2.0 * ((x / z_m) / t);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[z$95$m, 2.3e-25], N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.2999999999999999e-25

   1. Initial program 94.0%

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

      1. distribute-rgt-out-- 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in y around 0 50.6%

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

   if 2.2999999999999999e-25 < z

   1. Initial program 75.3%

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

      1. distribute-rgt-out-- 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in y around 0 47.1%

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

      1. associate-/l/ 58.3%

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

   7. Simplified 58.3%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{-25}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 92.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. distribute-rgt-out-- 90.8%

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

3. Simplified 90.8%

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

5. Taylor expanded in x around 0 90.8%

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

   1. associate-/r* 93.1%

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

7. Simplified 93.1%

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

Alternative 12: 53.0% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. distribute-rgt-out-- 90.8%

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

3. Simplified 90.8%

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

5. Taylor expanded in y around 0 49.7%

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

6. Final simplification 49.7%

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

Developer target: 96.7% 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 2024101 
(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))))