Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.8% → 97.1%

Time: 9.2s

Alternatives: 13

Speedup: 1.0×

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: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 9 \cdot 10^{-37}:\\ \;\;\;\;\frac{x \cdot 2}{z_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 9e-37)
    (/ (* x 2.0) (* z_m (- y t)))
    (* (/ x (- y t)) (/ 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 (z_m <= 9e-37) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x / (y - t)) * (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 (z_m <= 9d-37) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else
        tmp = (x / (y - t)) * (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 (z_m <= 9e-37) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else {
		tmp = (x / (y - t)) * (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 z_m <= 9e-37:
		tmp = (x * 2.0) / (z_m * (y - t))
	else:
		tmp = (x / (y - t)) * (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 (z_m <= 9e-37)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(x / Float64(y - t)) * 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 (z_m <= 9e-37)
		tmp = (x * 2.0) / (z_m * (y - t));
	else
		tmp = (x / (y - t)) * (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[z$95$m, 9e-37], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.00000000000000081e-37

   1. Initial program 90.2%

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

      1. distribute-rgt-out-- 92.0%

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

   3. Simplified 92.0%

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

   if 9.00000000000000081e-37 < z

   1. Initial program 82.9%

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

      1. distribute-rgt-out-- 84.2%

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

   3. Simplified 84.2%

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

      1. *-commutative 84.2%

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

      2. times-frac 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9 \cdot 10^{-37}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]

Alternative 2: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{+241}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z_m}{x}}\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z_m}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{z_m} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z_m} \cdot \frac{-x}{t}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -3.65e+241)
    (/ -2.0 (* t (/ z_m x)))
    (if (<= t -2.7e-37)
      (* -2.0 (/ (/ x t) z_m))
      (if (<= t 2.5e+54)
        (* (/ x z_m) (/ 2.0 y))
        (* (/ 2.0 z_m) (/ (- x) 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 (t <= -3.65e+241) {
		tmp = -2.0 / (t * (z_m / x));
	} else if (t <= -2.7e-37) {
		tmp = -2.0 * ((x / t) / z_m);
	} else if (t <= 2.5e+54) {
		tmp = (x / z_m) * (2.0 / y);
	} else {
		tmp = (2.0 / z_m) * (-x / 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 (t <= (-3.65d+241)) then
        tmp = (-2.0d0) / (t * (z_m / x))
    else if (t <= (-2.7d-37)) then
        tmp = (-2.0d0) * ((x / t) / z_m)
    else if (t <= 2.5d+54) then
        tmp = (x / z_m) * (2.0d0 / y)
    else
        tmp = (2.0d0 / z_m) * (-x / 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 (t <= -3.65e+241) {
		tmp = -2.0 / (t * (z_m / x));
	} else if (t <= -2.7e-37) {
		tmp = -2.0 * ((x / t) / z_m);
	} else if (t <= 2.5e+54) {
		tmp = (x / z_m) * (2.0 / y);
	} else {
		tmp = (2.0 / z_m) * (-x / 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 t <= -3.65e+241:
		tmp = -2.0 / (t * (z_m / x))
	elif t <= -2.7e-37:
		tmp = -2.0 * ((x / t) / z_m)
	elif t <= 2.5e+54:
		tmp = (x / z_m) * (2.0 / y)
	else:
		tmp = (2.0 / z_m) * (-x / 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 (t <= -3.65e+241)
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / x)));
	elseif (t <= -2.7e-37)
		tmp = Float64(-2.0 * Float64(Float64(x / t) / z_m));
	elseif (t <= 2.5e+54)
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(Float64(-x) / 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 (t <= -3.65e+241)
		tmp = -2.0 / (t * (z_m / x));
	elseif (t <= -2.7e-37)
		tmp = -2.0 * ((x / t) / z_m);
	elseif (t <= 2.5e+54)
		tmp = (x / z_m) * (2.0 / y);
	else
		tmp = (2.0 / z_m) * (-x / 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[t, -3.65e+241], N[(-2.0 / N[(t * N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.7e-37], N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.5e+54], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[((-x) / t), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < -3.6500000000000001e241

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-*l/ 71.8%

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

      3. *-commutative 71.8%

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

      4. distribute-rgt-out-- 71.8%

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

      5. associate-/l/ 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in y around 0 60.6%

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

      1. associate-/r* 60.8%

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

   6. Simplified 60.8%

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

      1. associate-*r/ 63.1%

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

      2. associate-*l/ 88.1%

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

      3. clear-num 88.1%

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

      4. frac-times 88.6%

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

      5. metadata-eval 88.6%

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

   8. Applied egg-rr 88.6%

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

   if -3.6500000000000001e241 < t < -2.70000000000000016e-37

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-*l/ 90.1%

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

      3. *-commutative 90.1%

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

      4. distribute-rgt-out-- 92.1%

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

      5. associate-/l/ 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in y around 0 72.6%

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

      1. associate-/r* 80.8%

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

   6. Simplified 80.8%

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

   if -2.70000000000000016e-37 < t < 2.50000000000000003e54

   1. Initial program 88.9%

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

      1. distribute-rgt-out-- 90.4%

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

      2. times-frac 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in y around inf 82.2%

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

   if 2.50000000000000003e54 < t

   1. Initial program 87.7%

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

      1. distribute-rgt-out-- 90.0%

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

   3. Simplified 90.0%

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

      1. *-commutative 90.0%

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

      2. times-frac 95.8%

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

   5. Applied egg-rr 95.8%

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

   6. Taylor expanded in y around 0 81.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{x}{t}\right)} \cdot \frac{2}{z} \]
   7. Step-by-step derivation

      1. associate-*r/ 81.7%

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

      2. neg-mul-1 81.7%

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

   8. Simplified 81.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{+241}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{-x}{t}\\ \end{array} \]

Alternative 3: 74.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{+241}:\\ \;\;\;\;\frac{x}{z_m} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-37} \lor \neg \left(t \leq 6 \cdot 10^{-23}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z_m} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -3.65e+241)
    (* (/ x z_m) (/ -2.0 t))
    (if (or (<= t -2.6e-37) (not (<= t 6e-23)))
      (* -2.0 (/ (/ x t) z_m))
      (* (/ 2.0 z_m) (/ x 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 (t <= -3.65e+241) {
		tmp = (x / z_m) * (-2.0 / t);
	} else if ((t <= -2.6e-37) || !(t <= 6e-23)) {
		tmp = -2.0 * ((x / t) / z_m);
	} else {
		tmp = (2.0 / z_m) * (x / 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 (t <= (-3.65d+241)) then
        tmp = (x / z_m) * ((-2.0d0) / t)
    else if ((t <= (-2.6d-37)) .or. (.not. (t <= 6d-23))) then
        tmp = (-2.0d0) * ((x / t) / z_m)
    else
        tmp = (2.0d0 / z_m) * (x / 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 (t <= -3.65e+241) {
		tmp = (x / z_m) * (-2.0 / t);
	} else if ((t <= -2.6e-37) || !(t <= 6e-23)) {
		tmp = -2.0 * ((x / t) / z_m);
	} else {
		tmp = (2.0 / z_m) * (x / 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 t <= -3.65e+241:
		tmp = (x / z_m) * (-2.0 / t)
	elif (t <= -2.6e-37) or not (t <= 6e-23):
		tmp = -2.0 * ((x / t) / z_m)
	else:
		tmp = (2.0 / z_m) * (x / 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 (t <= -3.65e+241)
		tmp = Float64(Float64(x / z_m) * Float64(-2.0 / t));
	elseif ((t <= -2.6e-37) || !(t <= 6e-23))
		tmp = Float64(-2.0 * Float64(Float64(x / t) / z_m));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x / 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 (t <= -3.65e+241)
		tmp = (x / z_m) * (-2.0 / t);
	elseif ((t <= -2.6e-37) || ~((t <= 6e-23)))
		tmp = -2.0 * ((x / t) / z_m);
	else
		tmp = (2.0 / z_m) * (x / 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[t, -3.65e+241], N[(N[(x / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -2.6e-37], N[Not[LessEqual[t, 6e-23]], $MachinePrecision]], N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -3.6500000000000001e241

   1. Initial program 71.8%

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

      1. distribute-rgt-out-- 71.8%

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

      2. times-frac 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in y around 0 88.1%

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

   if -3.6500000000000001e241 < t < -2.5999999999999998e-37 or 6.00000000000000006e-23 < t

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-*l/ 90.1%

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

      3. *-commutative 90.1%

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

      4. distribute-rgt-out-- 91.8%

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

      5. associate-/l/ 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in y around 0 69.6%

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

      1. associate-/r* 77.1%

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

   6. Simplified 77.1%

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

   if -2.5999999999999998e-37 < t < 6.00000000000000006e-23

   1. Initial program 87.7%

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

      1. distribute-rgt-out-- 89.4%

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

   3. Simplified 89.4%

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

      1. *-commutative 89.4%

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

      2. times-frac 92.3%

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

   5. Applied egg-rr 92.3%

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

   6. Taylor expanded in y around inf 81.6%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{+241}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-37} \lor \neg \left(t \leq 6 \cdot 10^{-23}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{+241}:\\ \;\;\;\;\frac{x}{z_m} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq -2.05 \cdot 10^{-37} \lor \neg \left(t \leq 6.8 \cdot 10^{+64}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z_m} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -3.65e+241)
    (* (/ x z_m) (/ -2.0 t))
    (if (or (<= t -2.05e-37) (not (<= t 6.8e+64)))
      (* -2.0 (/ (/ x 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 (t <= -3.65e+241) {
		tmp = (x / z_m) * (-2.0 / t);
	} else if ((t <= -2.05e-37) || !(t <= 6.8e+64)) {
		tmp = -2.0 * ((x / 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 (t <= (-3.65d+241)) then
        tmp = (x / z_m) * ((-2.0d0) / t)
    else if ((t <= (-2.05d-37)) .or. (.not. (t <= 6.8d+64))) then
        tmp = (-2.0d0) * ((x / 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 (t <= -3.65e+241) {
		tmp = (x / z_m) * (-2.0 / t);
	} else if ((t <= -2.05e-37) || !(t <= 6.8e+64)) {
		tmp = -2.0 * ((x / 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 t <= -3.65e+241:
		tmp = (x / z_m) * (-2.0 / t)
	elif (t <= -2.05e-37) or not (t <= 6.8e+64):
		tmp = -2.0 * ((x / 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 (t <= -3.65e+241)
		tmp = Float64(Float64(x / z_m) * Float64(-2.0 / t));
	elseif ((t <= -2.05e-37) || !(t <= 6.8e+64))
		tmp = Float64(-2.0 * Float64(Float64(x / 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 (t <= -3.65e+241)
		tmp = (x / z_m) * (-2.0 / t);
	elseif ((t <= -2.05e-37) || ~((t <= 6.8e+64)))
		tmp = -2.0 * ((x / 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[t, -3.65e+241], N[(N[(x / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -2.05e-37], N[Not[LessEqual[t, 6.8e+64]], $MachinePrecision]], N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $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 t < -3.6500000000000001e241

   1. Initial program 71.8%

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

      1. distribute-rgt-out-- 71.8%

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

      2. times-frac 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in y around 0 88.1%

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

   if -3.6500000000000001e241 < t < -2.0499999999999999e-37 or 6.8000000000000003e64 < t

   1. Initial program 88.5%

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

      1. *-commutative 88.5%

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

      2. associate-*l/ 88.4%

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

      3. *-commutative 88.4%

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

      4. distribute-rgt-out-- 90.6%

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

      5. associate-/l/ 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in y around 0 73.0%

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

      1. associate-/r* 82.5%

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

   6. Simplified 82.5%

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

   if -2.0499999999999999e-37 < t < 6.8000000000000003e64

   1. Initial program 89.2%

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

      1. distribute-rgt-out-- 90.6%

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

      2. times-frac 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in y around inf 81.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{+241}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \mathbf{elif}\;t \leq -2.05 \cdot 10^{-37} \lor \neg \left(t \leq 6.8 \cdot 10^{+64}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]

Alternative 5: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{+241}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z_m}{x}}\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-37} \lor \neg \left(t \leq 1.1 \cdot 10^{+65}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z_m} \cdot \frac{2}{y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -3.65e+241)
    (/ -2.0 (* t (/ z_m x)))
    (if (or (<= t -3.3e-37) (not (<= t 1.1e+65)))
      (* -2.0 (/ (/ x 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 (t <= -3.65e+241) {
		tmp = -2.0 / (t * (z_m / x));
	} else if ((t <= -3.3e-37) || !(t <= 1.1e+65)) {
		tmp = -2.0 * ((x / 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 (t <= (-3.65d+241)) then
        tmp = (-2.0d0) / (t * (z_m / x))
    else if ((t <= (-3.3d-37)) .or. (.not. (t <= 1.1d+65))) then
        tmp = (-2.0d0) * ((x / 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 (t <= -3.65e+241) {
		tmp = -2.0 / (t * (z_m / x));
	} else if ((t <= -3.3e-37) || !(t <= 1.1e+65)) {
		tmp = -2.0 * ((x / 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 t <= -3.65e+241:
		tmp = -2.0 / (t * (z_m / x))
	elif (t <= -3.3e-37) or not (t <= 1.1e+65):
		tmp = -2.0 * ((x / 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 (t <= -3.65e+241)
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / x)));
	elseif ((t <= -3.3e-37) || !(t <= 1.1e+65))
		tmp = Float64(-2.0 * Float64(Float64(x / 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 (t <= -3.65e+241)
		tmp = -2.0 / (t * (z_m / x));
	elseif ((t <= -3.3e-37) || ~((t <= 1.1e+65)))
		tmp = -2.0 * ((x / 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[t, -3.65e+241], N[(-2.0 / N[(t * N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -3.3e-37], N[Not[LessEqual[t, 1.1e+65]], $MachinePrecision]], N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $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 t < -3.6500000000000001e241

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-*l/ 71.8%

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

      3. *-commutative 71.8%

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

      4. distribute-rgt-out-- 71.8%

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

      5. associate-/l/ 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in y around 0 60.6%

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

      1. associate-/r* 60.8%

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

   6. Simplified 60.8%

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

      1. associate-*r/ 63.1%

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

      2. associate-*l/ 88.1%

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

      3. clear-num 88.1%

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

      4. frac-times 88.6%

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

      5. metadata-eval 88.6%

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

   8. Applied egg-rr 88.6%

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

   if -3.6500000000000001e241 < t < -3.29999999999999982e-37 or 1.0999999999999999e65 < t

   1. Initial program 88.5%

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

      1. *-commutative 88.5%

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

      2. associate-*l/ 88.4%

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

      3. *-commutative 88.4%

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

      4. distribute-rgt-out-- 90.6%

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

      5. associate-/l/ 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in y around 0 73.0%

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

      1. associate-/r* 82.5%

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

   6. Simplified 82.5%

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

   if -3.29999999999999982e-37 < t < 1.0999999999999999e65

   1. Initial program 89.2%

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

      1. distribute-rgt-out-- 90.6%

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

      2. times-frac 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in y around inf 81.4%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{+241}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-37} \lor \neg \left(t \leq 1.1 \cdot 10^{+65}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \end{array} \]

Alternative 6: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{+241}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z_m}{x}}\\ \mathbf{elif}\;t \leq -3.15 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z_m}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{z_m} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{t}}{z_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -3.65e+241)
    (/ -2.0 (* t (/ z_m x)))
    (if (<= t -3.15e-37)
      (* -2.0 (/ (/ x t) z_m))
      (if (<= t 6.2e+49) (* (/ x z_m) (/ 2.0 y)) (/ (/ (* x -2.0) t) z_m))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -3.65e+241) {
		tmp = -2.0 / (t * (z_m / x));
	} else if (t <= -3.15e-37) {
		tmp = -2.0 * ((x / t) / z_m);
	} else if (t <= 6.2e+49) {
		tmp = (x / z_m) * (2.0 / y);
	} else {
		tmp = ((x * -2.0) / t) / z_m;
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.65d+241)) then
        tmp = (-2.0d0) / (t * (z_m / x))
    else if (t <= (-3.15d-37)) then
        tmp = (-2.0d0) * ((x / t) / z_m)
    else if (t <= 6.2d+49) then
        tmp = (x / z_m) * (2.0d0 / y)
    else
        tmp = ((x * (-2.0d0)) / t) / z_m
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -3.65e+241) {
		tmp = -2.0 / (t * (z_m / x));
	} else if (t <= -3.15e-37) {
		tmp = -2.0 * ((x / t) / z_m);
	} else if (t <= 6.2e+49) {
		tmp = (x / z_m) * (2.0 / y);
	} else {
		tmp = ((x * -2.0) / t) / z_m;
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= -3.65e+241:
		tmp = -2.0 / (t * (z_m / x))
	elif t <= -3.15e-37:
		tmp = -2.0 * ((x / t) / z_m)
	elif t <= 6.2e+49:
		tmp = (x / z_m) * (2.0 / y)
	else:
		tmp = ((x * -2.0) / t) / z_m
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -3.65e+241)
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / x)));
	elseif (t <= -3.15e-37)
		tmp = Float64(-2.0 * Float64(Float64(x / t) / z_m));
	elseif (t <= 6.2e+49)
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	else
		tmp = Float64(Float64(Float64(x * -2.0) / t) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= -3.65e+241)
		tmp = -2.0 / (t * (z_m / x));
	elseif (t <= -3.15e-37)
		tmp = -2.0 * ((x / t) / z_m);
	elseif (t <= 6.2e+49)
		tmp = (x / z_m) * (2.0 / y);
	else
		tmp = ((x * -2.0) / t) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, -3.65e+241], N[(-2.0 / N[(t * N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.15e-37], N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.2e+49], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * -2.0), $MachinePrecision] / t), $MachinePrecision] / z$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < -3.6500000000000001e241

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-*l/ 71.8%

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

      3. *-commutative 71.8%

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

      4. distribute-rgt-out-- 71.8%

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

      5. associate-/l/ 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in y around 0 60.6%

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

      1. associate-/r* 60.8%

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

   6. Simplified 60.8%

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

      1. associate-*r/ 63.1%

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

      2. associate-*l/ 88.1%

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

      3. clear-num 88.1%

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

      4. frac-times 88.6%

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

      5. metadata-eval 88.6%

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

   8. Applied egg-rr 88.6%

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

   if -3.6500000000000001e241 < t < -3.15000000000000011e-37

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-*l/ 90.1%

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

      3. *-commutative 90.1%

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

      4. distribute-rgt-out-- 92.1%

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

      5. associate-/l/ 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in y around 0 72.6%

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

      1. associate-/r* 80.8%

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

   6. Simplified 80.8%

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

   if -3.15000000000000011e-37 < t < 6.19999999999999985e49

   1. Initial program 88.9%

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

      1. distribute-rgt-out-- 90.4%

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

      2. times-frac 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in y around inf 82.2%

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

   if 6.19999999999999985e49 < t

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. associate-*l/ 87.6%

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

      3. *-commutative 87.6%

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

      4. distribute-rgt-out-- 89.9%

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

      5. associate-/l/ 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in y around 0 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l/ 71.7%

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

      3. associate-/r* 81.7%

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

   6. Applied egg-rr 81.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.65 \cdot 10^{+241}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq -3.15 \cdot 10^{-37}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{t}}{z}\\ \end{array} \]

Alternative 7: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-37} \lor \neg \left(t \leq 7 \cdot 10^{-23}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= t -3.2e-37) (not (<= t 7e-23)))
    (* -2.0 (/ (/ x t) z_m))
    (* x (/ (/ 2.0 y) 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 ((t <= -3.2e-37) || !(t <= 7e-23)) {
		tmp = -2.0 * ((x / t) / z_m);
	} else {
		tmp = x * ((2.0 / y) / 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 ((t <= (-3.2d-37)) .or. (.not. (t <= 7d-23))) then
        tmp = (-2.0d0) * ((x / t) / z_m)
    else
        tmp = x * ((2.0d0 / y) / 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 ((t <= -3.2e-37) || !(t <= 7e-23)) {
		tmp = -2.0 * ((x / t) / z_m);
	} else {
		tmp = x * ((2.0 / y) / 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 (t <= -3.2e-37) or not (t <= 7e-23):
		tmp = -2.0 * ((x / t) / z_m)
	else:
		tmp = x * ((2.0 / y) / 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 ((t <= -3.2e-37) || !(t <= 7e-23))
		tmp = Float64(-2.0 * Float64(Float64(x / t) / z_m));
	else
		tmp = Float64(x * Float64(Float64(2.0 / y) / 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 ((t <= -3.2e-37) || ~((t <= 7e-23)))
		tmp = -2.0 * ((x / t) / z_m);
	else
		tmp = x * ((2.0 / y) / 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[Or[LessEqual[t, -3.2e-37], N[Not[LessEqual[t, 7e-23]], $MachinePrecision]], N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(2.0 / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -3.1999999999999999e-37 or 6.99999999999999987e-23 < t

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. associate-*l/ 87.8%

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

      3. *-commutative 87.8%

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

      4. distribute-rgt-out-- 89.4%

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

      5. associate-/l/ 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in y around 0 68.5%

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

      1. associate-/r* 75.3%

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

   6. Simplified 75.3%

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

   if -3.1999999999999999e-37 < t < 6.99999999999999987e-23

   1. Initial program 87.7%

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

      1. *-commutative 87.7%

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

      2. associate-*l/ 87.6%

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

      3. *-commutative 87.6%

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

      4. distribute-rgt-out-- 89.3%

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

      5. associate-/l/ 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in y around inf 79.3%

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

      1. associate-/r* 80.3%

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

   6. Simplified 80.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-37} \lor \neg \left(t \leq 7 \cdot 10^{-23}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \end{array} \]

Alternative 8: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-37} \lor \neg \left(t \leq 4.3 \cdot 10^{-23}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z_m} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= t -1.9e-37) (not (<= t 4.3e-23)))
    (* -2.0 (/ (/ x t) z_m))
    (* (/ 2.0 z_m) (/ x 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 ((t <= -1.9e-37) || !(t <= 4.3e-23)) {
		tmp = -2.0 * ((x / t) / z_m);
	} else {
		tmp = (2.0 / z_m) * (x / 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 ((t <= (-1.9d-37)) .or. (.not. (t <= 4.3d-23))) then
        tmp = (-2.0d0) * ((x / t) / z_m)
    else
        tmp = (2.0d0 / z_m) * (x / 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 ((t <= -1.9e-37) || !(t <= 4.3e-23)) {
		tmp = -2.0 * ((x / t) / z_m);
	} else {
		tmp = (2.0 / z_m) * (x / 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 (t <= -1.9e-37) or not (t <= 4.3e-23):
		tmp = -2.0 * ((x / t) / z_m)
	else:
		tmp = (2.0 / z_m) * (x / 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 ((t <= -1.9e-37) || !(t <= 4.3e-23))
		tmp = Float64(-2.0 * Float64(Float64(x / t) / z_m));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x / 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 ((t <= -1.9e-37) || ~((t <= 4.3e-23)))
		tmp = -2.0 * ((x / t) / z_m);
	else
		tmp = (2.0 / z_m) * (x / 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[Or[LessEqual[t, -1.9e-37], N[Not[LessEqual[t, 4.3e-23]], $MachinePrecision]], N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -1.9000000000000002e-37 or 4.30000000000000002e-23 < t

   1. Initial program 87.9%

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

      1. *-commutative 87.9%

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

      2. associate-*l/ 87.8%

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

      3. *-commutative 87.8%

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

      4. distribute-rgt-out-- 89.4%

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

      5. associate-/l/ 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in y around 0 68.5%

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

      1. associate-/r* 75.3%

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

   6. Simplified 75.3%

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

   if -1.9000000000000002e-37 < t < 4.30000000000000002e-23

   1. Initial program 87.7%

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

      1. distribute-rgt-out-- 89.4%

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

   3. Simplified 89.4%

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

      1. *-commutative 89.4%

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

      2. times-frac 92.3%

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

   5. Applied egg-rr 92.3%

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

   6. Taylor expanded in y around inf 81.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{-37} \lor \neg \left(t \leq 4.3 \cdot 10^{-23}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 9: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+252}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z_m}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -3.4e+252)
    (/ -2.0 (* t (/ z_m x)))
    (* x (/ (/ 2.0 (- y t)) z_m)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -3.4e+252) {
		tmp = -2.0 / (t * (z_m / x));
	} else {
		tmp = x * ((2.0 / (y - t)) / z_m);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.4d+252)) then
        tmp = (-2.0d0) / (t * (z_m / x))
    else
        tmp = x * ((2.0d0 / (y - t)) / z_m)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m, double t) {
	double tmp;
	if (t <= -3.4e+252) {
		tmp = -2.0 / (t * (z_m / x));
	} else {
		tmp = x * ((2.0 / (y - t)) / z_m);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m, t):
	tmp = 0
	if t <= -3.4e+252:
		tmp = -2.0 / (t * (z_m / x))
	else:
		tmp = x * ((2.0 / (y - t)) / z_m)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m, t)
	tmp = 0.0
	if (t <= -3.4e+252)
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / x)));
	else
		tmp = Float64(x * Float64(Float64(2.0 / Float64(y - t)) / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m, t)
	tmp = 0.0;
	if (t <= -3.4e+252)
		tmp = -2.0 / (t * (z_m / x));
	else
		tmp = x * ((2.0 / (y - t)) / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_, t_] := N[(z$95$s * If[LessEqual[t, -3.4e+252], N[(-2.0 / N[(t * N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -3.4e252

   1. Initial program 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*l/ 63.4%

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

      3. *-commutative 63.4%

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

      4. distribute-rgt-out-- 63.4%

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

      5. associate-/l/ 63.5%

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

   3. Simplified 63.5%

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

   4. Taylor expanded in y around 0 63.4%

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

      1. associate-/r* 63.5%

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

   6. Simplified 63.5%

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

      1. associate-*r/ 77.7%

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

      2. associate-*l/ 99.3%

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

      3. clear-num 99.3%

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

      4. frac-times 99.9%

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

      5. metadata-eval 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -3.4e252 < t

   1. Initial program 89.1%

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

      1. *-commutative 89.1%

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

      2. associate-*l/ 89.0%

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

      3. *-commutative 89.0%

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

      4. distribute-rgt-out-- 90.7%

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

      5. associate-/l/ 91.9%

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

   3. Simplified 91.9%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+252}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \end{array} \]

Alternative 10: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{2}{y - t}\\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 7.5 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \frac{t_1}{z_m}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{x}{z_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (- y t))))
   (* z_s (if (<= z_m 7.5e-51) (* x (/ t_1 z_m)) (* t_1 (/ x 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 t_1 = 2.0 / (y - t);
	double tmp;
	if (z_m <= 7.5e-51) {
		tmp = x * (t_1 / z_m);
	} else {
		tmp = t_1 * (x / 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) :: t_1
    real(8) :: tmp
    t_1 = 2.0d0 / (y - t)
    if (z_m <= 7.5d-51) then
        tmp = x * (t_1 / z_m)
    else
        tmp = t_1 * (x / 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 t_1 = 2.0 / (y - t);
	double tmp;
	if (z_m <= 7.5e-51) {
		tmp = x * (t_1 / z_m);
	} else {
		tmp = t_1 * (x / 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):
	t_1 = 2.0 / (y - t)
	tmp = 0
	if z_m <= 7.5e-51:
		tmp = x * (t_1 / z_m)
	else:
		tmp = t_1 * (x / 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)
	t_1 = Float64(2.0 / Float64(y - t))
	tmp = 0.0
	if (z_m <= 7.5e-51)
		tmp = Float64(x * Float64(t_1 / z_m));
	else
		tmp = Float64(t_1 * Float64(x / 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)
	t_1 = 2.0 / (y - t);
	tmp = 0.0;
	if (z_m <= 7.5e-51)
		tmp = x * (t_1 / z_m);
	else
		tmp = t_1 * (x / 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_] := Block[{t$95$1 = N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 7.5e-51], N[(x * N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.49999999999999976e-51

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-*l/ 90.0%

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

      3. *-commutative 90.0%

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

      4. distribute-rgt-out-- 91.8%

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

      5. associate-/l/ 93.2%

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

   3. Simplified 93.2%

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

   if 7.49999999999999976e-51 < z

   1. Initial program 83.3%

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

      1. distribute-rgt-out-- 84.6%

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

      2. times-frac 95.3%

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

   3. Simplified 95.3%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-51}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y - t} \cdot \frac{x}{z}\\ \end{array} \]

Alternative 11: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.3 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2.3e-37)
    (* x (/ (/ 2.0 (- y t)) z_m))
    (* (/ x (- y t)) (/ 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 (z_m <= 2.3e-37) {
		tmp = x * ((2.0 / (y - t)) / z_m);
	} else {
		tmp = (x / (y - t)) * (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 (z_m <= 2.3d-37) then
        tmp = x * ((2.0d0 / (y - t)) / z_m)
    else
        tmp = (x / (y - t)) * (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 (z_m <= 2.3e-37) {
		tmp = x * ((2.0 / (y - t)) / z_m);
	} else {
		tmp = (x / (y - t)) * (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 z_m <= 2.3e-37:
		tmp = x * ((2.0 / (y - t)) / z_m)
	else:
		tmp = (x / (y - t)) * (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 (z_m <= 2.3e-37)
		tmp = Float64(x * Float64(Float64(2.0 / Float64(y - t)) / z_m));
	else
		tmp = Float64(Float64(x / Float64(y - t)) * 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 (z_m <= 2.3e-37)
		tmp = x * ((2.0 / (y - t)) / z_m);
	else
		tmp = (x / (y - t)) * (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[z$95$m, 2.3e-37], N[(x * N[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.3e-37

   1. Initial program 90.2%

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

      1. *-commutative 90.2%

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

      2. associate-*l/ 90.1%

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

      3. *-commutative 90.1%

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

      4. distribute-rgt-out-- 91.9%

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

      5. associate-/l/ 93.2%

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

   3. Simplified 93.2%

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

   if 2.3e-37 < z

   1. Initial program 82.9%

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

      1. distribute-rgt-out-- 84.2%

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

   3. Simplified 84.2%

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

      1. *-commutative 84.2%

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

      2. times-frac 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]

Alternative 12: 53.7% 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 1 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 87.8%

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

   1. *-commutative 87.8%

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

   2. associate-*l/ 87.7%

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

   3. *-commutative 87.7%

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

   4. distribute-rgt-out-- 89.3%

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

   5. associate-/l/ 90.4%

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

3. Simplified 90.4%

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

4. Taylor expanded in y around 0 45.8%

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

5. Final simplification 45.8%

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

Alternative 13: 53.9% accurate, 1.6× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m t) :precision binary64 (* z_s (* -2.0 (/ (/ x t) z_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

   1. *-commutative 87.8%

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

   2. associate-*l/ 87.7%

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

   3. *-commutative 87.7%

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

   4. distribute-rgt-out-- 89.3%

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

   5. associate-/l/ 90.4%

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

3. Simplified 90.4%

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

4. Taylor expanded in y around 0 45.8%

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

   1. associate-/r* 49.6%

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

6. Simplified 49.6%

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

7. Final simplification 49.6%

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

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 2023334 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (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))))