Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 90.2% → 96.6%

Time: 9.7s

Alternatives: 11

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: 90.2% 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: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{elif}\;z\_m \leq 1.5 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z\_m}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z\_m \cdot 0.5}\\ \end{array} \end{array} \]

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 (<= z_m 3.5e-59)
    (/ (* x 2.0) (* z_m (- y t)))
    (if (<= z_m 1.5e+135)
      (* 2.0 (/ (/ x z_m) (- y t)))
      (/ (/ x (- y t)) (* z_m 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 (z_m <= 3.5e-59) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else if (z_m <= 1.5e+135) {
		tmp = 2.0 * ((x / z_m) / (y - t));
	} else {
		tmp = (x / (y - t)) / (z_m * 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 (z_m <= 3.5d-59) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else if (z_m <= 1.5d+135) then
        tmp = 2.0d0 * ((x / z_m) / (y - t))
    else
        tmp = (x / (y - t)) / (z_m * 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 (z_m <= 3.5e-59) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else if (z_m <= 1.5e+135) {
		tmp = 2.0 * ((x / z_m) / (y - t));
	} else {
		tmp = (x / (y - t)) / (z_m * 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 z_m <= 3.5e-59:
		tmp = (x * 2.0) / (z_m * (y - t))
	elif z_m <= 1.5e+135:
		tmp = 2.0 * ((x / z_m) / (y - t))
	else:
		tmp = (x / (y - t)) / (z_m * 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 (z_m <= 3.5e-59)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	elseif (z_m <= 1.5e+135)
		tmp = Float64(2.0 * Float64(Float64(x / z_m) / Float64(y - t)));
	else
		tmp = Float64(Float64(x / Float64(y - t)) / Float64(z_m * 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 (z_m <= 3.5e-59)
		tmp = (x * 2.0) / (z_m * (y - t));
	elseif (z_m <= 1.5e+135)
		tmp = 2.0 * ((x / z_m) / (y - t));
	else
		tmp = (x / (y - t)) / (z_m * 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[z$95$m, 3.5e-59], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.5e+135], N[(2.0 * N[(N[(x / z$95$m), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 3.5000000000000001e-59

   1. Initial program 89.3%

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

      1. distribute-rgt-out-- 92.2%

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

   3. Simplified 92.2%

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

   if 3.5000000000000001e-59 < z < 1.5e135

   1. Initial program 92.2%

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

      1. distribute-rgt-out-- 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in x around 0 92.4%

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

      1. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

   if 1.5e135 < z

   1. Initial program 78.3%

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

      1. distribute-rgt-out-- 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in x around 0 83.6%

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

      1. associate-/r* 93.2%

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

   7. Simplified 93.2%

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

      1. associate-*r/ 93.2%

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

      2. div-inv 93.2%

         \[\leadsto \frac{2 \cdot \color{blue}{\left(x \cdot \frac{1}{z}\right)}}{y - t} \]

      3. associate-*r* 93.2%

         \[\leadsto \frac{\color{blue}{\left(2 \cdot x\right) \cdot \frac{1}{z}}}{y - t} \]

      4. *-commutative 93.2%

         \[\leadsto \frac{\color{blue}{\left(x \cdot 2\right)} \cdot \frac{1}{z}}{y - t} \]

      5. associate-*r* 93.2%

         \[\leadsto \frac{\color{blue}{x \cdot \left(2 \cdot \frac{1}{z}\right)}}{y - t} \]

      6. div-inv 93.2%

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

      7. associate-*l/ 97.2%

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

      8. *-commutative 97.2%

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

      9. clear-num 97.2%

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

      10. frac-times 83.7%

          \[\leadsto \color{blue}{\frac{1 \cdot x}{\frac{z}{2} \cdot \left(y - t\right)}} \]

      11. *-un-lft-identity 83.7%

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

      12. div-inv 83.7%

          \[\leadsto \frac{x}{\color{blue}{\left(z \cdot \frac{1}{2}\right)} \cdot \left(y - t\right)} \]

      13. metadata-eval 83.7%

          \[\leadsto \frac{x}{\left(z \cdot \color{blue}{0.5}\right) \cdot \left(y - t\right)} \]

   9. Applied egg-rr 83.7%

      \[\leadsto \color{blue}{\frac{x}{\left(z \cdot 0.5\right) \cdot \left(y - t\right)}} \]
   10. Step-by-step derivation

       1. associate-/l/ 97.3%

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

   11. Simplified 97.3%

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

Alternative 2: 73.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x}{z\_m} \cdot \frac{2}{y}\\ t_2 := -2 \cdot \frac{x}{z\_m \cdot t}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -9.6 \cdot 10^{+20}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-190}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-190}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+95}:\\ \;\;\;\;\frac{-2}{z\_m \cdot \frac{t}{x}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+207} \lor \neg \left(t \leq 4.7 \cdot 10^{+253}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-2}{z\_m}}{t}\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_1 (* (/ x z_m) (/ 2.0 y))) (t_2 (* -2.0 (/ x (* z_m t)))))
   (*
    z_s
    (if (<= t -9.6e+20)
      t_2
      (if (<= t -1.85e-190)
        t_1
        (if (<= t -1.8e-190)
          t_2
          (if (<= t 4.6e+80)
            t_1
            (if (<= t 4.7e+95)
              (/ -2.0 (* z_m (/ t x)))
              (if (<= t 3.5e+96)
                (* x (/ 2.0 (* z_m y)))
                (if (or (<= t 4.2e+207) (not (<= t 4.7e+253)))
                  (* -2.0 (/ (/ x t) z_m))
                  (/ (* x (/ -2.0 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 t_1 = (x / z_m) * (2.0 / y);
	double t_2 = -2.0 * (x / (z_m * t));
	double tmp;
	if (t <= -9.6e+20) {
		tmp = t_2;
	} else if (t <= -1.85e-190) {
		tmp = t_1;
	} else if (t <= -1.8e-190) {
		tmp = t_2;
	} else if (t <= 4.6e+80) {
		tmp = t_1;
	} else if (t <= 4.7e+95) {
		tmp = -2.0 / (z_m * (t / x));
	} else if (t <= 3.5e+96) {
		tmp = x * (2.0 / (z_m * y));
	} else if ((t <= 4.2e+207) || !(t <= 4.7e+253)) {
		tmp = -2.0 * ((x / t) / z_m);
	} else {
		tmp = (x * (-2.0 / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x / z_m) * (2.0d0 / y)
    t_2 = (-2.0d0) * (x / (z_m * t))
    if (t <= (-9.6d+20)) then
        tmp = t_2
    else if (t <= (-1.85d-190)) then
        tmp = t_1
    else if (t <= (-1.8d-190)) then
        tmp = t_2
    else if (t <= 4.6d+80) then
        tmp = t_1
    else if (t <= 4.7d+95) then
        tmp = (-2.0d0) / (z_m * (t / x))
    else if (t <= 3.5d+96) then
        tmp = x * (2.0d0 / (z_m * y))
    else if ((t <= 4.2d+207) .or. (.not. (t <= 4.7d+253))) then
        tmp = (-2.0d0) * ((x / t) / z_m)
    else
        tmp = (x * ((-2.0d0) / 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 t_1 = (x / z_m) * (2.0 / y);
	double t_2 = -2.0 * (x / (z_m * t));
	double tmp;
	if (t <= -9.6e+20) {
		tmp = t_2;
	} else if (t <= -1.85e-190) {
		tmp = t_1;
	} else if (t <= -1.8e-190) {
		tmp = t_2;
	} else if (t <= 4.6e+80) {
		tmp = t_1;
	} else if (t <= 4.7e+95) {
		tmp = -2.0 / (z_m * (t / x));
	} else if (t <= 3.5e+96) {
		tmp = x * (2.0 / (z_m * y));
	} else if ((t <= 4.2e+207) || !(t <= 4.7e+253)) {
		tmp = -2.0 * ((x / t) / z_m);
	} else {
		tmp = (x * (-2.0 / 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):
	t_1 = (x / z_m) * (2.0 / y)
	t_2 = -2.0 * (x / (z_m * t))
	tmp = 0
	if t <= -9.6e+20:
		tmp = t_2
	elif t <= -1.85e-190:
		tmp = t_1
	elif t <= -1.8e-190:
		tmp = t_2
	elif t <= 4.6e+80:
		tmp = t_1
	elif t <= 4.7e+95:
		tmp = -2.0 / (z_m * (t / x))
	elif t <= 3.5e+96:
		tmp = x * (2.0 / (z_m * y))
	elif (t <= 4.2e+207) or not (t <= 4.7e+253):
		tmp = -2.0 * ((x / t) / z_m)
	else:
		tmp = (x * (-2.0 / 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)
	t_1 = Float64(Float64(x / z_m) * Float64(2.0 / y))
	t_2 = Float64(-2.0 * Float64(x / Float64(z_m * t)))
	tmp = 0.0
	if (t <= -9.6e+20)
		tmp = t_2;
	elseif (t <= -1.85e-190)
		tmp = t_1;
	elseif (t <= -1.8e-190)
		tmp = t_2;
	elseif (t <= 4.6e+80)
		tmp = t_1;
	elseif (t <= 4.7e+95)
		tmp = Float64(-2.0 / Float64(z_m * Float64(t / x)));
	elseif (t <= 3.5e+96)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	elseif ((t <= 4.2e+207) || !(t <= 4.7e+253))
		tmp = Float64(-2.0 * Float64(Float64(x / t) / z_m));
	else
		tmp = Float64(Float64(x * Float64(-2.0 / 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)
	t_1 = (x / z_m) * (2.0 / y);
	t_2 = -2.0 * (x / (z_m * t));
	tmp = 0.0;
	if (t <= -9.6e+20)
		tmp = t_2;
	elseif (t <= -1.85e-190)
		tmp = t_1;
	elseif (t <= -1.8e-190)
		tmp = t_2;
	elseif (t <= 4.6e+80)
		tmp = t_1;
	elseif (t <= 4.7e+95)
		tmp = -2.0 / (z_m * (t / x));
	elseif (t <= 3.5e+96)
		tmp = x * (2.0 / (z_m * y));
	elseif ((t <= 4.2e+207) || ~((t <= 4.7e+253)))
		tmp = -2.0 * ((x / t) / z_m);
	else
		tmp = (x * (-2.0 / 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_] := Block[{t$95$1 = N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t, -9.6e+20], t$95$2, If[LessEqual[t, -1.85e-190], t$95$1, If[LessEqual[t, -1.8e-190], t$95$2, If[LessEqual[t, 4.6e+80], t$95$1, If[LessEqual[t, 4.7e+95], N[(-2.0 / N[(z$95$m * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.5e+96], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 4.2e+207], N[Not[LessEqual[t, 4.7e+253]], $MachinePrecision]], N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 6 regimes

2. if t < -9.6e20 or -1.8500000000000001e-190 < t < -1.80000000000000003e-190

   1. Initial program 83.1%

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

      1. distribute-rgt-out-- 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in y around 0 81.2%

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

      1. *-commutative 81.2%

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

   7. Simplified 81.2%

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

   if -9.6e20 < t < -1.8500000000000001e-190 or -1.80000000000000003e-190 < t < 4.60000000000000008e80

   1. Initial program 89.1%

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

      1. distribute-rgt-out-- 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in y around inf 69.8%

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

      1. *-commutative 69.8%

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

   7. Simplified 69.8%

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

      1. times-frac 74.7%

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

   9. Applied egg-rr 74.7%

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

   if 4.60000000000000008e80 < t < 4.69999999999999972e95

   1. Initial program 99.6%

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

      1. distribute-rgt-out-- 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. associate-/l* 100.0%

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

   11. Simplified 100.0%

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

   if 4.69999999999999972e95 < t < 3.4999999999999999e96

   1. Initial program 100.0%

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

      1. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

   10. Simplified 100.0%

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

   if 3.4999999999999999e96 < t < 4.1999999999999999e207 or 4.70000000000000022e253 < t

   1. Initial program 90.1%

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

      1. distribute-rgt-out-- 90.2%

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

   3. Simplified 90.2%

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

   5. Taylor expanded in y around 0 78.0%

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

      1. *-commutative 78.0%

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

   7. Simplified 78.0%

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

   8. Taylor expanded in x around 0 78.0%

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

      1. associate-/r* 90.1%

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

   10. Simplified 90.1%

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

   if 4.1999999999999999e207 < t < 4.70000000000000022e253

   1. Initial program 88.0%

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

      1. distribute-rgt-out-- 88.0%

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

   3. Simplified 88.0%

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

   5. Taylor expanded in x around 0 88.0%

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

      1. associate-/r* 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around 0 76.5%

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

      1. associate-*r/ 76.5%

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

      2. *-commutative 76.5%

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

      3. associate-*r/ 76.5%

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

      4. associate-/r* 88.4%

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

      5. associate-*r/ 88.4%

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

      6. metadata-eval 88.4%

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

      7. distribute-lft-neg-in 88.4%

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

      8. associate-*r/ 88.4%

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

      9. *-commutative 88.4%

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

      10. associate-*r/ 88.4%

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

      11. distribute-rgt-neg-in 88.4%

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

      12. distribute-neg-frac 88.4%

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

      13. metadata-eval 88.4%

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

   10. Simplified 88.4%

       \[\leadsto \color{blue}{\frac{x \cdot \frac{-2}{z}}{t}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.6 \cdot 10^{+20}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-190}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-190}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+95}:\\ \;\;\;\;\frac{-2}{z \cdot \frac{t}{x}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+96}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+207} \lor \neg \left(t \leq 4.7 \cdot 10^{+253}\right):\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{-2}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := -2 \cdot \frac{x}{z\_m \cdot t}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.25 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-190}:\\ \;\;\;\;\frac{x}{z\_m} \cdot \frac{2}{y}\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-190}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{2}{y}}{\frac{z\_m}{x}}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+95}:\\ \;\;\;\;\frac{-2}{z\_m \cdot \frac{t}{x}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{+95}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z\_m}\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_1 (* -2.0 (/ x (* z_m t)))))
   (*
    z_s
    (if (<= t -3.25e+21)
      t_1
      (if (<= t -1.85e-190)
        (* (/ x z_m) (/ 2.0 y))
        (if (<= t -1.8e-190)
          t_1
          (if (<= t 5.2e+80)
            (/ (/ 2.0 y) (/ z_m x))
            (if (<= t 4.7e+95)
              (/ -2.0 (* z_m (/ t x)))
              (if (<= t 4.8e+95)
                (* x (/ 2.0 (* z_m y)))
                (* -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) {
	double t_1 = -2.0 * (x / (z_m * t));
	double tmp;
	if (t <= -3.25e+21) {
		tmp = t_1;
	} else if (t <= -1.85e-190) {
		tmp = (x / z_m) * (2.0 / y);
	} else if (t <= -1.8e-190) {
		tmp = t_1;
	} else if (t <= 5.2e+80) {
		tmp = (2.0 / y) / (z_m / x);
	} else if (t <= 4.7e+95) {
		tmp = -2.0 / (z_m * (t / x));
	} else if (t <= 4.8e+95) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = -2.0 * ((x / 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) :: t_1
    real(8) :: tmp
    t_1 = (-2.0d0) * (x / (z_m * t))
    if (t <= (-3.25d+21)) then
        tmp = t_1
    else if (t <= (-1.85d-190)) then
        tmp = (x / z_m) * (2.0d0 / y)
    else if (t <= (-1.8d-190)) then
        tmp = t_1
    else if (t <= 5.2d+80) then
        tmp = (2.0d0 / y) / (z_m / x)
    else if (t <= 4.7d+95) then
        tmp = (-2.0d0) / (z_m * (t / x))
    else if (t <= 4.8d+95) then
        tmp = x * (2.0d0 / (z_m * y))
    else
        tmp = (-2.0d0) * ((x / 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 t_1 = -2.0 * (x / (z_m * t));
	double tmp;
	if (t <= -3.25e+21) {
		tmp = t_1;
	} else if (t <= -1.85e-190) {
		tmp = (x / z_m) * (2.0 / y);
	} else if (t <= -1.8e-190) {
		tmp = t_1;
	} else if (t <= 5.2e+80) {
		tmp = (2.0 / y) / (z_m / x);
	} else if (t <= 4.7e+95) {
		tmp = -2.0 / (z_m * (t / x));
	} else if (t <= 4.8e+95) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = -2.0 * ((x / 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):
	t_1 = -2.0 * (x / (z_m * t))
	tmp = 0
	if t <= -3.25e+21:
		tmp = t_1
	elif t <= -1.85e-190:
		tmp = (x / z_m) * (2.0 / y)
	elif t <= -1.8e-190:
		tmp = t_1
	elif t <= 5.2e+80:
		tmp = (2.0 / y) / (z_m / x)
	elif t <= 4.7e+95:
		tmp = -2.0 / (z_m * (t / x))
	elif t <= 4.8e+95:
		tmp = x * (2.0 / (z_m * y))
	else:
		tmp = -2.0 * ((x / 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)
	t_1 = Float64(-2.0 * Float64(x / Float64(z_m * t)))
	tmp = 0.0
	if (t <= -3.25e+21)
		tmp = t_1;
	elseif (t <= -1.85e-190)
		tmp = Float64(Float64(x / z_m) * Float64(2.0 / y));
	elseif (t <= -1.8e-190)
		tmp = t_1;
	elseif (t <= 5.2e+80)
		tmp = Float64(Float64(2.0 / y) / Float64(z_m / x));
	elseif (t <= 4.7e+95)
		tmp = Float64(-2.0 / Float64(z_m * Float64(t / x)));
	elseif (t <= 4.8e+95)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x / 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)
	t_1 = -2.0 * (x / (z_m * t));
	tmp = 0.0;
	if (t <= -3.25e+21)
		tmp = t_1;
	elseif (t <= -1.85e-190)
		tmp = (x / z_m) * (2.0 / y);
	elseif (t <= -1.8e-190)
		tmp = t_1;
	elseif (t <= 5.2e+80)
		tmp = (2.0 / y) / (z_m / x);
	elseif (t <= 4.7e+95)
		tmp = -2.0 / (z_m * (t / x));
	elseif (t <= 4.8e+95)
		tmp = x * (2.0 / (z_m * y));
	else
		tmp = -2.0 * ((x / 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_] := Block[{t$95$1 = N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t, -3.25e+21], t$95$1, If[LessEqual[t, -1.85e-190], N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.8e-190], t$95$1, If[LessEqual[t, 5.2e+80], N[(N[(2.0 / y), $MachinePrecision] / N[(z$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.7e+95], N[(-2.0 / N[(z$95$m * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.8e+95], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]), $MachinePrecision]]

Derivation

1. Split input into 6 regimes

2. if t < -3.25e21 or -1.8500000000000001e-190 < t < -1.80000000000000003e-190

   1. Initial program 83.1%

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

      1. distribute-rgt-out-- 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in y around 0 81.2%

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

      1. *-commutative 81.2%

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

   7. Simplified 81.2%

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

   if -3.25e21 < t < -1.8500000000000001e-190

   1. Initial program 89.1%

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

      1. distribute-rgt-out-- 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in y around inf 60.4%

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

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

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

      1. times-frac 66.3%

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

   9. Applied egg-rr 66.3%

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

   if -1.80000000000000003e-190 < t < 5.19999999999999963e80

   1. Initial program 89.0%

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

      1. distribute-rgt-out-- 92.8%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in y around inf 74.1%

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

      1. *-commutative 74.1%

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

   7. Simplified 74.1%

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

      1. times-frac 78.5%

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

   9. Applied egg-rr 78.5%

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

       1. *-commutative 78.5%

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

       2. clear-num 78.7%

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

       3. un-div-inv 79.2%

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

   11. Applied egg-rr 79.2%

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

   if 5.19999999999999963e80 < t < 4.69999999999999972e95

   1. Initial program 99.6%

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

      1. distribute-rgt-out-- 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

       1. associate-/l* 100.0%

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

   11. Simplified 100.0%

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

   if 4.69999999999999972e95 < t < 4.8000000000000001e95

   1. Initial program 100.0%

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

      1. distribute-rgt-out-- 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-/r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 100.0%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r/ 100.0%

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

   10. Simplified 100.0%

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

   if 4.8000000000000001e95 < t

   1. Initial program 89.6%

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

      1. distribute-rgt-out-- 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in y around 0 77.7%

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

      1. *-commutative 77.7%

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

   7. Simplified 77.7%

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

   8. Taylor expanded in x around 0 77.7%

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

      1. associate-/r* 87.2%

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

   10. Simplified 87.2%

       \[\leadsto -2 \cdot \color{blue}{\frac{\frac{x}{t}}{z}} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x}{z\_m} \cdot \frac{2}{y}\\ t_2 := -2 \cdot \frac{x}{z\_m \cdot t}\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-190}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-190}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z\_m}\\ \end{array} \end{array} \end{array} \]

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
 (let* ((t_1 (* (/ x z_m) (/ 2.0 y))) (t_2 (* -2.0 (/ x (* z_m t)))))
   (*
    z_s
    (if (<= t -6.5e+20)
      t_2
      (if (<= t -1.85e-190)
        t_1
        (if (<= t -1.8e-190)
          t_2
          (if (<= t 4.8e-8) t_1 (* -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) {
	double t_1 = (x / z_m) * (2.0 / y);
	double t_2 = -2.0 * (x / (z_m * t));
	double tmp;
	if (t <= -6.5e+20) {
		tmp = t_2;
	} else if (t <= -1.85e-190) {
		tmp = t_1;
	} else if (t <= -1.8e-190) {
		tmp = t_2;
	} else if (t <= 4.8e-8) {
		tmp = t_1;
	} else {
		tmp = -2.0 * ((x / 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x / z_m) * (2.0d0 / y)
    t_2 = (-2.0d0) * (x / (z_m * t))
    if (t <= (-6.5d+20)) then
        tmp = t_2
    else if (t <= (-1.85d-190)) then
        tmp = t_1
    else if (t <= (-1.8d-190)) then
        tmp = t_2
    else if (t <= 4.8d-8) then
        tmp = t_1
    else
        tmp = (-2.0d0) * ((x / 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 t_1 = (x / z_m) * (2.0 / y);
	double t_2 = -2.0 * (x / (z_m * t));
	double tmp;
	if (t <= -6.5e+20) {
		tmp = t_2;
	} else if (t <= -1.85e-190) {
		tmp = t_1;
	} else if (t <= -1.8e-190) {
		tmp = t_2;
	} else if (t <= 4.8e-8) {
		tmp = t_1;
	} else {
		tmp = -2.0 * ((x / 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):
	t_1 = (x / z_m) * (2.0 / y)
	t_2 = -2.0 * (x / (z_m * t))
	tmp = 0
	if t <= -6.5e+20:
		tmp = t_2
	elif t <= -1.85e-190:
		tmp = t_1
	elif t <= -1.8e-190:
		tmp = t_2
	elif t <= 4.8e-8:
		tmp = t_1
	else:
		tmp = -2.0 * ((x / 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)
	t_1 = Float64(Float64(x / z_m) * Float64(2.0 / y))
	t_2 = Float64(-2.0 * Float64(x / Float64(z_m * t)))
	tmp = 0.0
	if (t <= -6.5e+20)
		tmp = t_2;
	elseif (t <= -1.85e-190)
		tmp = t_1;
	elseif (t <= -1.8e-190)
		tmp = t_2;
	elseif (t <= 4.8e-8)
		tmp = t_1;
	else
		tmp = Float64(-2.0 * Float64(Float64(x / 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)
	t_1 = (x / z_m) * (2.0 / y);
	t_2 = -2.0 * (x / (z_m * t));
	tmp = 0.0;
	if (t <= -6.5e+20)
		tmp = t_2;
	elseif (t <= -1.85e-190)
		tmp = t_1;
	elseif (t <= -1.8e-190)
		tmp = t_2;
	elseif (t <= 4.8e-8)
		tmp = t_1;
	else
		tmp = -2.0 * ((x / 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_] := Block[{t$95$1 = N[(N[(x / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[t, -6.5e+20], t$95$2, If[LessEqual[t, -1.85e-190], t$95$1, If[LessEqual[t, -1.8e-190], t$95$2, If[LessEqual[t, 4.8e-8], t$95$1, N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.5e20 or -1.8500000000000001e-190 < t < -1.80000000000000003e-190

   1. Initial program 83.1%

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

      1. distribute-rgt-out-- 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in y around 0 81.2%

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

      1. *-commutative 81.2%

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

   7. Simplified 81.2%

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

   if -6.5e20 < t < -1.8500000000000001e-190 or -1.80000000000000003e-190 < t < 4.79999999999999997e-8

   1. Initial program 87.4%

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

      1. distribute-rgt-out-- 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in y around inf 70.1%

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

      1. *-commutative 70.1%

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

   7. Simplified 70.1%

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

      1. times-frac 75.0%

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

   9. Applied egg-rr 75.0%

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

   if 4.79999999999999997e-8 < t

   1. Initial program 94.0%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in y around 0 72.8%

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

      1. *-commutative 72.8%

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

   7. Simplified 72.8%

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

   8. Taylor expanded in x around 0 72.8%

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

      1. associate-/r* 78.2%

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

   10. Simplified 78.2%

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

Alternative 5: 92.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-132} \lor \neg \left(x \cdot 2 \leq 10^{+259}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z\_m}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (or (<= (* x 2.0) 5e-132) (not (<= (* x 2.0) 1e+259)))
    (* 2.0 (/ (/ x 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 (((x * 2.0) <= 5e-132) || !((x * 2.0) <= 1e+259)) {
		tmp = 2.0 * ((x / 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 (((x * 2.0d0) <= 5d-132) .or. (.not. ((x * 2.0d0) <= 1d+259))) then
        tmp = 2.0d0 * ((x / 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 (((x * 2.0) <= 5e-132) || !((x * 2.0) <= 1e+259)) {
		tmp = 2.0 * ((x / 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 ((x * 2.0) <= 5e-132) or not ((x * 2.0) <= 1e+259):
		tmp = 2.0 * ((x / 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 ((Float64(x * 2.0) <= 5e-132) || !(Float64(x * 2.0) <= 1e+259))
		tmp = Float64(2.0 * Float64(Float64(x / 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 (((x * 2.0) <= 5e-132) || ~(((x * 2.0) <= 1e+259)))
		tmp = 2.0 * ((x / 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[Or[LessEqual[N[(x * 2.0), $MachinePrecision], 5e-132], N[Not[LessEqual[N[(x * 2.0), $MachinePrecision], 1e+259]], $MachinePrecision]], N[(2.0 * N[(N[(x / z$95$m), $MachinePrecision] / 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 (*.f64 x #s(literal 2 binary64)) < 4.9999999999999999e-132 or 9.999999999999999e258 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 90.8%

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

      1. distribute-rgt-out-- 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in x around 0 91.4%

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

      1. associate-/r* 95.6%

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

   7. Simplified 95.6%

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

   if 4.9999999999999999e-132 < (*.f64 x #s(literal 2 binary64)) < 9.999999999999999e258

   1. Initial program 82.4%

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

      1. *-commutative 90.0%

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

      2. times-frac 97.2%

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

   6. Applied egg-rr 97.2%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-132} \lor \neg \left(x \cdot 2 \leq 10^{+259}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-34}:\\ \;\;\;\;-2 \cdot \frac{x}{z\_m \cdot t}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x}{y}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -3.7e-34)
    (* -2.0 (/ x (* z_m t)))
    (if (<= t -4e-131)
      (* (/ 2.0 z_m) (/ x y))
      (if (<= t 2.35e-27) (* x (/ 2.0 (* z_m y))) (* -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) {
	double tmp;
	if (t <= -3.7e-34) {
		tmp = -2.0 * (x / (z_m * t));
	} else if (t <= -4e-131) {
		tmp = (2.0 / z_m) * (x / y);
	} else if (t <= 2.35e-27) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = -2.0 * ((x / 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 <= (-3.7d-34)) then
        tmp = (-2.0d0) * (x / (z_m * t))
    else if (t <= (-4d-131)) then
        tmp = (2.0d0 / z_m) * (x / y)
    else if (t <= 2.35d-27) then
        tmp = x * (2.0d0 / (z_m * y))
    else
        tmp = (-2.0d0) * ((x / 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 <= -3.7e-34) {
		tmp = -2.0 * (x / (z_m * t));
	} else if (t <= -4e-131) {
		tmp = (2.0 / z_m) * (x / y);
	} else if (t <= 2.35e-27) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = -2.0 * ((x / 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 <= -3.7e-34:
		tmp = -2.0 * (x / (z_m * t))
	elif t <= -4e-131:
		tmp = (2.0 / z_m) * (x / y)
	elif t <= 2.35e-27:
		tmp = x * (2.0 / (z_m * y))
	else:
		tmp = -2.0 * ((x / 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 <= -3.7e-34)
		tmp = Float64(-2.0 * Float64(x / Float64(z_m * t)));
	elseif (t <= -4e-131)
		tmp = Float64(Float64(2.0 / z_m) * Float64(x / y));
	elseif (t <= 2.35e-27)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x / 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 <= -3.7e-34)
		tmp = -2.0 * (x / (z_m * t));
	elseif (t <= -4e-131)
		tmp = (2.0 / z_m) * (x / y);
	elseif (t <= 2.35e-27)
		tmp = x * (2.0 / (z_m * y));
	else
		tmp = -2.0 * ((x / 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, -3.7e-34], N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4e-131], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.35e-27], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if t < -3.69999999999999988e-34

   1. Initial program 82.3%

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

      1. distribute-rgt-out-- 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in y around 0 76.0%

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

      1. *-commutative 76.0%

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

   7. Simplified 76.0%

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

   if -3.69999999999999988e-34 < t < -3.9999999999999999e-131

   1. Initial program 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in y around inf 64.0%

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

   if -3.9999999999999999e-131 < t < 2.35000000000000016e-27

   1. Initial program 87.1%

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

      1. distribute-rgt-out-- 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in x around 0 91.3%

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

      1. associate-/r* 95.2%

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

   7. Simplified 95.2%

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

   8. Taylor expanded in y around inf 75.2%

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

      1. associate-*r/ 75.2%

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

      2. *-commutative 75.2%

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

      3. *-commutative 75.2%

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

      4. associate-*r/ 75.1%

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

   10. Simplified 75.1%

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

   if 2.35000000000000016e-27 < t

   1. Initial program 94.2%

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

      1. distribute-rgt-out-- 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in y around 0 70.9%

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

      1. *-commutative 70.9%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in x around 0 70.9%

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

      1. associate-/r* 76.1%

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

   10. Simplified 76.1%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-34}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-131}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{x \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{elif}\;z\_m \leq 1.2 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z\_m}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= z_m 3.5e-59)
    (/ (* x 2.0) (* z_m (- y t)))
    (if (<= z_m 1.2e+135)
      (* 2.0 (/ (/ x 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 <= 3.5e-59) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else if (z_m <= 1.2e+135) {
		tmp = 2.0 * ((x / 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 <= 3.5d-59) then
        tmp = (x * 2.0d0) / (z_m * (y - t))
    else if (z_m <= 1.2d+135) then
        tmp = 2.0d0 * ((x / 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 <= 3.5e-59) {
		tmp = (x * 2.0) / (z_m * (y - t));
	} else if (z_m <= 1.2e+135) {
		tmp = 2.0 * ((x / 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 <= 3.5e-59:
		tmp = (x * 2.0) / (z_m * (y - t))
	elif z_m <= 1.2e+135:
		tmp = 2.0 * ((x / 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 <= 3.5e-59)
		tmp = Float64(Float64(x * 2.0) / Float64(z_m * Float64(y - t)));
	elseif (z_m <= 1.2e+135)
		tmp = Float64(2.0 * Float64(Float64(x / 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 <= 3.5e-59)
		tmp = (x * 2.0) / (z_m * (y - t));
	elseif (z_m <= 1.2e+135)
		tmp = 2.0 * ((x / 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, 3.5e-59], N[(N[(x * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.2e+135], N[(2.0 * N[(N[(x / z$95$m), $MachinePrecision] / 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 3 regimes

2. if z < 3.5000000000000001e-59

   1. Initial program 89.3%

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

      1. distribute-rgt-out-- 92.2%

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

   3. Simplified 92.2%

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

   if 3.5000000000000001e-59 < z < 1.19999999999999999e135

   1. Initial program 92.2%

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

      1. distribute-rgt-out-- 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in x around 0 92.4%

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

      1. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

   if 1.19999999999999999e135 < z

   1. Initial program 78.3%

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

      1. distribute-rgt-out-- 83.7%

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

   3. Simplified 83.7%

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

      1. *-commutative 83.7%

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

      2. times-frac 97.2%

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

   6. Applied egg-rr 97.2%

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

Alternative 8: 74.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-34}:\\ \;\;\;\;-2 \cdot \frac{x}{z\_m \cdot t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \frac{2}{z\_m \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m t)
 :precision binary64
 (*
  z_s
  (if (<= t -4.4e-34)
    (* -2.0 (/ x (* z_m t)))
    (if (<= t 1.15e-25) (* x (/ 2.0 (* z_m y))) (* -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) {
	double tmp;
	if (t <= -4.4e-34) {
		tmp = -2.0 * (x / (z_m * t));
	} else if (t <= 1.15e-25) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = -2.0 * ((x / 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 <= (-4.4d-34)) then
        tmp = (-2.0d0) * (x / (z_m * t))
    else if (t <= 1.15d-25) then
        tmp = x * (2.0d0 / (z_m * y))
    else
        tmp = (-2.0d0) * ((x / 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 <= -4.4e-34) {
		tmp = -2.0 * (x / (z_m * t));
	} else if (t <= 1.15e-25) {
		tmp = x * (2.0 / (z_m * y));
	} else {
		tmp = -2.0 * ((x / 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 <= -4.4e-34:
		tmp = -2.0 * (x / (z_m * t))
	elif t <= 1.15e-25:
		tmp = x * (2.0 / (z_m * y))
	else:
		tmp = -2.0 * ((x / 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 <= -4.4e-34)
		tmp = Float64(-2.0 * Float64(x / Float64(z_m * t)));
	elseif (t <= 1.15e-25)
		tmp = Float64(x * Float64(2.0 / Float64(z_m * y)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x / 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 <= -4.4e-34)
		tmp = -2.0 * (x / (z_m * t));
	elseif (t <= 1.15e-25)
		tmp = x * (2.0 / (z_m * y));
	else
		tmp = -2.0 * ((x / 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, -4.4e-34], N[(-2.0 * N[(x / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e-25], N[(x * N[(2.0 / N[(z$95$m * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -4.3999999999999998e-34

   1. Initial program 82.3%

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

      1. distribute-rgt-out-- 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in y around 0 76.0%

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

      1. *-commutative 76.0%

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

   7. Simplified 76.0%

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

   if -4.3999999999999998e-34 < t < 1.15e-25

   1. Initial program 87.9%

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

      1. distribute-rgt-out-- 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in x around 0 91.2%

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

      1. associate-/r* 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in y around inf 71.5%

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

      1. associate-*r/ 71.6%

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

      2. *-commutative 71.6%

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

      3. *-commutative 71.6%

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

      4. associate-*r/ 71.5%

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

   10. Simplified 71.5%

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

   if 1.15e-25 < t

   1. Initial program 94.2%

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

      1. distribute-rgt-out-- 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in y around 0 70.9%

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

      1. *-commutative 70.9%

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

   7. Simplified 70.9%

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

   8. Taylor expanded in x around 0 70.9%

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

      1. associate-/r* 76.1%

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

   10. Simplified 76.1%

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

Alternative 9: 91.6% accurate, 1.2× speedup

Math

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

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 88.2%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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. Taylor expanded in x around 0 91.0%

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

   1. associate-/r* 93.7%

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

7. Simplified 93.7%

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

Alternative 10: 53.7% accurate, 1.6× speedup

Math

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

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 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 88.2%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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. Taylor expanded in y around 0 50.1%

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

   1. *-commutative 50.1%

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

7. Simplified 50.1%

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

8. Taylor expanded in x around 0 50.1%

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

   1. associate-/r* 51.0%

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

10. Simplified 51.0%

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

Alternative 11: 53.6% 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 88.2%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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. Taylor expanded in y around 0 50.1%

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

   1. *-commutative 50.1%

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

7. Simplified 50.1%

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

Developer target: 97.2% 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 2024107 
(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))))