Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.3% → 95.9%

Time: 9.3s

Alternatives: 9

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 4 \cdot 10^{+96}:\\ \;\;\;\;\frac{2 \cdot \frac{x\_m}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 4e+96)
    (/ (* 2.0 (/ x_m z)) (- y t))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+96) {
		tmp = (2.0 * (x_m / z)) / (y - t);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 4d+96) then
        tmp = (2.0d0 * (x_m / z)) / (y - t)
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 4e+96) {
		tmp = (2.0 * (x_m / z)) / (y - t);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 4e+96:
		tmp = (2.0 * (x_m / z)) / (y - t)
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 4e+96)
		tmp = Float64(Float64(2.0 * Float64(x_m / z)) / Float64(y - t));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 4e+96)
		tmp = (2.0 * (x_m / z)) / (y - t);
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 4e+96], N[(N[(2.0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.0000000000000002e96

   1. Initial program 91.2%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 x around 0 91.7%

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

      1. associate-/r* 95.4%

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

      2. associate-*r/ 95.4%

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

   7. Simplified 95.4%

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

   if 4.0000000000000002e96 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 76.1%

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

      1. distribute-rgt-out-- 76.2%

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

   3. Simplified 76.2%

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

      1. *-commutative 76.2%

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

      2. times-frac 95.3%

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

   6. Applied egg-rr 95.3%

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

Alternative 2: 72.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;t \leq -3.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x\_m}}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-19}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{z \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -8.5e+118)
    (* (/ -2.0 z) (/ x_m t))
    (if (<= t -3.15e+49)
      (/ 2.0 (* y (/ z x_m)))
      (if (<= t -4.2e-19)
        (* -2.0 (/ x_m (* z t)))
        (if (<= t 2.15e+42)
          (/ (/ x_m y) (* z 0.5))
          (* -2.0 (/ (/ x_m z) t))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -8.5e+118) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (t <= -3.15e+49) {
		tmp = 2.0 / (y * (z / x_m));
	} else if (t <= -4.2e-19) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (t <= 2.15e+42) {
		tmp = (x_m / y) / (z * 0.5);
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-8.5d+118)) then
        tmp = ((-2.0d0) / z) * (x_m / t)
    else if (t <= (-3.15d+49)) then
        tmp = 2.0d0 / (y * (z / x_m))
    else if (t <= (-4.2d-19)) then
        tmp = (-2.0d0) * (x_m / (z * t))
    else if (t <= 2.15d+42) then
        tmp = (x_m / y) / (z * 0.5d0)
    else
        tmp = (-2.0d0) * ((x_m / z) / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -8.5e+118) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (t <= -3.15e+49) {
		tmp = 2.0 / (y * (z / x_m));
	} else if (t <= -4.2e-19) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (t <= 2.15e+42) {
		tmp = (x_m / y) / (z * 0.5);
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -8.5e+118:
		tmp = (-2.0 / z) * (x_m / t)
	elif t <= -3.15e+49:
		tmp = 2.0 / (y * (z / x_m))
	elif t <= -4.2e-19:
		tmp = -2.0 * (x_m / (z * t))
	elif t <= 2.15e+42:
		tmp = (x_m / y) / (z * 0.5)
	else:
		tmp = -2.0 * ((x_m / z) / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -8.5e+118)
		tmp = Float64(Float64(-2.0 / z) * Float64(x_m / t));
	elseif (t <= -3.15e+49)
		tmp = Float64(2.0 / Float64(y * Float64(z / x_m)));
	elseif (t <= -4.2e-19)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	elseif (t <= 2.15e+42)
		tmp = Float64(Float64(x_m / y) / Float64(z * 0.5));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -8.5e+118)
		tmp = (-2.0 / z) * (x_m / t);
	elseif (t <= -3.15e+49)
		tmp = 2.0 / (y * (z / x_m));
	elseif (t <= -4.2e-19)
		tmp = -2.0 * (x_m / (z * t));
	elseif (t <= 2.15e+42)
		tmp = (x_m / y) / (z * 0.5);
	else
		tmp = -2.0 * ((x_m / z) / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -8.5e+118], N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -3.15e+49], N[(2.0 / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.2e-19], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.15e+42], N[(N[(x$95$m / y), $MachinePrecision] / N[(z * 0.5), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if t < -8.50000000000000033e118

   1. Initial program 83.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 91.8%

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

   6. Applied egg-rr 91.8%

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

   7. Taylor expanded in y around 0 75.1%

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

      1. associate-/l/ 79.7%

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

      2. associate-*r/ 79.7%

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

      3. *-commutative 79.7%

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

      4. metadata-eval 79.7%

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

      5. distribute-rgt-neg-in 79.7%

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

      6. associate-*l/ 79.7%

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

      7. associate-*r/ 79.6%

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

      8. distribute-rgt-neg-in 79.6%

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

      9. distribute-neg-frac 79.6%

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

      10. metadata-eval 79.6%

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

   9. Simplified 79.6%

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

       1. *-commutative 79.6%

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

       2. associate-/l* 87.9%

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

   11. Applied egg-rr 87.9%

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

   if -8.50000000000000033e118 < t < -3.15000000000000003e49

   1. Initial program 93.9%

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

      1. distribute-rgt-out-- 99.8%

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

   3. Simplified 99.8%

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

      1. times-frac 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around inf 75.7%

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

      1. clear-num 75.7%

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

      2. frac-times 79.3%

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

      3. metadata-eval 79.3%

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

   9. Applied egg-rr 79.3%

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

   if -3.15000000000000003e49 < t < -4.1999999999999998e-19

   1. Initial program 99.8%

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

      1. distribute-rgt-out-- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 92.9%

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

      1. *-commutative 92.9%

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

   7. Simplified 92.9%

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

   if -4.1999999999999998e-19 < t < 2.1499999999999999e42

   1. Initial program 91.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 y around inf 75.1%

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

      1. associate-*r/ 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 78.0%

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

   7. Simplified 78.0%

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

      1. *-commutative 78.0%

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

      2. clear-num 78.0%

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

      3. un-div-inv 78.0%

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

      4. div-inv 78.0%

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

      5. metadata-eval 78.0%

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

   9. Applied egg-rr 78.0%

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

   if 2.1499999999999999e42 < t

   1. Initial program 80.5%

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

      1. distribute-rgt-out-- 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in y around 0 73.3%

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

      1. *-commutative 73.3%

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

      2. associate-/r* 85.3%

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

   7. Simplified 85.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{\frac{x}{z}}{t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq -3.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-19}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{x}{y}}{z \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;t \leq -1.62 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x\_m}}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-19}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -8.5e+118)
    (* (/ -2.0 z) (/ x_m t))
    (if (<= t -1.62e+50)
      (/ 2.0 (* y (/ z x_m)))
      (if (<= t -4.1e-19)
        (* -2.0 (/ x_m (* z t)))
        (if (<= t 3.3e+42)
          (* (/ 2.0 z) (/ x_m y))
          (* -2.0 (/ (/ x_m z) t))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -8.5e+118) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (t <= -1.62e+50) {
		tmp = 2.0 / (y * (z / x_m));
	} else if (t <= -4.1e-19) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (t <= 3.3e+42) {
		tmp = (2.0 / z) * (x_m / y);
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-8.5d+118)) then
        tmp = ((-2.0d0) / z) * (x_m / t)
    else if (t <= (-1.62d+50)) then
        tmp = 2.0d0 / (y * (z / x_m))
    else if (t <= (-4.1d-19)) then
        tmp = (-2.0d0) * (x_m / (z * t))
    else if (t <= 3.3d+42) then
        tmp = (2.0d0 / z) * (x_m / y)
    else
        tmp = (-2.0d0) * ((x_m / z) / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -8.5e+118) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (t <= -1.62e+50) {
		tmp = 2.0 / (y * (z / x_m));
	} else if (t <= -4.1e-19) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (t <= 3.3e+42) {
		tmp = (2.0 / z) * (x_m / y);
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -8.5e+118:
		tmp = (-2.0 / z) * (x_m / t)
	elif t <= -1.62e+50:
		tmp = 2.0 / (y * (z / x_m))
	elif t <= -4.1e-19:
		tmp = -2.0 * (x_m / (z * t))
	elif t <= 3.3e+42:
		tmp = (2.0 / z) * (x_m / y)
	else:
		tmp = -2.0 * ((x_m / z) / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -8.5e+118)
		tmp = Float64(Float64(-2.0 / z) * Float64(x_m / t));
	elseif (t <= -1.62e+50)
		tmp = Float64(2.0 / Float64(y * Float64(z / x_m)));
	elseif (t <= -4.1e-19)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	elseif (t <= 3.3e+42)
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -8.5e+118)
		tmp = (-2.0 / z) * (x_m / t);
	elseif (t <= -1.62e+50)
		tmp = 2.0 / (y * (z / x_m));
	elseif (t <= -4.1e-19)
		tmp = -2.0 * (x_m / (z * t));
	elseif (t <= 3.3e+42)
		tmp = (2.0 / z) * (x_m / y);
	else
		tmp = -2.0 * ((x_m / z) / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -8.5e+118], N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.62e+50], N[(2.0 / N[(y * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.1e-19], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.3e+42], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if t < -8.50000000000000033e118

   1. Initial program 83.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 91.8%

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

   6. Applied egg-rr 91.8%

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

   7. Taylor expanded in y around 0 75.1%

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

      1. associate-/l/ 79.7%

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

      2. associate-*r/ 79.7%

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

      3. *-commutative 79.7%

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

      4. metadata-eval 79.7%

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

      5. distribute-rgt-neg-in 79.7%

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

      6. associate-*l/ 79.7%

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

      7. associate-*r/ 79.6%

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

      8. distribute-rgt-neg-in 79.6%

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

      9. distribute-neg-frac 79.6%

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

      10. metadata-eval 79.6%

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

   9. Simplified 79.6%

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

       1. *-commutative 79.6%

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

       2. associate-/l* 87.9%

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

   11. Applied egg-rr 87.9%

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

   if -8.50000000000000033e118 < t < -1.61999999999999996e50

   1. Initial program 93.9%

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

      1. distribute-rgt-out-- 99.8%

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

   3. Simplified 99.8%

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

      1. times-frac 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around inf 75.7%

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

      1. clear-num 75.7%

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

      2. frac-times 79.3%

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

      3. metadata-eval 79.3%

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

   9. Applied egg-rr 79.3%

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

   if -1.61999999999999996e50 < t < -4.09999999999999985e-19

   1. Initial program 99.8%

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

      1. distribute-rgt-out-- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 92.9%

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

      1. *-commutative 92.9%

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

   7. Simplified 92.9%

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

   if -4.09999999999999985e-19 < t < 3.2999999999999999e42

   1. Initial program 91.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 y around inf 75.1%

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

      1. associate-*r/ 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 78.0%

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

   7. Simplified 78.0%

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

   if 3.2999999999999999e42 < t

   1. Initial program 80.5%

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

      1. distribute-rgt-out-- 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in y around 0 73.3%

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

      1. *-commutative 73.3%

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

      2. associate-/r* 85.3%

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

   7. Simplified 85.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{\frac{x}{z}}{t}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq -1.62 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{-19}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+42}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{+118}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{+48}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;t \leq -6.3 \cdot 10^{-21}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -8.5e+118)
    (* (/ -2.0 z) (/ x_m t))
    (if (<= t -1.7e+48)
      (* (/ x_m z) (/ 2.0 y))
      (if (<= t -6.3e-21)
        (* -2.0 (/ x_m (* z t)))
        (if (<= t 2.8e+42)
          (* (/ 2.0 z) (/ x_m y))
          (* -2.0 (/ (/ x_m z) t))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -8.5e+118) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (t <= -1.7e+48) {
		tmp = (x_m / z) * (2.0 / y);
	} else if (t <= -6.3e-21) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (t <= 2.8e+42) {
		tmp = (2.0 / z) * (x_m / y);
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-8.5d+118)) then
        tmp = ((-2.0d0) / z) * (x_m / t)
    else if (t <= (-1.7d+48)) then
        tmp = (x_m / z) * (2.0d0 / y)
    else if (t <= (-6.3d-21)) then
        tmp = (-2.0d0) * (x_m / (z * t))
    else if (t <= 2.8d+42) then
        tmp = (2.0d0 / z) * (x_m / y)
    else
        tmp = (-2.0d0) * ((x_m / z) / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -8.5e+118) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (t <= -1.7e+48) {
		tmp = (x_m / z) * (2.0 / y);
	} else if (t <= -6.3e-21) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (t <= 2.8e+42) {
		tmp = (2.0 / z) * (x_m / y);
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -8.5e+118:
		tmp = (-2.0 / z) * (x_m / t)
	elif t <= -1.7e+48:
		tmp = (x_m / z) * (2.0 / y)
	elif t <= -6.3e-21:
		tmp = -2.0 * (x_m / (z * t))
	elif t <= 2.8e+42:
		tmp = (2.0 / z) * (x_m / y)
	else:
		tmp = -2.0 * ((x_m / z) / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -8.5e+118)
		tmp = Float64(Float64(-2.0 / z) * Float64(x_m / t));
	elseif (t <= -1.7e+48)
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / y));
	elseif (t <= -6.3e-21)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	elseif (t <= 2.8e+42)
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -8.5e+118)
		tmp = (-2.0 / z) * (x_m / t);
	elseif (t <= -1.7e+48)
		tmp = (x_m / z) * (2.0 / y);
	elseif (t <= -6.3e-21)
		tmp = -2.0 * (x_m / (z * t));
	elseif (t <= 2.8e+42)
		tmp = (2.0 / z) * (x_m / y);
	else
		tmp = -2.0 * ((x_m / z) / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -8.5e+118], N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.7e+48], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -6.3e-21], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.8e+42], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if t < -8.50000000000000033e118

   1. Initial program 83.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 91.8%

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

   6. Applied egg-rr 91.8%

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

   7. Taylor expanded in y around 0 75.1%

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

      1. associate-/l/ 79.7%

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

      2. associate-*r/ 79.7%

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

      3. *-commutative 79.7%

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

      4. metadata-eval 79.7%

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

      5. distribute-rgt-neg-in 79.7%

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

      6. associate-*l/ 79.7%

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

      7. associate-*r/ 79.6%

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

      8. distribute-rgt-neg-in 79.6%

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

      9. distribute-neg-frac 79.6%

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

      10. metadata-eval 79.6%

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

   9. Simplified 79.6%

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

       1. *-commutative 79.6%

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

       2. associate-/l* 87.9%

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

   11. Applied egg-rr 87.9%

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

   if -8.50000000000000033e118 < t < -1.7000000000000002e48

   1. Initial program 93.9%

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

      1. distribute-rgt-out-- 99.8%

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

   3. Simplified 99.8%

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

      1. times-frac 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around inf 75.7%

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

   if -1.7000000000000002e48 < t < -6.3e-21

   1. Initial program 99.8%

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

      1. distribute-rgt-out-- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 92.9%

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

      1. *-commutative 92.9%

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

   7. Simplified 92.9%

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

   if -6.3e-21 < t < 2.7999999999999999e42

   1. Initial program 91.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 y around inf 75.1%

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

      1. associate-*r/ 75.1%

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

      2. *-commutative 75.1%

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

      3. times-frac 78.0%

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

   7. Simplified 78.0%

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

   if 2.7999999999999999e42 < t

   1. Initial program 80.5%

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

      1. distribute-rgt-out-- 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in y around 0 73.3%

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

      1. *-commutative 73.3%

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

      2. associate-/r* 85.3%

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

   7. Simplified 85.3%

      \[\leadsto \color{blue}{-2 \cdot \frac{\frac{x}{z}}{t}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{2}{z} \cdot \frac{x\_m}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+21}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* (/ 2.0 z) (/ x_m y))))
   (*
    x_s
    (if (<= t -1.1e-19)
      (* (/ -2.0 z) (/ x_m t))
      (if (<= t 9.5e-34)
        t_1
        (if (<= t 8.8e+21)
          (* -2.0 (/ x_m (* z t)))
          (if (<= t 1.5e+46) t_1 (* -2.0 (/ (/ x_m z) t)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (2.0 / z) * (x_m / y);
	double tmp;
	if (t <= -1.1e-19) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (t <= 9.5e-34) {
		tmp = t_1;
	} else if (t <= 8.8e+21) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (t <= 1.5e+46) {
		tmp = t_1;
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 / z) * (x_m / y)
    if (t <= (-1.1d-19)) then
        tmp = ((-2.0d0) / z) * (x_m / t)
    else if (t <= 9.5d-34) then
        tmp = t_1
    else if (t <= 8.8d+21) then
        tmp = (-2.0d0) * (x_m / (z * t))
    else if (t <= 1.5d+46) then
        tmp = t_1
    else
        tmp = (-2.0d0) * ((x_m / z) / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (2.0 / z) * (x_m / y);
	double tmp;
	if (t <= -1.1e-19) {
		tmp = (-2.0 / z) * (x_m / t);
	} else if (t <= 9.5e-34) {
		tmp = t_1;
	} else if (t <= 8.8e+21) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (t <= 1.5e+46) {
		tmp = t_1;
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (2.0 / z) * (x_m / y)
	tmp = 0
	if t <= -1.1e-19:
		tmp = (-2.0 / z) * (x_m / t)
	elif t <= 9.5e-34:
		tmp = t_1
	elif t <= 8.8e+21:
		tmp = -2.0 * (x_m / (z * t))
	elif t <= 1.5e+46:
		tmp = t_1
	else:
		tmp = -2.0 * ((x_m / z) / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(2.0 / z) * Float64(x_m / y))
	tmp = 0.0
	if (t <= -1.1e-19)
		tmp = Float64(Float64(-2.0 / z) * Float64(x_m / t));
	elseif (t <= 9.5e-34)
		tmp = t_1;
	elseif (t <= 8.8e+21)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	elseif (t <= 1.5e+46)
		tmp = t_1;
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (2.0 / z) * (x_m / y);
	tmp = 0.0;
	if (t <= -1.1e-19)
		tmp = (-2.0 / z) * (x_m / t);
	elseif (t <= 9.5e-34)
		tmp = t_1;
	elseif (t <= 8.8e+21)
		tmp = -2.0 * (x_m / (z * t));
	elseif (t <= 1.5e+46)
		tmp = t_1;
	else
		tmp = -2.0 * ((x_m / z) / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -1.1e-19], N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e-34], t$95$1, If[LessEqual[t, 8.8e+21], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.5e+46], t$95$1, N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < -1.0999999999999999e-19

   1. Initial program 89.7%

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

      1. *-commutative 91.3%

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

      2. times-frac 94.0%

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

   6. Applied egg-rr 94.0%

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

   7. Taylor expanded in y around 0 72.2%

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

      1. associate-/l/ 74.6%

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

      2. associate-*r/ 74.6%

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

      3. *-commutative 74.6%

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

      4. metadata-eval 74.6%

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

      5. distribute-rgt-neg-in 74.6%

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

      6. associate-*l/ 74.6%

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

      7. associate-*r/ 74.6%

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

      8. distribute-rgt-neg-in 74.6%

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

      9. distribute-neg-frac 74.6%

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

      10. metadata-eval 74.6%

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

   9. Simplified 74.6%

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

       1. *-commutative 74.6%

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

       2. associate-/l* 79.0%

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

   11. Applied egg-rr 79.0%

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

   if -1.0999999999999999e-19 < t < 9.49999999999999985e-34 or 8.8e21 < t < 1.50000000000000012e46

   1. Initial program 90.4%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 77.4%

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

      1. associate-*r/ 77.4%

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

      2. *-commutative 77.4%

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

      3. times-frac 80.7%

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

   7. Simplified 80.7%

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

   if 9.49999999999999985e-34 < t < 8.8e21

   1. Initial program 99.7%

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

      1. distribute-rgt-out-- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 83.9%

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

      1. *-commutative 83.9%

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

   7. Simplified 83.9%

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

   if 1.50000000000000012e46 < t

   1. Initial program 80.5%

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

      1. distribute-rgt-out-- 80.7%

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

   3. Simplified 80.7%

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

   5. Taylor expanded in y around 0 73.3%

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

      1. *-commutative 73.3%

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

      2. associate-/r* 85.3%

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

   7. Simplified 85.3%

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

Alternative 6: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 10^{+90}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 1e+90)
    (* (/ x_m z) (/ 2.0 (- y t)))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 1e+90) {
		tmp = (x_m / z) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 1d+90) then
        tmp = (x_m / z) * (2.0d0 / (y - t))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 1e+90) {
		tmp = (x_m / z) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 1e+90:
		tmp = (x_m / z) * (2.0 / (y - t))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 1e+90)
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 1e+90)
		tmp = (x_m / z) * (2.0 / (y - t));
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 1e+90], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 9.99999999999999966e89

   1. Initial program 91.2%

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

      1. times-frac 95.4%

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

   6. Applied egg-rr 95.4%

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

   if 9.99999999999999966e89 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 76.1%

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

      1. distribute-rgt-out-- 76.2%

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

   3. Simplified 76.2%

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

      1. *-commutative 76.2%

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

      2. times-frac 95.3%

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

   6. Applied egg-rr 95.3%

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

Alternative 7: 91.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. distribute-rgt-out-- 89.0%

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

3. Simplified 89.0%

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

   1. times-frac 94.2%

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

6. Applied egg-rr 94.2%

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

Alternative 8: 55.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. distribute-rgt-out-- 89.0%

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

3. Simplified 89.0%

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

5. Taylor expanded in y around 0 51.8%

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

   1. *-commutative 51.8%

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

   2. associate-/r* 58.3%

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

7. Simplified 58.3%

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

Alternative 9: 52.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. distribute-rgt-out-- 89.0%

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

3. Simplified 89.0%

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

5. Taylor expanded in y around 0 51.8%

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

   1. *-commutative 51.8%

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

7. Simplified 51.8%

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

Developer target: 97.0% 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 2024091 
(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))))