Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.5% → 96.3%

Time: 12.6s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

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) 5e-86)
    (/ (* x_m 2.0) (* 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) <= 5e-86) {
		tmp = (x_m * 2.0) / (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) <= 5d-86) then
        tmp = (x_m * 2.0d0) / (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) <= 5e-86) {
		tmp = (x_m * 2.0) / (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) <= 5e-86:
		tmp = (x_m * 2.0) / (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) <= 5e-86)
		tmp = Float64(Float64(x_m * 2.0) / Float64(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) <= 5e-86)
		tmp = (x_m * 2.0) / (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], 5e-86], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * 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)) < 4.9999999999999999e-86

   1. Initial program 91.9%

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

      1. distribute-rgt-out-- 93.1%

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

   3. Simplified 93.1%

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

   if 4.9999999999999999e-86 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 86.3%

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

      1. distribute-rgt-out-- 87.6%

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

   3. Simplified 87.6%

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

      1. *-commutative 87.6%

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

      2. times-frac 97.4%

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

   6. Applied egg-rr 97.4%

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

Alternative 2: 73.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{-2}{z} \cdot \frac{x\_m}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 13500000:\\ \;\;\;\;\frac{x\_m \cdot 2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{z \cdot \frac{t}{x\_m}}\\ \end{array} \end{array} \end{array} \]

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 t))))
   (*
    x_s
    (if (<= t -1.08e+20)
      t_1
      (if (<= t 4e-185)
        (* 2.0 (/ (/ x_m z) y))
        (if (<= t 8.5e-168)
          t_1
          (if (<= t 13500000.0)
            (/ (* x_m 2.0) (* z y))
            (/ -2.0 (* z (/ t x_m))))))))))

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 / t);
	double tmp;
	if (t <= -1.08e+20) {
		tmp = t_1;
	} else if (t <= 4e-185) {
		tmp = 2.0 * ((x_m / z) / y);
	} else if (t <= 8.5e-168) {
		tmp = t_1;
	} else if (t <= 13500000.0) {
		tmp = (x_m * 2.0) / (z * y);
	} else {
		tmp = -2.0 / (z * (t / x_m));
	}
	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 / t)
    if (t <= (-1.08d+20)) then
        tmp = t_1
    else if (t <= 4d-185) then
        tmp = 2.0d0 * ((x_m / z) / y)
    else if (t <= 8.5d-168) then
        tmp = t_1
    else if (t <= 13500000.0d0) then
        tmp = (x_m * 2.0d0) / (z * y)
    else
        tmp = (-2.0d0) / (z * (t / x_m))
    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 / t);
	double tmp;
	if (t <= -1.08e+20) {
		tmp = t_1;
	} else if (t <= 4e-185) {
		tmp = 2.0 * ((x_m / z) / y);
	} else if (t <= 8.5e-168) {
		tmp = t_1;
	} else if (t <= 13500000.0) {
		tmp = (x_m * 2.0) / (z * y);
	} else {
		tmp = -2.0 / (z * (t / x_m));
	}
	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 / t)
	tmp = 0
	if t <= -1.08e+20:
		tmp = t_1
	elif t <= 4e-185:
		tmp = 2.0 * ((x_m / z) / y)
	elif t <= 8.5e-168:
		tmp = t_1
	elif t <= 13500000.0:
		tmp = (x_m * 2.0) / (z * y)
	else:
		tmp = -2.0 / (z * (t / x_m))
	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 / t))
	tmp = 0.0
	if (t <= -1.08e+20)
		tmp = t_1;
	elseif (t <= 4e-185)
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / y));
	elseif (t <= 8.5e-168)
		tmp = t_1;
	elseif (t <= 13500000.0)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z * y));
	else
		tmp = Float64(-2.0 / Float64(z * Float64(t / x_m)));
	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 / t);
	tmp = 0.0;
	if (t <= -1.08e+20)
		tmp = t_1;
	elseif (t <= 4e-185)
		tmp = 2.0 * ((x_m / z) / y);
	elseif (t <= 8.5e-168)
		tmp = t_1;
	elseif (t <= 13500000.0)
		tmp = (x_m * 2.0) / (z * y);
	else
		tmp = -2.0 / (z * (t / x_m));
	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 / t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -1.08e+20], t$95$1, If[LessEqual[t, 4e-185], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.5e-168], t$95$1, If[LessEqual[t, 13500000.0], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z * y), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(z * N[(t / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if t < -1.08e20 or 4e-185 < t < 8.4999999999999994e-168

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around 0 79.9%

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

      1. associate-*r* 79.9%

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

      2. neg-mul-1 79.9%

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

      3. *-commutative 79.9%

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

   7. Simplified 79.9%

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

      1. distribute-rgt-neg-out 79.9%

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

      2. distribute-frac-neg2 79.9%

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

      3. distribute-frac-neg 79.9%

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

      4. *-commutative 79.9%

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

      5. distribute-lft-neg-in 79.9%

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

      6. metadata-eval 79.9%

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

      7. times-frac 82.0%

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

   9. Applied egg-rr 82.0%

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

   if -1.08e20 < t < 4e-185

   1. Initial program 93.0%

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

      1. distribute-rgt-out-- 94.1%

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

   3. Simplified 94.1%

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

   5. Taylor expanded in y around inf 76.9%

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

      1. distribute-lft-out 76.9%

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

      2. associate-/l* 76.9%

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

      3. times-frac 80.1%

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

      4. distribute-rgt1-in 83.6%

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

   7. Simplified 83.6%

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

   8. Taylor expanded in t around 0 84.6%

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

   if 8.4999999999999994e-168 < t < 1.35e7

   1. Initial program 98.0%

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

      1. distribute-rgt-out-- 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in y around inf 87.1%

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

      1. *-commutative 87.1%

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

   7. Simplified 87.1%

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

   if 1.35e7 < t

   1. Initial program 83.8%

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

      1. distribute-rgt-out-- 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in y around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. neg-mul-1 74.8%

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

      3. *-commutative 74.8%

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

   7. Simplified 74.8%

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

      1. distribute-rgt-neg-out 74.8%

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

      2. distribute-frac-neg2 74.8%

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

      3. distribute-frac-neg 74.8%

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

      4. *-commutative 74.8%

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

      5. distribute-lft-neg-in 74.8%

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

      6. metadata-eval 74.8%

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

      7. times-frac 79.7%

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

   9. Applied egg-rr 79.7%

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

       1. *-commutative 79.7%

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

       2. clear-num 80.7%

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

       3. frac-times 80.6%

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

       4. metadata-eval 80.6%

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

   11. Applied egg-rr 80.6%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{+20}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq 13500000:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{z \cdot \frac{t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := 2 \cdot \frac{\frac{x\_m}{z}}{y}\\ t_2 := \frac{-2}{z} \cdot \frac{x\_m}{t}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-185}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-168}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 80000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{z \cdot \frac{t}{x\_m}}\\ \end{array} \end{array} \end{array} \]

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 (/ (/ x_m z) y))) (t_2 (* (/ -2.0 z) (/ x_m t))))
   (*
    x_s
    (if (<= t -2.4e+18)
      t_2
      (if (<= t 4e-185)
        t_1
        (if (<= t 8.5e-168)
          t_2
          (if (<= t 80000000.0) t_1 (/ -2.0 (* z (/ t x_m))))))))))

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 * ((x_m / z) / y);
	double t_2 = (-2.0 / z) * (x_m / t);
	double tmp;
	if (t <= -2.4e+18) {
		tmp = t_2;
	} else if (t <= 4e-185) {
		tmp = t_1;
	} else if (t <= 8.5e-168) {
		tmp = t_2;
	} else if (t <= 80000000.0) {
		tmp = t_1;
	} else {
		tmp = -2.0 / (z * (t / x_m));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = 2.0d0 * ((x_m / z) / y)
    t_2 = ((-2.0d0) / z) * (x_m / t)
    if (t <= (-2.4d+18)) then
        tmp = t_2
    else if (t <= 4d-185) then
        tmp = t_1
    else if (t <= 8.5d-168) then
        tmp = t_2
    else if (t <= 80000000.0d0) then
        tmp = t_1
    else
        tmp = (-2.0d0) / (z * (t / x_m))
    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 * ((x_m / z) / y);
	double t_2 = (-2.0 / z) * (x_m / t);
	double tmp;
	if (t <= -2.4e+18) {
		tmp = t_2;
	} else if (t <= 4e-185) {
		tmp = t_1;
	} else if (t <= 8.5e-168) {
		tmp = t_2;
	} else if (t <= 80000000.0) {
		tmp = t_1;
	} else {
		tmp = -2.0 / (z * (t / x_m));
	}
	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 * ((x_m / z) / y)
	t_2 = (-2.0 / z) * (x_m / t)
	tmp = 0
	if t <= -2.4e+18:
		tmp = t_2
	elif t <= 4e-185:
		tmp = t_1
	elif t <= 8.5e-168:
		tmp = t_2
	elif t <= 80000000.0:
		tmp = t_1
	else:
		tmp = -2.0 / (z * (t / x_m))
	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(2.0 * Float64(Float64(x_m / z) / y))
	t_2 = Float64(Float64(-2.0 / z) * Float64(x_m / t))
	tmp = 0.0
	if (t <= -2.4e+18)
		tmp = t_2;
	elseif (t <= 4e-185)
		tmp = t_1;
	elseif (t <= 8.5e-168)
		tmp = t_2;
	elseif (t <= 80000000.0)
		tmp = t_1;
	else
		tmp = Float64(-2.0 / Float64(z * Float64(t / x_m)));
	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 * ((x_m / z) / y);
	t_2 = (-2.0 / z) * (x_m / t);
	tmp = 0.0;
	if (t <= -2.4e+18)
		tmp = t_2;
	elseif (t <= 4e-185)
		tmp = t_1;
	elseif (t <= 8.5e-168)
		tmp = t_2;
	elseif (t <= 80000000.0)
		tmp = t_1;
	else
		tmp = -2.0 / (z * (t / x_m));
	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[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t, -2.4e+18], t$95$2, If[LessEqual[t, 4e-185], t$95$1, If[LessEqual[t, 8.5e-168], t$95$2, If[LessEqual[t, 80000000.0], t$95$1, N[(-2.0 / N[(z * N[(t / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.4e18 or 4e-185 < t < 8.4999999999999994e-168

   1. Initial program 89.9%

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

      1. distribute-rgt-out-- 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around 0 79.9%

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

      1. associate-*r* 79.9%

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

      2. neg-mul-1 79.9%

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

      3. *-commutative 79.9%

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

   7. Simplified 79.9%

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

      1. distribute-rgt-neg-out 79.9%

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

      2. distribute-frac-neg2 79.9%

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

      3. distribute-frac-neg 79.9%

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

      4. *-commutative 79.9%

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

      5. distribute-lft-neg-in 79.9%

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

      6. metadata-eval 79.9%

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

      7. times-frac 82.0%

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

   9. Applied egg-rr 82.0%

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

   if -2.4e18 < t < 4e-185 or 8.4999999999999994e-168 < t < 8e7

   1. Initial program 94.3%

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

      1. distribute-rgt-out-- 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in y around inf 74.5%

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

      1. distribute-lft-out 74.5%

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

      2. associate-/l* 74.5%

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

      3. times-frac 76.7%

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

      4. distribute-rgt1-in 79.5%

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

   7. Simplified 79.5%

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

   8. Taylor expanded in t around 0 82.0%

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

   if 8e7 < t

   1. Initial program 83.8%

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

      1. distribute-rgt-out-- 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in y around 0 74.8%

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

      1. associate-*r* 74.8%

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

      2. neg-mul-1 74.8%

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

      3. *-commutative 74.8%

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

   7. Simplified 74.8%

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

      1. distribute-rgt-neg-out 74.8%

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

      2. distribute-frac-neg2 74.8%

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

      3. distribute-frac-neg 74.8%

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

      4. *-commutative 74.8%

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

      5. distribute-lft-neg-in 74.8%

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

      6. metadata-eval 74.8%

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

      7. times-frac 79.7%

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

   9. Applied egg-rr 79.7%

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

       1. *-commutative 79.7%

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

       2. clear-num 80.7%

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

       3. frac-times 80.6%

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

       4. metadata-eval 80.6%

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

   11. Applied egg-rr 80.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.4 \cdot 10^{+18}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-185}:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-168}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq 80000000:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{z \cdot \frac{t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+27} \lor \neg \left(y \leq 3.6 \cdot 10^{+29}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x\_m}{t}\\ \end{array} \end{array} \]

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 (or (<= y -3.6e+27) (not (<= y 3.6e+29)))
    (* 2.0 (/ (/ x_m z) y))
    (* (/ -2.0 z) (/ x_m 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 ((y <= -3.6e+27) || !(y <= 3.6e+29)) {
		tmp = 2.0 * ((x_m / z) / y);
	} else {
		tmp = (-2.0 / z) * (x_m / 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 ((y <= (-3.6d+27)) .or. (.not. (y <= 3.6d+29))) then
        tmp = 2.0d0 * ((x_m / z) / y)
    else
        tmp = ((-2.0d0) / z) * (x_m / 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 ((y <= -3.6e+27) || !(y <= 3.6e+29)) {
		tmp = 2.0 * ((x_m / z) / y);
	} else {
		tmp = (-2.0 / z) * (x_m / 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 (y <= -3.6e+27) or not (y <= 3.6e+29):
		tmp = 2.0 * ((x_m / z) / y)
	else:
		tmp = (-2.0 / z) * (x_m / 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 ((y <= -3.6e+27) || !(y <= 3.6e+29))
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / y));
	else
		tmp = Float64(Float64(-2.0 / z) * Float64(x_m / 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 ((y <= -3.6e+27) || ~((y <= 3.6e+29)))
		tmp = 2.0 * ((x_m / z) / y);
	else
		tmp = (-2.0 / z) * (x_m / 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[Or[LessEqual[y, -3.6e+27], N[Not[LessEqual[y, 3.6e+29]], $MachinePrecision]], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -3.59999999999999983e27 or 3.59999999999999976e29 < y

   1. Initial program 87.4%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 74.9%

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

      1. distribute-lft-out 74.9%

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

      2. associate-/l* 74.8%

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

      3. times-frac 77.2%

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

      4. distribute-rgt1-in 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in t around 0 79.2%

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

   if -3.59999999999999983e27 < y < 3.59999999999999976e29

   1. Initial program 92.6%

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

      1. distribute-rgt-out-- 93.3%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in y around 0 73.4%

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

      1. associate-*r* 73.4%

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

      2. neg-mul-1 73.4%

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

      3. *-commutative 73.4%

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

   7. Simplified 73.4%

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

      1. distribute-rgt-neg-out 73.4%

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

      2. distribute-frac-neg2 73.4%

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

      3. distribute-frac-neg 73.4%

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

      4. *-commutative 73.4%

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

      5. distribute-lft-neg-in 73.4%

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

      6. metadata-eval 73.4%

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

      7. times-frac 77.5%

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

   9. Applied egg-rr 77.5%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+27} \lor \neg \left(y \leq 3.6 \cdot 10^{+29}\right):\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+27}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x\_m}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \end{array} \end{array} \]

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 (<= y -9.2e+27)
    (* (/ x_m z) (/ 2.0 y))
    (if (<= y 2.2e+27) (* (/ -2.0 z) (/ x_m t)) (* 2.0 (/ (/ x_m z) y))))))

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 (y <= -9.2e+27) {
		tmp = (x_m / z) * (2.0 / y);
	} else if (y <= 2.2e+27) {
		tmp = (-2.0 / z) * (x_m / t);
	} else {
		tmp = 2.0 * ((x_m / z) / y);
	}
	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 (y <= (-9.2d+27)) then
        tmp = (x_m / z) * (2.0d0 / y)
    else if (y <= 2.2d+27) then
        tmp = ((-2.0d0) / z) * (x_m / t)
    else
        tmp = 2.0d0 * ((x_m / z) / y)
    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 (y <= -9.2e+27) {
		tmp = (x_m / z) * (2.0 / y);
	} else if (y <= 2.2e+27) {
		tmp = (-2.0 / z) * (x_m / t);
	} else {
		tmp = 2.0 * ((x_m / z) / y);
	}
	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 y <= -9.2e+27:
		tmp = (x_m / z) * (2.0 / y)
	elif y <= 2.2e+27:
		tmp = (-2.0 / z) * (x_m / t)
	else:
		tmp = 2.0 * ((x_m / z) / y)
	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 (y <= -9.2e+27)
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / y));
	elseif (y <= 2.2e+27)
		tmp = Float64(Float64(-2.0 / z) * Float64(x_m / t));
	else
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / y));
	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 (y <= -9.2e+27)
		tmp = (x_m / z) * (2.0 / y);
	elseif (y <= 2.2e+27)
		tmp = (-2.0 / z) * (x_m / t);
	else
		tmp = 2.0 * ((x_m / z) / y);
	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[y, -9.2e+27], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+27], N[(N[(-2.0 / z), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -9.2000000000000002e27

   1. Initial program 91.7%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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 81.1%

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

      1. associate-*r/ 81.1%

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

      2. *-commutative 81.1%

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

      3. *-commutative 81.1%

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

      4. times-frac 76.5%

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

   7. Simplified 76.5%

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

   if -9.2000000000000002e27 < y < 2.1999999999999999e27

   1. Initial program 92.6%

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

      1. distribute-rgt-out-- 93.3%

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

   3. Simplified 93.3%

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

   5. Taylor expanded in y around 0 73.4%

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

      1. associate-*r* 73.4%

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

      2. neg-mul-1 73.4%

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

      3. *-commutative 73.4%

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

   7. Simplified 73.4%

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

      1. distribute-rgt-neg-out 73.4%

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

      2. distribute-frac-neg2 73.4%

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

      3. distribute-frac-neg 73.4%

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

      4. *-commutative 73.4%

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

      5. distribute-lft-neg-in 73.4%

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

      6. metadata-eval 73.4%

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

      7. times-frac 77.5%

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

   9. Applied egg-rr 77.5%

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

   if 2.1999999999999999e27 < y

   1. Initial program 83.2%

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

      1. distribute-rgt-out-- 86.7%

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

   3. Simplified 86.7%

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

   5. Taylor expanded in y around inf 75.2%

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

      1. distribute-lft-out 75.2%

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

      2. associate-/l* 75.2%

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

      3. times-frac 78.6%

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

      4. distribute-rgt1-in 80.1%

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

   7. Simplified 80.1%

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

   8. Taylor expanded in t around 0 81.9%

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

Alternative 6: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+18}:\\ \;\;\;\;x\_m \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;t \leq 24000000:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

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 -4.8e+18)
    (* x_m (/ -2.0 (* z t)))
    (if (<= t 24000000.0) (* 2.0 (/ (/ x_m z) 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 <= -4.8e+18) {
		tmp = x_m * (-2.0 / (z * t));
	} else if (t <= 24000000.0) {
		tmp = 2.0 * ((x_m / z) / 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 <= (-4.8d+18)) then
        tmp = x_m * ((-2.0d0) / (z * t))
    else if (t <= 24000000.0d0) then
        tmp = 2.0d0 * ((x_m / z) / 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 <= -4.8e+18) {
		tmp = x_m * (-2.0 / (z * t));
	} else if (t <= 24000000.0) {
		tmp = 2.0 * ((x_m / z) / 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 <= -4.8e+18:
		tmp = x_m * (-2.0 / (z * t))
	elif t <= 24000000.0:
		tmp = 2.0 * ((x_m / z) / 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 <= -4.8e+18)
		tmp = Float64(x_m * Float64(-2.0 / Float64(z * t)));
	elseif (t <= 24000000.0)
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / 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 <= -4.8e+18)
		tmp = x_m * (-2.0 / (z * t));
	elseif (t <= 24000000.0)
		tmp = 2.0 * ((x_m / z) / 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, -4.8e+18], N[(x$95$m * N[(-2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 24000000.0], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -4.8e18

   1. Initial program 90.6%

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

      1. distribute-rgt-out-- 90.6%

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

   3. Simplified 90.6%

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

      1. *-un-lft-identity 90.6%

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

      2. *-commutative 90.6%

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

      3. times-frac 90.5%

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

      4. associate-/l* 90.5%

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

   6. Applied egg-rr 90.5%

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

   7. Taylor expanded in y around 0 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-*l/ 79.4%

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

      3. associate-/l* 79.4%

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

   9. Simplified 79.4%

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

   if -4.8e18 < t < 2.4e7

   1. Initial program 93.8%

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

      1. distribute-rgt-out-- 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in y around inf 71.0%

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

      1. distribute-lft-out 71.0%

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

      2. associate-/l* 71.0%

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

      3. times-frac 73.1%

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

      4. distribute-rgt1-in 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in t around 0 79.0%

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

   if 2.4e7 < t

   1. Initial program 83.8%

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

      1. distribute-rgt-out-- 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in y around 0 74.8%

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

      1. *-commutative 74.8%

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

      2. associate-/r* 77.5%

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

   7. Simplified 77.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{elif}\;t \leq 24000000:\\ \;\;\;\;2 \cdot \frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{+20}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \mathbf{elif}\;t \leq 370000000:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

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 -1.08e+20)
    (* -2.0 (/ x_m (* z t)))
    (if (<= t 370000000.0) (* 2.0 (/ (/ x_m z) 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 <= -1.08e+20) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (t <= 370000000.0) {
		tmp = 2.0 * ((x_m / z) / 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 <= (-1.08d+20)) then
        tmp = (-2.0d0) * (x_m / (z * t))
    else if (t <= 370000000.0d0) then
        tmp = 2.0d0 * ((x_m / z) / 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 <= -1.08e+20) {
		tmp = -2.0 * (x_m / (z * t));
	} else if (t <= 370000000.0) {
		tmp = 2.0 * ((x_m / z) / 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 <= -1.08e+20:
		tmp = -2.0 * (x_m / (z * t))
	elif t <= 370000000.0:
		tmp = 2.0 * ((x_m / z) / 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 <= -1.08e+20)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	elseif (t <= 370000000.0)
		tmp = Float64(2.0 * Float64(Float64(x_m / z) / 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 <= -1.08e+20)
		tmp = -2.0 * (x_m / (z * t));
	elseif (t <= 370000000.0)
		tmp = 2.0 * ((x_m / z) / 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, -1.08e+20], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 370000000.0], N[(2.0 * N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.08e20

   1. Initial program 90.6%

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

      1. distribute-rgt-out-- 90.6%

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

   3. Simplified 90.6%

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

   5. Taylor expanded in y around 0 79.3%

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

      1. *-commutative 79.3%

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

   7. Simplified 79.3%

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

   if -1.08e20 < t < 3.7e8

   1. Initial program 93.8%

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

      1. distribute-rgt-out-- 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in y around inf 71.0%

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

      1. distribute-lft-out 71.0%

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

      2. associate-/l* 71.0%

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

      3. times-frac 73.1%

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

      4. distribute-rgt1-in 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in t around 0 79.0%

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

   if 3.7e8 < t

   1. Initial program 83.8%

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

      1. distribute-rgt-out-- 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in y around 0 74.8%

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

      1. *-commutative 74.8%

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

      2. associate-/r* 77.5%

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

   7. Simplified 77.5%

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

Alternative 8: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 10^{-79}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

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) 1e-79)
    (* x_m (/ (/ 2.0 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) <= 1e-79) {
		tmp = x_m * ((2.0 / 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) <= 1d-79) then
        tmp = x_m * ((2.0d0 / 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) <= 1e-79) {
		tmp = x_m * ((2.0 / 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) <= 1e-79:
		tmp = x_m * ((2.0 / 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) <= 1e-79)
		tmp = Float64(x_m * Float64(Float64(2.0 / 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) <= 1e-79)
		tmp = x_m * ((2.0 / 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], 1e-79], N[(x$95$m * N[(N[(2.0 / z), $MachinePrecision] / 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)) < 1e-79

   1. Initial program 92.0%

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

      1. distribute-rgt-out-- 93.2%

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

   3. Simplified 93.2%

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

      1. *-un-lft-identity 93.2%

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

      2. *-commutative 93.2%

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

      3. times-frac 91.2%

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

      4. associate-/l* 91.3%

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

   6. Applied egg-rr 91.3%

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

      1. associate-*l/ 91.2%

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

      2. *-un-lft-identity 91.2%

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

      3. frac-2neg 91.2%

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

      4. clear-num 91.2%

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

      5. un-div-inv 91.3%

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

      6. div-inv 91.3%

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

      7. metadata-eval 91.3%

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

      8. sub-neg 91.3%

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

      9. distribute-neg-in 91.3%

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

      10. add-sqr-sqrt 40.0%

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

      11. sqrt-unprod 68.2%

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

      12. sqr-neg 68.2%

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

      13. sqrt-unprod 34.8%

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

      14. add-sqr-sqrt 61.8%

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

      15. add-sqr-sqrt 27.0%

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

      16. sqrt-unprod 70.8%

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

      17. sqr-neg 70.8%

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

      18. sqrt-unprod 51.1%

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

      19. add-sqr-sqrt 91.3%

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

   8. Applied egg-rr 91.3%

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

      1. distribute-frac-neg 91.3%

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

      2. distribute-neg-frac2 91.3%

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

      3. distribute-neg-in 91.3%

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

      4. remove-double-neg 91.3%

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

      5. sub-neg 91.3%

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

   10. Simplified 91.3%

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot 0.5}}{y - t}} \]
   11. Step-by-step derivation

       1. div-inv 91.2%

          \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{z \cdot 0.5}}}{y - t} \]

       2. associate-/l* 92.8%

          \[\leadsto \color{blue}{x \cdot \frac{\frac{1}{z \cdot 0.5}}{y - t}} \]

       3. *-commutative 92.8%

          \[\leadsto x \cdot \frac{\frac{1}{\color{blue}{0.5 \cdot z}}}{y - t} \]

       4. associate-/r* 92.8%

          \[\leadsto x \cdot \frac{\color{blue}{\frac{\frac{1}{0.5}}{z}}}{y - t} \]

       5. metadata-eval 92.8%

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

   12. Applied egg-rr 92.8%

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

   if 1e-79 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 86.0%

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

      1. distribute-rgt-out-- 87.4%

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

   3. Simplified 87.4%

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

      1. *-commutative 87.4%

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

      2. times-frac 97.3%

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

   6. Applied egg-rr 97.3%

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

Alternative 9: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-21}:\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{z}}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y - t}\\ \end{array} \end{array} \]

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 (<= z 7.5e-21)
    (* x_m (/ (/ 2.0 z) (- y t)))
    (* (/ 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) {
	double tmp;
	if (z <= 7.5e-21) {
		tmp = x_m * ((2.0 / z) / (y - t));
	} else {
		tmp = (x_m / z) * (2.0 / (y - 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 (z <= 7.5d-21) then
        tmp = x_m * ((2.0d0 / z) / (y - t))
    else
        tmp = (x_m / z) * (2.0d0 / (y - 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 (z <= 7.5e-21) {
		tmp = x_m * ((2.0 / z) / (y - t));
	} else {
		tmp = (x_m / z) * (2.0 / (y - 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 z <= 7.5e-21:
		tmp = x_m * ((2.0 / z) / (y - t))
	else:
		tmp = (x_m / z) * (2.0 / (y - 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 (z <= 7.5e-21)
		tmp = Float64(x_m * Float64(Float64(2.0 / z) / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / Float64(y - 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 (z <= 7.5e-21)
		tmp = x_m * ((2.0 / z) / (y - t));
	else
		tmp = (x_m / z) * (2.0 / (y - 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[z, 7.5e-21], N[(x$95$m * N[(N[(2.0 / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.50000000000000072e-21

   1. Initial program 90.0%

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

      1. distribute-rgt-out-- 91.6%

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

   3. Simplified 91.6%

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

      1. *-un-lft-identity 91.6%

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

      2. *-commutative 91.6%

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

      3. times-frac 89.9%

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

      4. associate-/l* 90.0%

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

   6. Applied egg-rr 90.0%

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

      1. associate-*l/ 89.9%

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

      2. *-un-lft-identity 89.9%

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

      3. frac-2neg 89.9%

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

      4. clear-num 89.9%

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

      5. un-div-inv 90.1%

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

      6. div-inv 90.1%

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

      7. metadata-eval 90.1%

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

      8. sub-neg 90.1%

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

      9. distribute-neg-in 90.1%

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

      10. add-sqr-sqrt 41.1%

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

      11. sqrt-unprod 66.7%

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

      12. sqr-neg 66.7%

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

      13. sqrt-unprod 30.5%

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

      14. add-sqr-sqrt 58.1%

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

      15. add-sqr-sqrt 27.6%

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

      16. sqrt-unprod 67.7%

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

      17. sqr-neg 67.7%

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

      18. sqrt-unprod 48.9%

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

      19. add-sqr-sqrt 90.1%

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

   8. Applied egg-rr 90.1%

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

      1. distribute-frac-neg 90.1%

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

      2. distribute-neg-frac2 90.1%

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

      3. distribute-neg-in 90.1%

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

      4. remove-double-neg 90.1%

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

      5. sub-neg 90.1%

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

   10. Simplified 90.1%

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot 0.5}}{y - t}} \]
   11. Step-by-step derivation

       1. div-inv 89.9%

          \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{z \cdot 0.5}}}{y - t} \]

       2. associate-/l* 91.3%

          \[\leadsto \color{blue}{x \cdot \frac{\frac{1}{z \cdot 0.5}}{y - t}} \]

       3. *-commutative 91.3%

          \[\leadsto x \cdot \frac{\frac{1}{\color{blue}{0.5 \cdot z}}}{y - t} \]

       4. associate-/r* 91.3%

          \[\leadsto x \cdot \frac{\color{blue}{\frac{\frac{1}{0.5}}{z}}}{y - t} \]

       5. metadata-eval 91.3%

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

   12. Applied egg-rr 91.3%

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

   if 7.50000000000000072e-21 < z

   1. Initial program 90.1%

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

      1. distribute-rgt-out-- 90.3%

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

   3. Simplified 90.3%

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

      1. times-frac 98.1%

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

   6. Applied egg-rr 98.1%

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

Alternative 10: 54.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2000000000:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

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 (<= z 2000000000.0) (* -2.0 (/ x_m (* z t))) (* -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 (z <= 2000000000.0) {
		tmp = -2.0 * (x_m / (z * t));
	} 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 (z <= 2000000000.0d0) then
        tmp = (-2.0d0) * (x_m / (z * t))
    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 (z <= 2000000000.0) {
		tmp = -2.0 * (x_m / (z * t));
	} 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 z <= 2000000000.0:
		tmp = -2.0 * (x_m / (z * t))
	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 (z <= 2000000000.0)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	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 (z <= 2000000000.0)
		tmp = -2.0 * (x_m / (z * t));
	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[z, 2000000000.0], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2e9

   1. Initial program 90.2%

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

      1. distribute-rgt-out-- 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in y around 0 53.7%

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

      1. *-commutative 53.7%

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

   7. Simplified 53.7%

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

   if 2e9 < z

   1. Initial program 89.2%

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

      1. distribute-rgt-out-- 89.4%

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

   3. Simplified 89.4%

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

   5. Taylor expanded in y around 0 41.2%

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

      1. *-commutative 41.2%

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

      2. associate-/r* 50.9%

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

   7. Simplified 50.9%

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

Alternative 11: 91.5% accurate, 1.2× speedup

Math

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

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 (/ (/ 2.0 z) (- 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 * ((2.0 / z) / (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 * ((2.0d0 / z) / (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 * ((2.0 / z) / (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 * ((2.0 / z) / (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(x_m * Float64(Float64(2.0 / z) / 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 * ((2.0 / z) / (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[(x$95$m * N[(N[(2.0 / z), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.0%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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. *-un-lft-identity 91.3%

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

   2. *-commutative 91.3%

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

   3. times-frac 91.8%

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

   4. associate-/l* 91.8%

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

6. Applied egg-rr 91.8%

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

   1. associate-*l/ 91.8%

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

   2. *-un-lft-identity 91.8%

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

   3. frac-2neg 91.8%

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

   4. clear-num 91.8%

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

   5. un-div-inv 91.9%

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

   6. div-inv 91.9%

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

   7. metadata-eval 91.9%

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

   8. sub-neg 91.9%

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

   9. distribute-neg-in 91.9%

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

   10. add-sqr-sqrt 40.3%

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

   11. sqrt-unprod 68.9%

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

   12. sqr-neg 68.9%

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

   13. sqrt-unprod 33.3%

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

   14. add-sqr-sqrt 61.7%

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

   15. add-sqr-sqrt 28.4%

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

   16. sqrt-unprod 71.1%

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

   17. sqr-neg 71.1%

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

   18. sqrt-unprod 51.4%

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

   19. add-sqr-sqrt 91.9%

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

8. Applied egg-rr 91.9%

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

   1. distribute-frac-neg 91.9%

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

   2. distribute-neg-frac2 91.9%

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

   3. distribute-neg-in 91.9%

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

   4. remove-double-neg 91.9%

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

   5. sub-neg 91.9%

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

10. Simplified 91.9%

    \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot 0.5}}{y - t}} \]
11. Step-by-step derivation

    1. div-inv 91.8%

       \[\leadsto \frac{\color{blue}{x \cdot \frac{1}{z \cdot 0.5}}}{y - t} \]

    2. associate-/l* 91.0%

       \[\leadsto \color{blue}{x \cdot \frac{\frac{1}{z \cdot 0.5}}{y - t}} \]

    3. *-commutative 91.0%

       \[\leadsto x \cdot \frac{\frac{1}{\color{blue}{0.5 \cdot z}}}{y - t} \]

    4. associate-/r* 91.0%

       \[\leadsto x \cdot \frac{\color{blue}{\frac{\frac{1}{0.5}}{z}}}{y - t} \]

    5. metadata-eval 91.0%

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

12. Applied egg-rr 91.0%

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

Alternative 12: 52.0% 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 90.0%

   \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
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 0 51.1%

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

   1. *-commutative 51.1%

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

7. Simplified 51.1%

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

Developer target: 97.1% 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 2024110 
(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))))