Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.9% → 96.6%

Time: 11.6s

Alternatives: 15

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: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \sqrt{x\_m \cdot 2}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot 2}{y \cdot z\_m - z\_m \cdot t} \leq -4 \cdot 10^{-304}:\\ \;\;\;\;\frac{x\_m \cdot 2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{y - t} \cdot \frac{t\_1}{z\_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (sqrt (* x_m 2.0))))
   (*
    z_s
    (*
     x_s
     (if (<= (/ (* x_m 2.0) (- (* y z_m) (* z_m t))) -4e-304)
       (/ (* x_m 2.0) (* z_m (- y t)))
       (* (/ t_1 (- y t)) (/ t_1 z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = sqrt((x_m * 2.0));
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -4e-304) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (t_1 / (y - t)) * (t_1 / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((x_m * 2.0d0))
    if (((x_m * 2.0d0) / ((y * z_m) - (z_m * t))) <= (-4d-304)) then
        tmp = (x_m * 2.0d0) / (z_m * (y - t))
    else
        tmp = (t_1 / (y - t)) * (t_1 / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double t_1 = Math.sqrt((x_m * 2.0));
	double tmp;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -4e-304) {
		tmp = (x_m * 2.0) / (z_m * (y - t));
	} else {
		tmp = (t_1 / (y - t)) * (t_1 / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	t_1 = math.sqrt((x_m * 2.0))
	tmp = 0
	if ((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -4e-304:
		tmp = (x_m * 2.0) / (z_m * (y - t))
	else:
		tmp = (t_1 / (y - t)) * (t_1 / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	t_1 = sqrt(Float64(x_m * 2.0))
	tmp = 0.0
	if (Float64(Float64(x_m * 2.0) / Float64(Float64(y * z_m) - Float64(z_m * t))) <= -4e-304)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z_m * Float64(y - t)));
	else
		tmp = Float64(Float64(t_1 / Float64(y - t)) * Float64(t_1 / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	t_1 = sqrt((x_m * 2.0));
	tmp = 0.0;
	if (((x_m * 2.0) / ((y * z_m) - (z_m * t))) <= -4e-304)
		tmp = (x_m * 2.0) / (z_m * (y - t));
	else
		tmp = (t_1 / (y - t)) * (t_1 / z_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := Block[{t$95$1 = N[Sqrt[N[(x$95$m * 2.0), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-304], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -3.99999999999999988e-304

   1. Initial program 95.0%

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

      1. distribute-rgt-out-- 97.4%

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

   3. Simplified 97.4%

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

   if -3.99999999999999988e-304 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 88.4%

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

      1. distribute-rgt-out-- 90.8%

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

   3. Simplified 90.8%

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

      1. add-sqr-sqrt 51.9%

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

      2. *-commutative 51.9%

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

      3. times-frac 55.7%

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

   6. Applied egg-rr 55.7%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -4 \cdot 10^{-304}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x \cdot 2}}{y - t} \cdot \frac{\sqrt{x \cdot 2}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+19} \lor \neg \left(y \leq 7.6 \cdot 10^{-16}\right):\\ \;\;\;\;x\_m \cdot \frac{2}{y \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (or (<= y -1.05e+19) (not (<= y 7.6e-16)))
     (* x_m (/ 2.0 (* y z_m)))
     (* -2.0 (/ x_m (* z_m t)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((y <= -1.05e+19) || !(y <= 7.6e-16)) {
		tmp = x_m * (2.0 / (y * z_m));
	} else {
		tmp = -2.0 * (x_m / (z_m * t));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-1.05d+19)) .or. (.not. (y <= 7.6d-16))) then
        tmp = x_m * (2.0d0 / (y * z_m))
    else
        tmp = (-2.0d0) * (x_m / (z_m * t))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((y <= -1.05e+19) || !(y <= 7.6e-16)) {
		tmp = x_m * (2.0 / (y * z_m));
	} else {
		tmp = -2.0 * (x_m / (z_m * t));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (y <= -1.05e+19) or not (y <= 7.6e-16):
		tmp = x_m * (2.0 / (y * z_m))
	else:
		tmp = -2.0 * (x_m / (z_m * t))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if ((y <= -1.05e+19) || !(y <= 7.6e-16))
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z_m)));
	else
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((y <= -1.05e+19) || ~((y <= 7.6e-16)))
		tmp = x_m * (2.0 / (y * z_m));
	else
		tmp = -2.0 * (x_m / (z_m * t));
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[Or[LessEqual[y, -1.05e+19], N[Not[LessEqual[y, 7.6e-16]], $MachinePrecision]], N[(x$95$m * N[(2.0 / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e19 or 7.60000000000000024e-16 < y

   1. Initial program 90.2%

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

      1. distribute-rgt-out-- 93.4%

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

   3. Simplified 93.4%

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

      1. distribute-rgt-out-- 90.2%

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

      2. associate-/l* 90.1%

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

      3. *-commutative 90.1%

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

      4. distribute-rgt-out-- 93.3%

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

   6. Applied egg-rr 93.3%

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

   7. Taylor expanded in y around inf 82.2%

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

      1. *-commutative 82.2%

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

   9. Simplified 82.2%

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

   if -1.05e19 < y < 7.60000000000000024e-16

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around 0 75.3%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+19} \lor \neg \left(y \leq 7.6 \cdot 10^{-16}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+23}:\\ \;\;\;\;x\_m \cdot \frac{2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-16}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -9e+23)
     (* x_m (/ 2.0 (* y z_m)))
     (if (<= y 4.8e-16)
       (* -2.0 (/ x_m (* z_m t)))
       (* (/ 2.0 z_m) (/ x_m y)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -9e+23) {
		tmp = x_m * (2.0 / (y * z_m));
	} else if (y <= 4.8e-16) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-9d+23)) then
        tmp = x_m * (2.0d0 / (y * z_m))
    else if (y <= 4.8d-16) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else
        tmp = (2.0d0 / z_m) * (x_m / y)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -9e+23) {
		tmp = x_m * (2.0 / (y * z_m));
	} else if (y <= 4.8e-16) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if y <= -9e+23:
		tmp = x_m * (2.0 / (y * z_m))
	elif y <= 4.8e-16:
		tmp = -2.0 * (x_m / (z_m * t))
	else:
		tmp = (2.0 / z_m) * (x_m / y)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (y <= -9e+23)
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z_m)));
	elseif (y <= 4.8e-16)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (y <= -9e+23)
		tmp = x_m * (2.0 / (y * z_m));
	elseif (y <= 4.8e-16)
		tmp = -2.0 * (x_m / (z_m * t));
	else
		tmp = (2.0 / z_m) * (x_m / y);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[y, -9e+23], N[(x$95$m * N[(2.0 / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e-16], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -8.99999999999999958e23

   1. Initial program 91.9%

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

      1. distribute-rgt-out-- 93.7%

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

   3. Simplified 93.7%

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

      1. distribute-rgt-out-- 91.9%

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

      2. associate-/l* 91.9%

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

      3. *-commutative 91.9%

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

      4. distribute-rgt-out-- 93.7%

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

   6. Applied egg-rr 93.7%

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

   7. Taylor expanded in y around inf 87.0%

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

      1. *-commutative 87.0%

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

   9. Simplified 87.0%

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

   if -8.99999999999999958e23 < y < 4.8000000000000001e-16

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around 0 75.3%

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

   if 4.8000000000000001e-16 < y

   1. Initial program 88.7%

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

      1. *-commutative 93.1%

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

      2. times-frac 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in y around inf 79.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+23}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-16}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+21}:\\ \;\;\;\;x\_m \cdot \frac{2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z\_m}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -1.85e+21)
     (* x_m (/ 2.0 (* y z_m)))
     (if (<= y 1.05e-15)
       (/ -2.0 (* t (/ z_m x_m)))
       (* (/ 2.0 z_m) (/ x_m y)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.85e+21) {
		tmp = x_m * (2.0 / (y * z_m));
	} else if (y <= 1.05e-15) {
		tmp = -2.0 / (t * (z_m / x_m));
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.85d+21)) then
        tmp = x_m * (2.0d0 / (y * z_m))
    else if (y <= 1.05d-15) then
        tmp = (-2.0d0) / (t * (z_m / x_m))
    else
        tmp = (2.0d0 / z_m) * (x_m / y)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.85e+21) {
		tmp = x_m * (2.0 / (y * z_m));
	} else if (y <= 1.05e-15) {
		tmp = -2.0 / (t * (z_m / x_m));
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if y <= -1.85e+21:
		tmp = x_m * (2.0 / (y * z_m))
	elif y <= 1.05e-15:
		tmp = -2.0 / (t * (z_m / x_m))
	else:
		tmp = (2.0 / z_m) * (x_m / y)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (y <= -1.85e+21)
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z_m)));
	elseif (y <= 1.05e-15)
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / x_m)));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (y <= -1.85e+21)
		tmp = x_m * (2.0 / (y * z_m));
	elseif (y <= 1.05e-15)
		tmp = -2.0 / (t * (z_m / x_m));
	else
		tmp = (2.0 / z_m) * (x_m / y);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[y, -1.85e+21], N[(x$95$m * N[(2.0 / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e-15], N[(-2.0 / N[(t * N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.85e21

   1. Initial program 91.9%

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

      1. distribute-rgt-out-- 93.7%

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

   3. Simplified 93.7%

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

      1. distribute-rgt-out-- 91.9%

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

      2. associate-/l* 91.9%

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

      3. *-commutative 91.9%

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

      4. distribute-rgt-out-- 93.7%

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

   6. Applied egg-rr 93.7%

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

   7. Taylor expanded in y around inf 87.0%

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

      1. *-commutative 87.0%

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

   9. Simplified 87.0%

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

   if -1.85e21 < y < 1.0499999999999999e-15

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around 0 75.3%

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

      1. clear-num 74.4%

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

      2. un-div-inv 74.4%

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

      3. associate-/l* 75.8%

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

   7. Applied egg-rr 75.8%

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

   if 1.0499999999999999e-15 < y

   1. Initial program 88.7%

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

      1. *-commutative 93.1%

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

      2. times-frac 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in y around inf 79.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-15}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+26}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z\_m}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -1.1e+26)
     (/ (* x_m 2.0) (* y z_m))
     (if (<= y 3.5e-16)
       (/ -2.0 (* t (/ z_m x_m)))
       (* (/ 2.0 z_m) (/ x_m y)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.1e+26) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 3.5e-16) {
		tmp = -2.0 / (t * (z_m / x_m));
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.1d+26)) then
        tmp = (x_m * 2.0d0) / (y * z_m)
    else if (y <= 3.5d-16) then
        tmp = (-2.0d0) / (t * (z_m / x_m))
    else
        tmp = (2.0d0 / z_m) * (x_m / y)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.1e+26) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 3.5e-16) {
		tmp = -2.0 / (t * (z_m / x_m));
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if y <= -1.1e+26:
		tmp = (x_m * 2.0) / (y * z_m)
	elif y <= 3.5e-16:
		tmp = -2.0 / (t * (z_m / x_m))
	else:
		tmp = (2.0 / z_m) * (x_m / y)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (y <= -1.1e+26)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	elseif (y <= 3.5e-16)
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / x_m)));
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (y <= -1.1e+26)
		tmp = (x_m * 2.0) / (y * z_m);
	elseif (y <= 3.5e-16)
		tmp = -2.0 / (t * (z_m / x_m));
	else
		tmp = (2.0 / z_m) * (x_m / y);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[y, -1.1e+26], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e-16], N[(-2.0 / N[(t * N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.10000000000000004e26

   1. Initial program 91.9%

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

      1. distribute-rgt-out-- 93.7%

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

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   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 -1.10000000000000004e26 < y < 3.50000000000000017e-16

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around 0 75.3%

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

      1. clear-num 74.4%

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

      2. un-div-inv 74.4%

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

      3. associate-/l* 75.8%

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

   7. Applied egg-rr 75.8%

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

   if 3.50000000000000017e-16 < y

   1. Initial program 88.7%

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

      1. *-commutative 93.1%

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

      2. times-frac 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in y around inf 79.7%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+26}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+19}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 10^{-15}:\\ \;\;\;\;\frac{x\_m \cdot \frac{-2}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -1.7e+19)
     (/ (* x_m 2.0) (* y z_m))
     (if (<= y 1e-15) (/ (* x_m (/ -2.0 z_m)) t) (* (/ 2.0 z_m) (/ x_m y)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.7e+19) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 1e-15) {
		tmp = (x_m * (-2.0 / z_m)) / t;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.7d+19)) then
        tmp = (x_m * 2.0d0) / (y * z_m)
    else if (y <= 1d-15) then
        tmp = (x_m * ((-2.0d0) / z_m)) / t
    else
        tmp = (2.0d0 / z_m) * (x_m / y)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -1.7e+19) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 1e-15) {
		tmp = (x_m * (-2.0 / z_m)) / t;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if y <= -1.7e+19:
		tmp = (x_m * 2.0) / (y * z_m)
	elif y <= 1e-15:
		tmp = (x_m * (-2.0 / z_m)) / t
	else:
		tmp = (2.0 / z_m) * (x_m / y)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (y <= -1.7e+19)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	elseif (y <= 1e-15)
		tmp = Float64(Float64(x_m * Float64(-2.0 / z_m)) / t);
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (y <= -1.7e+19)
		tmp = (x_m * 2.0) / (y * z_m);
	elseif (y <= 1e-15)
		tmp = (x_m * (-2.0 / z_m)) / t;
	else
		tmp = (2.0 / z_m) * (x_m / y);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[y, -1.7e+19], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e-15], N[(N[(x$95$m * N[(-2.0 / z$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.7e19

   1. Initial program 91.9%

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

      1. distribute-rgt-out-- 93.7%

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

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   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 -1.7e19 < y < 1.0000000000000001e-15

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around 0 75.3%

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

      1. associate-*r/ 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-*l/ 75.3%

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

      4. associate-/r* 75.3%

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

      5. associate-*l/ 77.2%

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

   7. Applied egg-rr 77.2%

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

   if 1.0000000000000001e-15 < y

   1. Initial program 88.7%

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

      1. *-commutative 93.1%

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

      2. times-frac 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in y around inf 79.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 10^{-15}:\\ \;\;\;\;\frac{x \cdot \frac{-2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+23}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z\_m} \cdot \frac{x\_m}{y}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -6e+23)
     (/ (* x_m 2.0) (* y z_m))
     (if (<= y 7.8e-16)
       (/ (/ (* x_m -2.0) z_m) t)
       (* (/ 2.0 z_m) (/ x_m y)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -6e+23) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 7.8e-16) {
		tmp = ((x_m * -2.0) / z_m) / t;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-6d+23)) then
        tmp = (x_m * 2.0d0) / (y * z_m)
    else if (y <= 7.8d-16) then
        tmp = ((x_m * (-2.0d0)) / z_m) / t
    else
        tmp = (2.0d0 / z_m) * (x_m / y)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -6e+23) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 7.8e-16) {
		tmp = ((x_m * -2.0) / z_m) / t;
	} else {
		tmp = (2.0 / z_m) * (x_m / y);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if y <= -6e+23:
		tmp = (x_m * 2.0) / (y * z_m)
	elif y <= 7.8e-16:
		tmp = ((x_m * -2.0) / z_m) / t
	else:
		tmp = (2.0 / z_m) * (x_m / y)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (y <= -6e+23)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	elseif (y <= 7.8e-16)
		tmp = Float64(Float64(Float64(x_m * -2.0) / z_m) / t);
	else
		tmp = Float64(Float64(2.0 / z_m) * Float64(x_m / y));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (y <= -6e+23)
		tmp = (x_m * 2.0) / (y * z_m);
	elseif (y <= 7.8e-16)
		tmp = ((x_m * -2.0) / z_m) / t;
	else
		tmp = (2.0 / z_m) * (x_m / y);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[y, -6e+23], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.8e-16], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / z$95$m), $MachinePrecision] / t), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -6.0000000000000002e23

   1. Initial program 91.9%

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

      1. distribute-rgt-out-- 93.7%

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

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   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 -6.0000000000000002e23 < y < 7.79999999999999954e-16

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around 0 75.3%

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

      1. associate-*r/ 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-/r* 77.2%

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

      4. *-commutative 77.2%

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

   7. Applied egg-rr 77.2%

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

   if 7.79999999999999954e-16 < y

   1. Initial program 88.7%

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

      1. *-commutative 93.1%

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

      2. times-frac 92.9%

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

   6. Applied egg-rr 92.9%

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

   7. Taylor expanded in y around inf 79.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+23}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{z\_m}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot 2}{y}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -7.5e+18)
     (/ (* x_m 2.0) (* y z_m))
     (if (<= y 6e-16) (/ (/ (* x_m -2.0) z_m) t) (/ (/ (* x_m 2.0) y) z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -7.5e+18) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 6e-16) {
		tmp = ((x_m * -2.0) / z_m) / t;
	} else {
		tmp = ((x_m * 2.0) / y) / z_m;
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-7.5d+18)) then
        tmp = (x_m * 2.0d0) / (y * z_m)
    else if (y <= 6d-16) then
        tmp = ((x_m * (-2.0d0)) / z_m) / t
    else
        tmp = ((x_m * 2.0d0) / y) / z_m
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (y <= -7.5e+18) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else if (y <= 6e-16) {
		tmp = ((x_m * -2.0) / z_m) / t;
	} else {
		tmp = ((x_m * 2.0) / y) / z_m;
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if y <= -7.5e+18:
		tmp = (x_m * 2.0) / (y * z_m)
	elif y <= 6e-16:
		tmp = ((x_m * -2.0) / z_m) / t
	else:
		tmp = ((x_m * 2.0) / y) / z_m
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (y <= -7.5e+18)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	elseif (y <= 6e-16)
		tmp = Float64(Float64(Float64(x_m * -2.0) / z_m) / t);
	else
		tmp = Float64(Float64(Float64(x_m * 2.0) / y) / z_m);
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (y <= -7.5e+18)
		tmp = (x_m * 2.0) / (y * z_m);
	elseif (y <= 6e-16)
		tmp = ((x_m * -2.0) / z_m) / t;
	else
		tmp = ((x_m * 2.0) / y) / z_m;
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[y, -7.5e+18], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e-16], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / z$95$m), $MachinePrecision] / t), $MachinePrecision], N[(N[(N[(x$95$m * 2.0), $MachinePrecision] / y), $MachinePrecision] / z$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -7.5e18

   1. Initial program 91.9%

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

      1. distribute-rgt-out-- 93.7%

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

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   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 -7.5e18 < y < 5.99999999999999987e-16

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in y around 0 75.3%

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

      1. associate-*r/ 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-/r* 77.2%

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

      4. *-commutative 77.2%

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

   7. Applied egg-rr 77.2%

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

   if 5.99999999999999987e-16 < y

   1. Initial program 88.7%

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

      1. distribute-rgt-out-- 88.7%

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

      2. associate-/l* 88.6%

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

      3. *-commutative 88.6%

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

      4. distribute-rgt-out-- 93.0%

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

   6. Applied egg-rr 93.0%

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

   7. Taylor expanded in y around inf 78.3%

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

      1. *-commutative 78.3%

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

   9. Simplified 78.3%

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

       1. *-commutative 78.3%

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

       2. associate-*r/ 78.4%

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

       3. metadata-eval 78.4%

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

       4. distribute-rgt-neg-in 78.4%

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

       5. *-commutative 78.4%

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

       6. associate-/r* 79.8%

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

       7. distribute-rgt-neg-in 79.8%

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

       8. metadata-eval 79.8%

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

   11. Applied egg-rr 79.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+18}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{y}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 5e-55)
     (* (/ x_m z_m) (/ 2.0 (- y t)))
     (* (/ x_m (- y t)) (/ 2.0 z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-55) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-55) then
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-55) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-55:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-55)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-55)
		tmp = (x_m / z_m) * (2.0 / (y - t));
	else
		tmp = (x_m / (y - t)) * (2.0 / z_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-55], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 5.0000000000000002e-55

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 94.3%

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

   3. Simplified 94.3%

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

      1. times-frac 90.8%

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

   6. Applied egg-rr 90.8%

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

   if 5.0000000000000002e-55 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 89.6%

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

      1. distribute-rgt-out-- 89.5%

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

   3. Simplified 89.5%

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

      1. *-commutative 89.5%

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

      2. times-frac 95.7%

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

   6. Applied egg-rr 95.7%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z\_m}}{\frac{y - t}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 5e-55)
     (* (/ x_m z_m) (/ 2.0 (- y t)))
     (/ (/ 2.0 z_m) (/ (- y t) x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-55) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (2.0 / z_m) / ((y - t) / x_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-55) then
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    else
        tmp = (2.0d0 / z_m) / ((y - t) / x_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-55) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (2.0 / z_m) / ((y - t) / x_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-55:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	else:
		tmp = (2.0 / z_m) / ((y - t) / x_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-55)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(2.0 / z_m) / Float64(Float64(y - t) / x_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-55)
		tmp = (x_m / z_m) * (2.0 / (y - t));
	else
		tmp = (2.0 / z_m) / ((y - t) / x_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-55], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z$95$m), $MachinePrecision] / N[(N[(y - t), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 5.0000000000000002e-55

   1. Initial program 90.9%

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

      1. distribute-rgt-out-- 94.3%

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

   3. Simplified 94.3%

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

      1. times-frac 90.8%

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

   6. Applied egg-rr 90.8%

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

   if 5.0000000000000002e-55 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 89.6%

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

      1. distribute-rgt-out-- 89.5%

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

   3. Simplified 89.5%

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

      1. add-sqr-sqrt 89.2%

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

      2. *-commutative 89.2%

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

      3. times-frac 95.6%

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

   6. Applied egg-rr 95.6%

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

      1. frac-times 89.2%

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

      2. add-sqr-sqrt 89.5%

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

      3. frac-times 95.7%

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

      4. *-commutative 95.7%

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

      5. clear-num 95.6%

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

      6. un-div-inv 95.8%

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

   8. Applied egg-rr 95.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{z}}{\frac{y - t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y - t}}{z\_m \cdot 0.5}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 5e-15)
     (* (/ x_m z_m) (/ 2.0 (- y t)))
     (/ (/ x_m (- y t)) (* z_m 0.5))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-15) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) / (z_m * 0.5);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d-15) then
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    else
        tmp = (x_m / (y - t)) / (z_m * 0.5d0)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e-15) {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) / (z_m * 0.5);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if (x_m * 2.0) <= 5e-15:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	else:
		tmp = (x_m / (y - t)) / (z_m * 0.5)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e-15)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) / Float64(z_m * 0.5));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e-15)
		tmp = (x_m / z_m) * (2.0 / (y - t));
	else
		tmp = (x_m / (y - t)) / (z_m * 0.5);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e-15], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.99999999999999999e-15

   1. Initial program 91.1%

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

      1. distribute-rgt-out-- 94.4%

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

   3. Simplified 94.4%

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

      1. times-frac 90.9%

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

   6. Applied egg-rr 90.9%

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

   if 4.99999999999999999e-15 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 89.0%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. add-sqr-sqrt 88.6%

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

      2. *-commutative 88.6%

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

      3. times-frac 95.4%

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

   6. Applied egg-rr 95.4%

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

      1. frac-times 88.6%

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

      2. add-sqr-sqrt 89.0%

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

      3. frac-times 95.5%

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

      4. clear-num 95.5%

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

      5. un-div-inv 95.7%

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

      6. div-inv 95.7%

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

      7. metadata-eval 95.7%

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

   8. Applied egg-rr 95.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{y - t}}{z \cdot 0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y - t}}{z \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.05 \cdot 10^{+22}:\\ \;\;\;\;x\_m \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 2.05e+22)
     (* x_m (/ 2.0 (* z_m (- y t))))
     (* (/ x_m z_m) (/ 2.0 (- y t)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.05e+22) {
		tmp = x_m * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 2.05d+22) then
        tmp = x_m * (2.0d0 / (z_m * (y - t)))
    else
        tmp = (x_m / z_m) * (2.0d0 / (y - t))
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 2.05e+22) {
		tmp = x_m * (2.0 / (z_m * (y - t)));
	} else {
		tmp = (x_m / z_m) * (2.0 / (y - t));
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if z_m <= 2.05e+22:
		tmp = x_m * (2.0 / (z_m * (y - t)))
	else:
		tmp = (x_m / z_m) * (2.0 / (y - t))
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (z_m <= 2.05e+22)
		tmp = Float64(x_m * Float64(2.0 / Float64(z_m * Float64(y - t))));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / Float64(y - t)));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 2.05e+22)
		tmp = x_m * (2.0 / (z_m * (y - t)));
	else
		tmp = (x_m / z_m) * (2.0 / (y - t));
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.05e+22], N[(x$95$m * N[(2.0 / N[(z$95$m * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.0499999999999999e22

   1. Initial program 91.8%

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

      1. distribute-rgt-out-- 91.8%

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

      2. associate-/l* 91.6%

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

      3. *-commutative 91.6%

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

      4. distribute-rgt-out-- 93.2%

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

   6. Applied egg-rr 93.2%

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

   if 2.0499999999999999e22 < z

   1. Initial program 86.0%

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

      1. distribute-rgt-out-- 91.5%

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

   3. Simplified 91.5%

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

      1. times-frac 97.6%

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

   6. Applied egg-rr 97.6%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.05 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{-36}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{t}}{z\_m}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 5e-36) (* -2.0 (/ x_m (* z_m t))) (* -2.0 (/ (/ x_m t) z_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 5e-36) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = -2.0 * ((x_m / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x_s, x_m, y, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 5d-36) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else
        tmp = (-2.0d0) * ((x_m / t) / z_m)
    end if
    code = z_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x_s, double x_m, double y, double z_m, double t) {
	double tmp;
	if (z_m <= 5e-36) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else {
		tmp = -2.0 * ((x_m / t) / z_m);
	}
	return z_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x_s, x_m, y, z_m, t):
	tmp = 0
	if z_m <= 5e-36:
		tmp = -2.0 * (x_m / (z_m * t))
	else:
		tmp = -2.0 * ((x_m / t) / z_m)
	return z_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0
	if (z_m <= 5e-36)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z_m));
	end
	return Float64(z_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x_s, x_m, y, z_m, t)
	tmp = 0.0;
	if (z_m <= 5e-36)
		tmp = -2.0 * (x_m / (z_m * t));
	else
		tmp = -2.0 * ((x_m / t) / z_m);
	end
	tmp_2 = z_s * (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]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x$95$s_, x$95$m_, y_, z$95$m_, t_] := N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 5e-36], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.00000000000000004e-36

   1. Initial program 91.2%

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

      1. distribute-rgt-out-- 92.8%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in y around 0 55.2%

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

   if 5.00000000000000004e-36 < z

   1. Initial program 88.7%

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

   5. Taylor expanded in y around 0 53.3%

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

      1. associate-/r* 54.8%

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

   7. Simplified 54.8%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-36}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 92.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 90.5%

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

   1. distribute-rgt-out-- 92.9%

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

3. Simplified 92.9%

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

   1. distribute-rgt-out-- 90.5%

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

   2. associate-/l* 90.4%

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

   3. *-commutative 90.4%

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

   4. distribute-rgt-out-- 92.8%

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

6. Applied egg-rr 92.8%

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

7. Final simplification 92.8%

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

Alternative 15: 52.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 90.5%

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

   1. distribute-rgt-out-- 92.9%

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

3. Simplified 92.9%

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

5. Taylor expanded in y around 0 54.7%

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

6. Final simplification 54.7%

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

Developer target: 97.3% 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 2024130 
(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))))