Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 90.0% → 95.5%

Time: 8.9s

Alternatives: 13

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x_m \cdot 2}{y \cdot z_m - z_m \cdot t} \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{x_m \cdot 2}{z_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{y - t} \cdot \frac{2}{z_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (/ (* x_m 2.0) (- (* y z_m) (* z_m t))) -5e-309)
     (/ (* x_m 2.0) (* z_m (- 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) / ((y * z_m) - (z_m * t))) <= -5e-309) {
		tmp = (x_m * 2.0) / (z_m * (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) / ((y * z_m) - (z_m * t))) <= (-5d-309)) then
        tmp = (x_m * 2.0d0) / (z_m * (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) / ((y * z_m) - (z_m * t))) <= -5e-309) {
		tmp = (x_m * 2.0) / (z_m * (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) / ((y * z_m) - (z_m * t))) <= -5e-309:
		tmp = (x_m * 2.0) / (z_m * (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(Float64(x_m * 2.0) / Float64(Float64(y * z_m) - Float64(z_m * t))) <= -5e-309)
		tmp = Float64(Float64(x_m * 2.0) / Float64(z_m * 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) / ((y * z_m) - (z_m * t))) <= -5e-309)
		tmp = (x_m * 2.0) / (z_m * (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[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-309], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(z$95$m * 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 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z))) < -4.9999999999999995e-309

   1. Initial program 97.3%

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

      1. distribute-rgt-out-- 98.5%

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

   3. Simplified 98.5%

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

   if -4.9999999999999995e-309 < (/.f64 (*.f64 x 2) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 84.2%

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

      1. distribute-rgt-out-- 86.0%

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

   3. Simplified 86.0%

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

      1. *-commutative 86.0%

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

      2. times-frac 94.4%

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

   5. Applied egg-rr 94.4%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -5 \cdot 10^{-309}:\\ \;\;\;\;\frac{x \cdot 2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]

Alternative 2: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \cdot 2 \leq 10^{-92}:\\ \;\;\;\;\frac{x_m}{z_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{y - t} \cdot \frac{2}{z_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= (* x_m 2.0) 1e-92)
     (* (/ 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) <= 1e-92) {
		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) <= 1d-92) 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) <= 1e-92) {
		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) <= 1e-92:
		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) <= 1e-92)
		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) <= 1e-92)
		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], 1e-92], 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 2) < 9.99999999999999988e-93

   1. Initial program 89.8%

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

      1. distribute-rgt-out-- 91.1%

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

      2. times-frac 94.3%

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

   3. Simplified 94.3%

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

   if 9.99999999999999988e-93 < (*.f64 x 2)

   1. Initial program 86.7%

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

      1. *-commutative 89.0%

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

      2. times-frac 97.4%

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

   5. Applied egg-rr 97.4%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot 2 \leq 10^{-92}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]

Alternative 3: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2700000 \lor \neg \left(y \leq 1.55 \cdot 10^{-44}\right):\\ \;\;\;\;x_m \cdot \frac{2}{y \cdot z_m}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x_m}{t}}{z_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (or (<= y -2700000.0) (not (<= y 1.55e-44)))
     (* x_m (/ 2.0 (* y z_m)))
     (* -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 ((y <= -2700000.0) || !(y <= 1.55e-44)) {
		tmp = x_m * (2.0 / (y * z_m));
	} 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 ((y <= (-2700000.0d0)) .or. (.not. (y <= 1.55d-44))) then
        tmp = x_m * (2.0d0 / (y * z_m))
    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 ((y <= -2700000.0) || !(y <= 1.55e-44)) {
		tmp = x_m * (2.0 / (y * z_m));
	} 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 (y <= -2700000.0) or not (y <= 1.55e-44):
		tmp = x_m * (2.0 / (y * z_m))
	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 ((y <= -2700000.0) || !(y <= 1.55e-44))
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z_m)));
	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 ((y <= -2700000.0) || ~((y <= 1.55e-44)))
		tmp = x_m * (2.0 / (y * z_m));
	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[Or[LessEqual[y, -2700000.0], N[Not[LessEqual[y, 1.55e-44]], $MachinePrecision]], N[(x$95$m * N[(2.0 / N[(y * z$95$m), $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 y < -2.7e6 or 1.54999999999999992e-44 < y

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 88.9%

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

      3. *-commutative 88.9%

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

      4. distribute-rgt-out-- 90.7%

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

      5. associate-/l/ 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in y around inf 76.5%

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

      1. *-commutative 76.5%

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

   6. Simplified 76.5%

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

   if -2.7e6 < y < 1.54999999999999992e-44

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l/ 86.9%

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

      3. *-commutative 86.9%

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

      4. distribute-rgt-out-- 88.3%

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

      5. associate-/l/ 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around 0 70.7%

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

      1. associate-/r* 74.6%

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

   6. Simplified 74.6%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2700000 \lor \neg \left(y \leq 1.55 \cdot 10^{-44}\right):\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 4: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5500000 \lor \neg \left(y \leq 1.2 \cdot 10^{-44}\right):\\ \;\;\;\;x_m \cdot \frac{\frac{2}{y}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x_m}{t}}{z_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (or (<= y -5500000.0) (not (<= y 1.2e-44)))
     (* x_m (/ (/ 2.0 y) z_m))
     (* -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 ((y <= -5500000.0) || !(y <= 1.2e-44)) {
		tmp = x_m * ((2.0 / y) / z_m);
	} 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 ((y <= (-5500000.0d0)) .or. (.not. (y <= 1.2d-44))) then
        tmp = x_m * ((2.0d0 / y) / z_m)
    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 ((y <= -5500000.0) || !(y <= 1.2e-44)) {
		tmp = x_m * ((2.0 / y) / z_m);
	} 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 (y <= -5500000.0) or not (y <= 1.2e-44):
		tmp = x_m * ((2.0 / y) / z_m)
	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 ((y <= -5500000.0) || !(y <= 1.2e-44))
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z_m));
	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 ((y <= -5500000.0) || ~((y <= 1.2e-44)))
		tmp = x_m * ((2.0 / y) / z_m);
	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[Or[LessEqual[y, -5500000.0], N[Not[LessEqual[y, 1.2e-44]], $MachinePrecision]], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z$95$m), $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 y < -5.5e6 or 1.20000000000000004e-44 < y

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 88.9%

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

      3. *-commutative 88.9%

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

      4. distribute-rgt-out-- 90.7%

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

      5. associate-/l/ 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in y around inf 76.5%

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

      1. associate-/r* 77.7%

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

   6. Simplified 77.7%

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

   if -5.5e6 < y < 1.20000000000000004e-44

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l/ 86.9%

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

      3. *-commutative 86.9%

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

      4. distribute-rgt-out-- 88.3%

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

      5. associate-/l/ 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around 0 70.7%

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

      1. associate-/r* 74.6%

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

   6. Simplified 74.6%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5500000 \lor \neg \left(y \leq 1.2 \cdot 10^{-44}\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 5: 71.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.45 \cdot 10^{-48}:\\ \;\;\;\;-2 \cdot \frac{\frac{x_m}{t}}{z_m}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-144}:\\ \;\;\;\;x_m \cdot \frac{\frac{2}{y}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z_m} \cdot \frac{-2}{t}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -4.45e-48)
     (* -2.0 (/ (/ x_m t) z_m))
     (if (<= t 1.15e-144)
       (* x_m (/ (/ 2.0 y) z_m))
       (* (/ x_m z_m) (/ -2.0 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 (t <= -4.45e-48) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else if (t <= 1.15e-144) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else {
		tmp = (x_m / z_m) * (-2.0 / 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 (t <= (-4.45d-48)) then
        tmp = (-2.0d0) * ((x_m / t) / z_m)
    else if (t <= 1.15d-144) then
        tmp = x_m * ((2.0d0 / y) / z_m)
    else
        tmp = (x_m / z_m) * ((-2.0d0) / 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 (t <= -4.45e-48) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else if (t <= 1.15e-144) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else {
		tmp = (x_m / z_m) * (-2.0 / 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 t <= -4.45e-48:
		tmp = -2.0 * ((x_m / t) / z_m)
	elif t <= 1.15e-144:
		tmp = x_m * ((2.0 / y) / z_m)
	else:
		tmp = (x_m / z_m) * (-2.0 / 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 (t <= -4.45e-48)
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z_m));
	elseif (t <= 1.15e-144)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z_m));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(-2.0 / 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 (t <= -4.45e-48)
		tmp = -2.0 * ((x_m / t) / z_m);
	elseif (t <= 1.15e-144)
		tmp = x_m * ((2.0 / y) / z_m);
	else
		tmp = (x_m / z_m) * (-2.0 / 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[t, -4.45e-48], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e-144], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -4.45000000000000007e-48

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l/ 87.8%

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

      3. *-commutative 87.8%

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

      4. distribute-rgt-out-- 89.0%

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

      5. associate-/l/ 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in y around 0 69.6%

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

      1. associate-/r* 75.5%

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

   6. Simplified 75.5%

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

   if -4.45000000000000007e-48 < t < 1.15e-144

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

      2. associate-*l/ 90.0%

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

      3. *-commutative 90.0%

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

      4. distribute-rgt-out-- 91.2%

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

      5. associate-/l/ 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in y around inf 79.7%

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

      1. associate-/r* 80.1%

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

   6. Simplified 80.1%

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

   if 1.15e-144 < t

   1. Initial program 86.5%

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

      1. distribute-rgt-out-- 88.8%

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

      2. times-frac 94.0%

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

   3. Simplified 94.0%

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

   4. Taylor expanded in y around 0 73.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.45 \cdot 10^{-48}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-144}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]

Alternative 6: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-47}:\\ \;\;\;\;-2 \cdot \frac{\frac{x_m}{t}}{z_m}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{x_m}{z_m} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z_m} \cdot \frac{-2}{t}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -8e-47)
     (* -2.0 (/ (/ x_m t) z_m))
     (if (<= t 3.4e-57)
       (* (/ x_m z_m) (/ 2.0 y))
       (* (/ x_m z_m) (/ -2.0 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 (t <= -8e-47) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else if (t <= 3.4e-57) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else {
		tmp = (x_m / z_m) * (-2.0 / 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 (t <= (-8d-47)) then
        tmp = (-2.0d0) * ((x_m / t) / z_m)
    else if (t <= 3.4d-57) then
        tmp = (x_m / z_m) * (2.0d0 / y)
    else
        tmp = (x_m / z_m) * ((-2.0d0) / 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 (t <= -8e-47) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else if (t <= 3.4e-57) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else {
		tmp = (x_m / z_m) * (-2.0 / 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 t <= -8e-47:
		tmp = -2.0 * ((x_m / t) / z_m)
	elif t <= 3.4e-57:
		tmp = (x_m / z_m) * (2.0 / y)
	else:
		tmp = (x_m / z_m) * (-2.0 / 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 (t <= -8e-47)
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z_m));
	elseif (t <= 3.4e-57)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / y));
	else
		tmp = Float64(Float64(x_m / z_m) * Float64(-2.0 / 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 (t <= -8e-47)
		tmp = -2.0 * ((x_m / t) / z_m);
	elseif (t <= 3.4e-57)
		tmp = (x_m / z_m) * (2.0 / y);
	else
		tmp = (x_m / z_m) * (-2.0 / 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[t, -8e-47], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.4e-57], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(-2.0 / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -7.9999999999999998e-47

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l/ 87.8%

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

      3. *-commutative 87.8%

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

      4. distribute-rgt-out-- 89.0%

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

      5. associate-/l/ 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in y around 0 69.6%

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

      1. associate-/r* 75.5%

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

   6. Simplified 75.5%

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

   if -7.9999999999999998e-47 < t < 3.40000000000000016e-57

   1. Initial program 92.6%

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

      1. distribute-rgt-out-- 93.7%

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

      2. times-frac 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in y around inf 79.4%

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

   if 3.40000000000000016e-57 < t

   1. Initial program 84.8%

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

      1. distribute-rgt-out-- 87.4%

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

      2. times-frac 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in y around 0 76.7%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-47}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{-2}{t}\\ \end{array} \]

Alternative 7: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-46}:\\ \;\;\;\;-2 \cdot \frac{\frac{x_m}{t}}{z_m}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{x_m}{z_m} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z_m}{x_m}}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -4.1e-46)
     (* -2.0 (/ (/ x_m t) z_m))
     (if (<= t 2.5e-58)
       (* (/ x_m z_m) (/ 2.0 y))
       (/ -2.0 (* t (/ z_m 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 (t <= -4.1e-46) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else if (t <= 2.5e-58) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else {
		tmp = -2.0 / (t * (z_m / 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 (t <= (-4.1d-46)) then
        tmp = (-2.0d0) * ((x_m / t) / z_m)
    else if (t <= 2.5d-58) then
        tmp = (x_m / z_m) * (2.0d0 / y)
    else
        tmp = (-2.0d0) / (t * (z_m / 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 (t <= -4.1e-46) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else if (t <= 2.5e-58) {
		tmp = (x_m / z_m) * (2.0 / y);
	} else {
		tmp = -2.0 / (t * (z_m / 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 t <= -4.1e-46:
		tmp = -2.0 * ((x_m / t) / z_m)
	elif t <= 2.5e-58:
		tmp = (x_m / z_m) * (2.0 / y)
	else:
		tmp = -2.0 / (t * (z_m / 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 (t <= -4.1e-46)
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z_m));
	elseif (t <= 2.5e-58)
		tmp = Float64(Float64(x_m / z_m) * Float64(2.0 / y));
	else
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / 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 (t <= -4.1e-46)
		tmp = -2.0 * ((x_m / t) / z_m);
	elseif (t <= 2.5e-58)
		tmp = (x_m / z_m) * (2.0 / y);
	else
		tmp = -2.0 / (t * (z_m / 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[t, -4.1e-46], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.5e-58], N[(N[(x$95$m / z$95$m), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(t * N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -4.0999999999999999e-46

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l/ 87.8%

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

      3. *-commutative 87.8%

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

      4. distribute-rgt-out-- 89.0%

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

      5. associate-/l/ 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in y around 0 69.6%

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

      1. associate-/r* 75.5%

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

   6. Simplified 75.5%

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

   if -4.0999999999999999e-46 < t < 2.49999999999999989e-58

   1. Initial program 92.6%

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

      1. distribute-rgt-out-- 93.7%

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

      2. times-frac 94.3%

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

   3. Simplified 94.3%

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

   4. Taylor expanded in y around inf 79.4%

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

   if 2.49999999999999989e-58 < t

   1. Initial program 84.8%

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

      1. distribute-rgt-out-- 87.4%

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

      2. times-frac 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in y around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. clear-num 76.5%

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

      3. frac-times 77.3%

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

      4. metadata-eval 77.3%

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

   6. Applied egg-rr 77.3%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-46}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 8: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -82000:\\ \;\;\;\;x_m \cdot \frac{\frac{2}{y}}{z_m}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \frac{\frac{x_m}{t}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z_m \cdot \frac{y}{x_m}}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= y -82000.0)
     (* x_m (/ (/ 2.0 y) z_m))
     (if (<= y 8.5e-45)
       (* -2.0 (/ (/ x_m t) z_m))
       (/ 2.0 (* z_m (/ y 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 (y <= -82000.0) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else if (y <= 8.5e-45) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else {
		tmp = 2.0 / (z_m * (y / 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 (y <= (-82000.0d0)) then
        tmp = x_m * ((2.0d0 / y) / z_m)
    else if (y <= 8.5d-45) then
        tmp = (-2.0d0) * ((x_m / t) / z_m)
    else
        tmp = 2.0d0 / (z_m * (y / 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 (y <= -82000.0) {
		tmp = x_m * ((2.0 / y) / z_m);
	} else if (y <= 8.5e-45) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else {
		tmp = 2.0 / (z_m * (y / 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 y <= -82000.0:
		tmp = x_m * ((2.0 / y) / z_m)
	elif y <= 8.5e-45:
		tmp = -2.0 * ((x_m / t) / z_m)
	else:
		tmp = 2.0 / (z_m * (y / 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 (y <= -82000.0)
		tmp = Float64(x_m * Float64(Float64(2.0 / y) / z_m));
	elseif (y <= 8.5e-45)
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z_m));
	else
		tmp = Float64(2.0 / Float64(z_m * Float64(y / 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 (y <= -82000.0)
		tmp = x_m * ((2.0 / y) / z_m);
	elseif (y <= 8.5e-45)
		tmp = -2.0 * ((x_m / t) / z_m);
	else
		tmp = 2.0 / (z_m * (y / 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[y, -82000.0], N[(x$95$m * N[(N[(2.0 / y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e-45], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(z$95$m * N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -82000

   1. Initial program 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*l/ 93.9%

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

      3. *-commutative 93.9%

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

      4. distribute-rgt-out-- 96.0%

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

      5. associate-/l/ 97.4%

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

   3. Simplified 97.4%

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

   4. Taylor expanded in y around inf 81.9%

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

      1. associate-/r* 83.4%

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

   6. Simplified 83.4%

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

   if -82000 < y < 8.50000000000000041e-45

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l/ 86.9%

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

      3. *-commutative 86.9%

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

      4. distribute-rgt-out-- 88.3%

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

      5. associate-/l/ 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in y around 0 70.7%

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

      1. associate-/r* 74.6%

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

   6. Simplified 74.6%

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

   if 8.50000000000000041e-45 < y

   1. Initial program 86.5%

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

      1. *-commutative 86.5%

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

      2. associate-*l/ 85.1%

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

      3. *-commutative 85.1%

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

      4. distribute-rgt-out-- 86.7%

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

      5. associate-/l/ 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in y around inf 72.4%

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

      1. *-commutative 72.4%

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

   6. Simplified 72.4%

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

      1. associate-*r/ 73.8%

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

      2. frac-times 75.2%

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

      3. clear-num 75.2%

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

      4. frac-times 77.7%

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

      5. metadata-eval 77.7%

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

   8. Applied egg-rr 77.7%

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

   9. Taylor expanded in z around 0 73.6%

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

       1. associate-*l/ 79.6%

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

   11. Simplified 79.6%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -82000:\\ \;\;\;\;x \cdot \frac{\frac{2}{y}}{z}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-45}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 9: 73.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-48}:\\ \;\;\;\;-2 \cdot \frac{\frac{x_m}{t}}{z_m}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z_m}{x_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z_m}{x_m}}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= t -3.6e-48)
     (* -2.0 (/ (/ x_m t) z_m))
     (if (<= t 5e-58) (/ 2.0 (* y (/ z_m x_m))) (/ -2.0 (* t (/ z_m 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 (t <= -3.6e-48) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else if (t <= 5e-58) {
		tmp = 2.0 / (y * (z_m / x_m));
	} else {
		tmp = -2.0 / (t * (z_m / 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 (t <= (-3.6d-48)) then
        tmp = (-2.0d0) * ((x_m / t) / z_m)
    else if (t <= 5d-58) then
        tmp = 2.0d0 / (y * (z_m / x_m))
    else
        tmp = (-2.0d0) / (t * (z_m / 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 (t <= -3.6e-48) {
		tmp = -2.0 * ((x_m / t) / z_m);
	} else if (t <= 5e-58) {
		tmp = 2.0 / (y * (z_m / x_m));
	} else {
		tmp = -2.0 / (t * (z_m / 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 t <= -3.6e-48:
		tmp = -2.0 * ((x_m / t) / z_m)
	elif t <= 5e-58:
		tmp = 2.0 / (y * (z_m / x_m))
	else:
		tmp = -2.0 / (t * (z_m / 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 (t <= -3.6e-48)
		tmp = Float64(-2.0 * Float64(Float64(x_m / t) / z_m));
	elseif (t <= 5e-58)
		tmp = Float64(2.0 / Float64(y * Float64(z_m / x_m)));
	else
		tmp = Float64(-2.0 / Float64(t * Float64(z_m / 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 (t <= -3.6e-48)
		tmp = -2.0 * ((x_m / t) / z_m);
	elseif (t <= 5e-58)
		tmp = 2.0 / (y * (z_m / x_m));
	else
		tmp = -2.0 / (t * (z_m / 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[t, -3.6e-48], N[(-2.0 * N[(N[(x$95$m / t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e-58], N[(2.0 / N[(y * N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 / N[(t * N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -3.6000000000000002e-48

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l/ 87.8%

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

      3. *-commutative 87.8%

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

      4. distribute-rgt-out-- 89.0%

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

      5. associate-/l/ 91.2%

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

   3. Simplified 91.2%

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

   4. Taylor expanded in y around 0 69.6%

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

      1. associate-/r* 75.5%

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

   6. Simplified 75.5%

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

   if -3.6000000000000002e-48 < t < 4.99999999999999977e-58

   1. Initial program 92.6%

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

      1. *-commutative 92.6%

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

      2. associate-*l/ 91.0%

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

      3. *-commutative 91.0%

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

      4. distribute-rgt-out-- 92.1%

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

      5. associate-/l/ 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in y around inf 76.3%

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

      1. *-commutative 76.3%

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

   6. Simplified 76.3%

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

      1. associate-*r/ 77.9%

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

      2. frac-times 79.4%

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

      3. clear-num 79.3%

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

      4. frac-times 80.3%

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

      5. metadata-eval 80.3%

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

   8. Applied egg-rr 80.3%

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

   if 4.99999999999999977e-58 < t

   1. Initial program 84.8%

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

      1. distribute-rgt-out-- 87.4%

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

      2. times-frac 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in y around 0 76.7%

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

      1. *-commutative 76.7%

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

      2. clear-num 76.5%

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

      3. frac-times 77.3%

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

      4. metadata-eval 77.3%

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

   6. Applied egg-rr 77.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-48}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{y \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{t \cdot \frac{z}{x}}\\ \end{array} \]

Alternative 10: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{2}{y - t}\\ z_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2 \cdot 10^{-71}:\\ \;\;\;\;x_m \cdot \frac{t_1}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z_m} \cdot t_1\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (- y t))))
   (*
    z_s
    (* x_s (if (<= z_m 2e-71) (* x_m (/ t_1 z_m)) (* (/ x_m z_m) t_1))))))

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 = 2.0 / (y - t);
	double tmp;
	if (z_m <= 2e-71) {
		tmp = x_m * (t_1 / z_m);
	} else {
		tmp = (x_m / z_m) * t_1;
	}
	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 = 2.0d0 / (y - t)
    if (z_m <= 2d-71) then
        tmp = x_m * (t_1 / z_m)
    else
        tmp = (x_m / z_m) * t_1
    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 = 2.0 / (y - t);
	double tmp;
	if (z_m <= 2e-71) {
		tmp = x_m * (t_1 / z_m);
	} else {
		tmp = (x_m / z_m) * t_1;
	}
	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 = 2.0 / (y - t)
	tmp = 0
	if z_m <= 2e-71:
		tmp = x_m * (t_1 / z_m)
	else:
		tmp = (x_m / z_m) * t_1
	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 = Float64(2.0 / Float64(y - t))
	tmp = 0.0
	if (z_m <= 2e-71)
		tmp = Float64(x_m * Float64(t_1 / z_m));
	else
		tmp = Float64(Float64(x_m / z_m) * t_1);
	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 = 2.0 / (y - t);
	tmp = 0.0;
	if (z_m <= 2e-71)
		tmp = x_m * (t_1 / z_m);
	else
		tmp = (x_m / z_m) * t_1;
	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[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2e-71], N[(x$95$m * N[(t$95$1 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z$95$m), $MachinePrecision] * t$95$1), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.9999999999999998e-71

   1. Initial program 89.6%

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

      1. *-commutative 89.6%

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

      2. associate-*l/ 88.2%

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

      3. *-commutative 88.2%

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

      4. distribute-rgt-out-- 89.4%

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

      5. associate-/l/ 90.6%

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

   3. Simplified 90.6%

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

   if 1.9999999999999998e-71 < z

   1. Initial program 87.1%

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

      1. distribute-rgt-out-- 89.3%

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

      2. times-frac 98.1%

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

   3. Simplified 98.1%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \frac{\frac{2}{y - t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]

Alternative 11: 55.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 4 \cdot 10^{+58}:\\ \;\;\;\;-2 \cdot \frac{x_m}{z_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x_m}{t}}{z_m}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (*
  z_s
  (*
   x_s
   (if (<= z_m 4e+58) (* -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 <= 4e+58) {
		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 <= 4d+58) 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 <= 4e+58) {
		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 <= 4e+58:
		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 <= 4e+58)
		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 <= 4e+58)
		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, 4e+58], 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 < 3.99999999999999978e58

   1. Initial program 90.4%

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

      1. *-commutative 90.4%

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

      2. associate-*l/ 89.2%

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

      3. *-commutative 89.2%

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

      4. distribute-rgt-out-- 90.2%

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

      5. associate-/l/ 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in y around 0 54.2%

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

   if 3.99999999999999978e58 < z

   1. Initial program 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l/ 82.8%

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

      3. *-commutative 82.8%

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

      4. distribute-rgt-out-- 86.3%

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

      5. associate-/l/ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in y around 0 57.4%

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

      1. associate-/r* 65.9%

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

   6. Simplified 65.9%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+58}:\\ \;\;\;\;-2 \cdot \frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 12: 92.5% accurate, 1.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x_s x_m y z_m t)
 :precision binary64
 (* z_s (* x_s (* x_m (/ (/ 2.0 (- y 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) {
	return z_s * (x_s * (x_m * ((2.0 / (y - t)) / z_m)));
}

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 / (y - t)) / z_m)))
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 / (y - t)) / z_m)));
}

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 / (y - t)) / z_m)))

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(Float64(2.0 / Float64(y - t)) / z_m))))
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 / (y - t)) / z_m)));
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[(N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.7%

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

   1. *-commutative 88.7%

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

   2. associate-*l/ 87.8%

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

   3. *-commutative 87.8%

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

   4. distribute-rgt-out-- 89.4%

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

   5. associate-/l/ 90.9%

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

3. Simplified 90.9%

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

4. Final simplification 90.9%

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

Alternative 13: 53.7% 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 1 x)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 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 88.7%

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

   1. *-commutative 88.7%

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

   2. associate-*l/ 87.8%

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

   3. *-commutative 87.8%

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

   4. distribute-rgt-out-- 89.4%

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

   5. associate-/l/ 90.9%

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

3. Simplified 90.9%

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

4. Taylor expanded in y around 0 54.9%

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

5. Final simplification 54.9%

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

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 2023322 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))