Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.7% → 97.0%

Time: 10.6s

Alternatives: 14

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: 89.7% 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: 97.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x\_m \cdot 2}{y \cdot z\_m - z\_m \cdot t}\\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-316}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 10^{-167}:\\ \;\;\;\;\frac{-2 \cdot \frac{x\_m}{z\_m}}{t - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z\_m}\\ \end{array}\right) \end{array} \end{array} \]

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 (/ (* x_m 2.0) (- (* y z_m) (* z_m t)))))
   (*
    z_s
    (*
     x_s
     (if (<= t_1 -5e-316)
       t_1
       (if (<= t_1 1e-167)
         (/ (* -2.0 (/ x_m z_m)) (- t y))
         (* (/ 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 t_1 = (x_m * 2.0) / ((y * z_m) - (z_m * t));
	double tmp;
	if (t_1 <= -5e-316) {
		tmp = t_1;
	} else if (t_1 <= 1e-167) {
		tmp = (-2.0 * (x_m / z_m)) / (t - y);
	} 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) :: t_1
    real(8) :: tmp
    t_1 = (x_m * 2.0d0) / ((y * z_m) - (z_m * t))
    if (t_1 <= (-5d-316)) then
        tmp = t_1
    else if (t_1 <= 1d-167) then
        tmp = ((-2.0d0) * (x_m / z_m)) / (t - y)
    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 t_1 = (x_m * 2.0) / ((y * z_m) - (z_m * t));
	double tmp;
	if (t_1 <= -5e-316) {
		tmp = t_1;
	} else if (t_1 <= 1e-167) {
		tmp = (-2.0 * (x_m / z_m)) / (t - y);
	} 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):
	t_1 = (x_m * 2.0) / ((y * z_m) - (z_m * t))
	tmp = 0
	if t_1 <= -5e-316:
		tmp = t_1
	elif t_1 <= 1e-167:
		tmp = (-2.0 * (x_m / z_m)) / (t - y)
	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)
	t_1 = Float64(Float64(x_m * 2.0) / Float64(Float64(y * z_m) - Float64(z_m * t)))
	tmp = 0.0
	if (t_1 <= -5e-316)
		tmp = t_1;
	elseif (t_1 <= 1e-167)
		tmp = Float64(Float64(-2.0 * Float64(x_m / z_m)) / Float64(t - y));
	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)
	t_1 = (x_m * 2.0) / ((y * z_m) - (z_m * t));
	tmp = 0.0;
	if (t_1 <= -5e-316)
		tmp = t_1;
	elseif (t_1 <= 1e-167)
		tmp = (-2.0 * (x_m / z_m)) / (t - y);
	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_] := Block[{t$95$1 = N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(N[(y * z$95$m), $MachinePrecision] - N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(x$95$s * If[LessEqual[t$95$1, -5e-316], t$95$1, If[LessEqual[t$95$1, 1e-167], N[(N[(-2.0 * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision] / N[(t - y), $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 3 regimes

2. if (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < -5.000000017e-316

   1. Initial program 92.0%

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

   if -5.000000017e-316 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z))) < 1e-167

   1. Initial program 85.7%

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

      1. distribute-rgt-out-- 85.7%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in x around 0 85.6%

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

      1. associate-*r/ 85.7%

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

      2. metadata-eval 85.7%

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

      3. distribute-lft-neg-in 85.7%

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

      4. *-commutative 85.7%

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

      5. distribute-neg-frac 85.7%

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

      6. associate-/r* 98.9%

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

      7. *-commutative 98.9%

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

      8. associate-*r/ 98.9%

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

      9. distribute-neg-frac2 98.9%

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

      10. neg-sub0 98.9%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{\color{blue}{0 - \left(y - t\right)}} \]

      11. sub-neg 98.9%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{0 - \color{blue}{\left(y + \left(-t\right)\right)}} \]

      12. +-commutative 98.9%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{0 - \color{blue}{\left(\left(-t\right) + y\right)}} \]

      13. associate--r+ 98.9%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{\color{blue}{\left(0 - \left(-t\right)\right) - y}} \]

      14. neg-sub0 98.9%

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

      15. remove-double-neg 98.9%

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

   7. Simplified 98.9%

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

   if 1e-167 < (/.f64 (*.f64 x #s(literal 2 binary64)) (-.f64 (*.f64 y z) (*.f64 t z)))

   1. Initial program 92.7%

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

      1. distribute-rgt-out-- 97.7%

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

   3. Simplified 97.7%

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

      1. *-commutative 97.7%

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

      2. times-frac 91.3%

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

   6. Applied egg-rr 91.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq -5 \cdot 10^{-316}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z - z \cdot t}\\ \mathbf{elif}\;\frac{x \cdot 2}{y \cdot z - z \cdot t} \leq 10^{-167}:\\ \;\;\;\;\frac{-2 \cdot \frac{x}{z}}{t - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - t} \cdot \frac{2}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -80000000000000 \lor \neg \left(y \leq 11600\right):\\ \;\;\;\;x\_m \cdot \frac{\frac{2}{z\_m}}{y + t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{t}}{z\_m}\\ \end{array}\right) \end{array} \]

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 (or (<= y -80000000000000.0) (not (<= y 11600.0)))
     (* x_m (/ (/ 2.0 z_m) (+ y t)))
     (/ (/ (* x_m -2.0) 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 <= -80000000000000.0) || !(y <= 11600.0)) {
		tmp = x_m * ((2.0 / z_m) / (y + t));
	} else {
		tmp = ((x_m * -2.0) / 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 <= (-80000000000000.0d0)) .or. (.not. (y <= 11600.0d0))) then
        tmp = x_m * ((2.0d0 / z_m) / (y + t))
    else
        tmp = ((x_m * (-2.0d0)) / 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 <= -80000000000000.0) || !(y <= 11600.0)) {
		tmp = x_m * ((2.0 / z_m) / (y + t));
	} else {
		tmp = ((x_m * -2.0) / 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 <= -80000000000000.0) or not (y <= 11600.0):
		tmp = x_m * ((2.0 / z_m) / (y + t))
	else:
		tmp = ((x_m * -2.0) / 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 <= -80000000000000.0) || !(y <= 11600.0))
		tmp = Float64(x_m * Float64(Float64(2.0 / z_m) / Float64(y + t)));
	else
		tmp = Float64(Float64(Float64(x_m * -2.0) / 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 <= -80000000000000.0) || ~((y <= 11600.0)))
		tmp = x_m * ((2.0 / z_m) / (y + t));
	else
		tmp = ((x_m * -2.0) / 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, -80000000000000.0], N[Not[LessEqual[y, 11600.0]], $MachinePrecision]], N[(x$95$m * N[(N[(2.0 / z$95$m), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / t), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -8e13 or 11600 < 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-- 92.5%

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

   3. Simplified 92.5%

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

      1. clear-num 91.6%

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

      2. distribute-rgt-out-- 89.2%

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

      3. inv-pow 89.2%

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

      4. distribute-rgt-out-- 91.6%

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

      5. *-commutative 91.6%

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

      6. associate-/l* 91.4%

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

   6. Applied egg-rr 91.4%

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

      1. unpow-1 91.4%

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

      2. *-commutative 91.4%

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

      3. associate-*l/ 91.6%

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

      4. clear-num 92.5%

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

      5. frac-times 91.0%

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

      6. associate-*l/ 88.8%

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

      7. associate-/l* 92.7%

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

      8. sub-neg 92.7%

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

      9. add-sqr-sqrt 51.8%

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

      10. sqrt-unprod 79.4%

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

      11. sqr-neg 79.4%

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

      12. sqrt-unprod 35.6%

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

      13. add-sqr-sqrt 81.0%

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

   8. Applied egg-rr 81.0%

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

      1. associate-/r* 81.0%

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

      2. associate-/l/ 80.9%

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

      3. +-commutative 80.9%

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

   10. Simplified 80.9%

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

   if -8e13 < y < 11600

   1. Initial program 89.4%

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

      1. distribute-rgt-out-- 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in y around 0 76.4%

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

      1. *-commutative 76.4%

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

   7. Simplified 76.4%

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

      1. *-commutative 76.4%

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

      2. associate-*l/ 76.4%

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

      3. metadata-eval 76.4%

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

      4. distribute-rgt-neg-in 76.4%

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

      5. *-commutative 76.4%

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

      6. associate-/r* 79.1%

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

      7. distribute-rgt-neg-in 79.1%

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

      8. metadata-eval 79.1%

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

   9. Applied egg-rr 79.1%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -80000000000000 \lor \neg \left(y \leq 11600\right):\\ \;\;\;\;x \cdot \frac{\frac{2}{z}}{y + t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{t}}{z\_m}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{z\_m}}{t}\\ \end{array}\right) \end{array} \]

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 (<= t -3.3e+59)
     (/ (/ (* x_m -2.0) t) z_m)
     (if (<= t 1.6e-7)
       (/ (* x_m 2.0) (* y z_m))
       (/ (/ (* x_m -2.0) 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 (t <= -3.3e+59) {
		tmp = ((x_m * -2.0) / t) / z_m;
	} else if (t <= 1.6e-7) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else {
		tmp = ((x_m * -2.0) / 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 (t <= (-3.3d+59)) then
        tmp = ((x_m * (-2.0d0)) / t) / z_m
    else if (t <= 1.6d-7) then
        tmp = (x_m * 2.0d0) / (y * z_m)
    else
        tmp = ((x_m * (-2.0d0)) / 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 (t <= -3.3e+59) {
		tmp = ((x_m * -2.0) / t) / z_m;
	} else if (t <= 1.6e-7) {
		tmp = (x_m * 2.0) / (y * z_m);
	} else {
		tmp = ((x_m * -2.0) / 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 t <= -3.3e+59:
		tmp = ((x_m * -2.0) / t) / z_m
	elif t <= 1.6e-7:
		tmp = (x_m * 2.0) / (y * z_m)
	else:
		tmp = ((x_m * -2.0) / 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 (t <= -3.3e+59)
		tmp = Float64(Float64(Float64(x_m * -2.0) / t) / z_m);
	elseif (t <= 1.6e-7)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	else
		tmp = Float64(Float64(Float64(x_m * -2.0) / 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 (t <= -3.3e+59)
		tmp = ((x_m * -2.0) / t) / z_m;
	elseif (t <= 1.6e-7)
		tmp = (x_m * 2.0) / (y * z_m);
	else
		tmp = ((x_m * -2.0) / 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[LessEqual[t, -3.3e+59], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / t), $MachinePrecision] / z$95$m), $MachinePrecision], If[LessEqual[t, 1.6e-7], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / z$95$m), $MachinePrecision] / t), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -3.2999999999999999e59

   1. Initial program 88.3%

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

      1. distribute-rgt-out-- 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 76.6%

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

      1. *-commutative 76.6%

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

   7. Simplified 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-*l/ 76.7%

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

      3. metadata-eval 76.7%

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

      4. distribute-rgt-neg-in 76.7%

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

      5. *-commutative 76.7%

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

      6. associate-/r* 78.3%

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

      7. distribute-rgt-neg-in 78.3%

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

      8. metadata-eval 78.3%

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

   9. Applied egg-rr 78.3%

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

   if -3.2999999999999999e59 < t < 1.6e-7

   1. Initial program 91.7%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in y around inf 78.1%

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

      1. *-commutative 78.1%

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

   7. Simplified 78.1%

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

   if 1.6e-7 < t

   1. Initial program 87.4%

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

      1. distribute-rgt-out-- 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around 0 79.2%

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

      1. *-commutative 79.2%

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

   7. Simplified 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*l/ 79.2%

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

      3. metadata-eval 79.2%

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

      4. distribute-rgt-neg-in 79.2%

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

      5. associate-/r* 83.5%

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

      6. distribute-rgt-neg-in 83.5%

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

      7. metadata-eval 83.5%

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

   9. Applied egg-rr 83.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{t}}{z}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{t}}{z\_m}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z\_m}}{t}\\ \end{array}\right) \end{array} \]

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 (<= t -3.3e+54)
     (/ (/ (* x_m -2.0) t) z_m)
     (if (<= t 1.95e-5)
       (/ (* 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 (t <= -3.3e+54) {
		tmp = ((x_m * -2.0) / t) / z_m;
	} else if (t <= 1.95e-5) {
		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 (t <= (-3.3d+54)) then
        tmp = ((x_m * (-2.0d0)) / t) / z_m
    else if (t <= 1.95d-5) 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 (t <= -3.3e+54) {
		tmp = ((x_m * -2.0) / t) / z_m;
	} else if (t <= 1.95e-5) {
		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 t <= -3.3e+54:
		tmp = ((x_m * -2.0) / t) / z_m
	elif t <= 1.95e-5:
		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 (t <= -3.3e+54)
		tmp = Float64(Float64(Float64(x_m * -2.0) / t) / z_m);
	elseif (t <= 1.95e-5)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / 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 (t <= -3.3e+54)
		tmp = ((x_m * -2.0) / t) / z_m;
	elseif (t <= 1.95e-5)
		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[LessEqual[t, -3.3e+54], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / t), $MachinePrecision] / z$95$m), $MachinePrecision], If[LessEqual[t, 1.95e-5], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -3.3e54

   1. Initial program 88.3%

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

      1. distribute-rgt-out-- 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 76.6%

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

      1. *-commutative 76.6%

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

   7. Simplified 76.6%

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

      1. *-commutative 76.6%

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

      2. associate-*l/ 76.7%

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

      3. metadata-eval 76.7%

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

      4. distribute-rgt-neg-in 76.7%

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

      5. *-commutative 76.7%

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

      6. associate-/r* 78.3%

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

      7. distribute-rgt-neg-in 78.3%

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

      8. metadata-eval 78.3%

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

   9. Applied egg-rr 78.3%

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

   if -3.3e54 < t < 1.95e-5

   1. Initial program 91.7%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in y around inf 78.1%

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

      1. *-commutative 78.1%

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

   7. Simplified 78.1%

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

   if 1.95e-5 < t

   1. Initial program 87.4%

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

      1. distribute-rgt-out-- 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/r* 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{x \cdot -2}{t}}{z}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{-2}{z\_m} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x\_m \cdot 2}{y \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z\_m}}{t}\\ \end{array}\right) \end{array} \]

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 (<= t -4.2e+57)
     (* (/ -2.0 z_m) (/ x_m t))
     (if (<= t 7.5e-6)
       (/ (* 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 (t <= -4.2e+57) {
		tmp = (-2.0 / z_m) * (x_m / t);
	} else if (t <= 7.5e-6) {
		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 (t <= (-4.2d+57)) then
        tmp = ((-2.0d0) / z_m) * (x_m / t)
    else if (t <= 7.5d-6) 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 (t <= -4.2e+57) {
		tmp = (-2.0 / z_m) * (x_m / t);
	} else if (t <= 7.5e-6) {
		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 t <= -4.2e+57:
		tmp = (-2.0 / z_m) * (x_m / t)
	elif t <= 7.5e-6:
		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 (t <= -4.2e+57)
		tmp = Float64(Float64(-2.0 / z_m) * Float64(x_m / t));
	elseif (t <= 7.5e-6)
		tmp = Float64(Float64(x_m * 2.0) / Float64(y * z_m));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / 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 (t <= -4.2e+57)
		tmp = (-2.0 / z_m) * (x_m / t);
	elseif (t <= 7.5e-6)
		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[LessEqual[t, -4.2e+57], N[(N[(-2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.5e-6], N[(N[(x$95$m * 2.0), $MachinePrecision] / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -4.19999999999999982e57

   1. Initial program 88.3%

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

      1. distribute-rgt-out-- 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 76.7%

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

      1. associate-*r* 76.7%

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

      2. neg-mul-1 76.7%

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

      3. *-commutative 76.7%

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

   7. Simplified 76.7%

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

      1. frac-2neg 76.7%

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

      2. *-commutative 76.7%

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

      3. distribute-lft-neg-in 76.7%

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

      4. metadata-eval 76.7%

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

      5. distribute-rgt-neg-out 76.7%

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

      6. remove-double-neg 76.7%

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

      7. times-frac 78.2%

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

   9. Applied egg-rr 78.2%

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

   if -4.19999999999999982e57 < t < 7.50000000000000019e-6

   1. Initial program 91.7%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in y around inf 78.1%

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

      1. *-commutative 78.1%

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

   7. Simplified 78.1%

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

   if 7.50000000000000019e-6 < t

   1. Initial program 87.4%

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

      1. distribute-rgt-out-- 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/r* 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot 2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{-2}{z\_m} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;x\_m \cdot \frac{2}{y \cdot z\_m}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z\_m}}{t}\\ \end{array}\right) \end{array} \]

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 (<= t -7.5e+54)
     (* (/ -2.0 z_m) (/ x_m t))
     (if (<= t 1.15e-8)
       (* 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 (t <= -7.5e+54) {
		tmp = (-2.0 / z_m) * (x_m / t);
	} else if (t <= 1.15e-8) {
		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 (t <= (-7.5d+54)) then
        tmp = ((-2.0d0) / z_m) * (x_m / t)
    else if (t <= 1.15d-8) 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 (t <= -7.5e+54) {
		tmp = (-2.0 / z_m) * (x_m / t);
	} else if (t <= 1.15e-8) {
		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 t <= -7.5e+54:
		tmp = (-2.0 / z_m) * (x_m / t)
	elif t <= 1.15e-8:
		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 (t <= -7.5e+54)
		tmp = Float64(Float64(-2.0 / z_m) * Float64(x_m / t));
	elseif (t <= 1.15e-8)
		tmp = Float64(x_m * Float64(2.0 / Float64(y * z_m)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / 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 (t <= -7.5e+54)
		tmp = (-2.0 / z_m) * (x_m / t);
	elseif (t <= 1.15e-8)
		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[LessEqual[t, -7.5e+54], N[(N[(-2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e-8], N[(x$95$m * N[(2.0 / N[(y * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -7.50000000000000042e54

   1. Initial program 88.3%

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

      1. distribute-rgt-out-- 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 76.7%

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

      1. associate-*r* 76.7%

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

      2. neg-mul-1 76.7%

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

      3. *-commutative 76.7%

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

   7. Simplified 76.7%

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

      1. frac-2neg 76.7%

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

      2. *-commutative 76.7%

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

      3. distribute-lft-neg-in 76.7%

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

      4. metadata-eval 76.7%

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

      5. distribute-rgt-neg-out 76.7%

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

      6. remove-double-neg 76.7%

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

      7. times-frac 78.2%

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

   9. Applied egg-rr 78.2%

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

   if -7.50000000000000042e54 < t < 1.15e-8

   1. Initial program 91.7%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

   5. Taylor expanded in y around inf 78.1%

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

      1. *-commutative 78.1%

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

   7. Simplified 78.1%

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

      1. associate-/l* 78.1%

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

      2. *-commutative 78.1%

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

      3. *-commutative 78.1%

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

   9. Applied egg-rr 78.1%

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

   if 1.15e-8 < t

   1. Initial program 87.4%

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

      1. distribute-rgt-out-- 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/r* 83.5%

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

   7. Simplified 83.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{-2}{z} \cdot \frac{x}{t}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{2}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{-2}{z\_m} \cdot \frac{x\_m}{t}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-7}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z\_m}}{t}\\ \end{array}\right) \end{array} \]

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 (<= t -7.5e+54)
     (* (/ -2.0 z_m) (/ x_m t))
     (if (<= t 2e-7) (* 2.0 (/ (/ x_m z_m) y)) (* -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 (t <= -7.5e+54) {
		tmp = (-2.0 / z_m) * (x_m / t);
	} else if (t <= 2e-7) {
		tmp = 2.0 * ((x_m / z_m) / y);
	} 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 (t <= (-7.5d+54)) then
        tmp = ((-2.0d0) / z_m) * (x_m / t)
    else if (t <= 2d-7) then
        tmp = 2.0d0 * ((x_m / z_m) / y)
    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 (t <= -7.5e+54) {
		tmp = (-2.0 / z_m) * (x_m / t);
	} else if (t <= 2e-7) {
		tmp = 2.0 * ((x_m / z_m) / y);
	} 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 t <= -7.5e+54:
		tmp = (-2.0 / z_m) * (x_m / t)
	elif t <= 2e-7:
		tmp = 2.0 * ((x_m / z_m) / y)
	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 (t <= -7.5e+54)
		tmp = Float64(Float64(-2.0 / z_m) * Float64(x_m / t));
	elseif (t <= 2e-7)
		tmp = Float64(2.0 * Float64(Float64(x_m / z_m) / y));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / 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 (t <= -7.5e+54)
		tmp = (-2.0 / z_m) * (x_m / t);
	elseif (t <= 2e-7)
		tmp = 2.0 * ((x_m / z_m) / y);
	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[LessEqual[t, -7.5e+54], N[(N[(-2.0 / z$95$m), $MachinePrecision] * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2e-7], N[(2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -7.50000000000000042e54

   1. Initial program 88.3%

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

      1. distribute-rgt-out-- 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 76.7%

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

      1. associate-*r* 76.7%

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

      2. neg-mul-1 76.7%

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

      3. *-commutative 76.7%

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

   7. Simplified 76.7%

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

      1. frac-2neg 76.7%

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

      2. *-commutative 76.7%

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

      3. distribute-lft-neg-in 76.7%

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

      4. metadata-eval 76.7%

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

      5. distribute-rgt-neg-out 76.7%

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

      6. remove-double-neg 76.7%

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

      7. times-frac 78.2%

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

   9. Applied egg-rr 78.2%

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

   if -7.50000000000000042e54 < t < 1.9999999999999999e-7

   1. Initial program 91.7%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

      1. *-commutative 94.0%

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

      2. times-frac 90.9%

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

   6. Applied egg-rr 90.9%

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

   7. Taylor expanded in y around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-/r* 77.3%

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

   9. Simplified 77.3%

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

   if 1.9999999999999999e-7 < t

   1. Initial program 87.4%

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

      1. distribute-rgt-out-- 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/r* 83.5%

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

   7. Simplified 83.5%

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

Alternative 8: 74.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.32 \cdot 10^{+55}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \frac{\frac{x\_m}{z\_m}}{y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z\_m}}{t}\\ \end{array}\right) \end{array} \]

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 (<= t -1.32e+55)
     (* -2.0 (/ x_m (* z_m t)))
     (if (<= t 1.45e-6)
       (* 2.0 (/ (/ x_m z_m) y))
       (* -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 (t <= -1.32e+55) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (t <= 1.45e-6) {
		tmp = 2.0 * ((x_m / z_m) / y);
	} 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 (t <= (-1.32d+55)) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    else if (t <= 1.45d-6) then
        tmp = 2.0d0 * ((x_m / z_m) / y)
    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 (t <= -1.32e+55) {
		tmp = -2.0 * (x_m / (z_m * t));
	} else if (t <= 1.45e-6) {
		tmp = 2.0 * ((x_m / z_m) / y);
	} 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 t <= -1.32e+55:
		tmp = -2.0 * (x_m / (z_m * t))
	elif t <= 1.45e-6:
		tmp = 2.0 * ((x_m / z_m) / y)
	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 (t <= -1.32e+55)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	elseif (t <= 1.45e-6)
		tmp = Float64(2.0 * Float64(Float64(x_m / z_m) / y));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / 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 (t <= -1.32e+55)
		tmp = -2.0 * (x_m / (z_m * t));
	elseif (t <= 1.45e-6)
		tmp = 2.0 * ((x_m / z_m) / y);
	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[LessEqual[t, -1.32e+55], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e-6], N[(2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.32e55

   1. Initial program 88.3%

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

      1. distribute-rgt-out-- 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in y around 0 76.6%

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

      1. *-commutative 76.6%

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

   7. Simplified 76.6%

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

   if -1.32e55 < t < 1.4500000000000001e-6

   1. Initial program 91.7%

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

      1. distribute-rgt-out-- 94.0%

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

   3. Simplified 94.0%

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

      1. *-commutative 94.0%

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

      2. times-frac 90.9%

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

   6. Applied egg-rr 90.9%

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

   7. Taylor expanded in y around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-/r* 77.3%

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

   9. Simplified 77.3%

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

   if 1.4500000000000001e-6 < t

   1. Initial program 87.4%

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

      1. distribute-rgt-out-- 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-/r* 83.5%

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

   7. Simplified 83.5%

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

Alternative 9: 97.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 2000:\\ \;\;\;\;\frac{-2 \cdot \frac{x\_m}{z\_m}}{t - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z\_m}\\ \end{array}\right) \end{array} \]

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) 2000.0)
     (/ (* -2.0 (/ x_m z_m)) (- t y))
     (* (/ 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) <= 2000.0) {
		tmp = (-2.0 * (x_m / z_m)) / (t - y);
	} 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) <= 2000.0d0) then
        tmp = ((-2.0d0) * (x_m / z_m)) / (t - y)
    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) <= 2000.0) {
		tmp = (-2.0 * (x_m / z_m)) / (t - y);
	} 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) <= 2000.0:
		tmp = (-2.0 * (x_m / z_m)) / (t - y)
	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) <= 2000.0)
		tmp = Float64(Float64(-2.0 * Float64(x_m / z_m)) / Float64(t - y));
	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) <= 2000.0)
		tmp = (-2.0 * (x_m / z_m)) / (t - y);
	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], 2000.0], N[(N[(-2.0 * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision] / N[(t - y), $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)) < 2e3

   1. Initial program 92.2%

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

      1. distribute-rgt-out-- 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in x around 0 93.2%

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

      1. associate-*r/ 93.2%

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

      2. metadata-eval 93.2%

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

      3. distribute-lft-neg-in 93.2%

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

      4. *-commutative 93.2%

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

      5. distribute-neg-frac 93.2%

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

      6. associate-/r* 95.3%

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

      7. *-commutative 95.3%

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

      8. associate-*r/ 95.3%

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

      9. distribute-neg-frac2 95.3%

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

      10. neg-sub0 95.3%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{\color{blue}{0 - \left(y - t\right)}} \]

      11. sub-neg 95.3%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{0 - \color{blue}{\left(y + \left(-t\right)\right)}} \]

      12. +-commutative 95.3%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{0 - \color{blue}{\left(\left(-t\right) + y\right)}} \]

      13. associate--r+ 95.3%

          \[\leadsto \frac{-2 \cdot \frac{x}{z}}{\color{blue}{\left(0 - \left(-t\right)\right) - y}} \]

      14. neg-sub0 95.3%

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

      15. remove-double-neg 95.3%

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

   7. Simplified 95.3%

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

   if 2e3 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 81.7%

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

      1. distribute-rgt-out-- 86.8%

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

   3. Simplified 86.8%

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

      1. *-commutative 86.8%

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

      2. times-frac 96.3%

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

   6. Applied egg-rr 96.3%

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

Alternative 10: 96.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 3 \cdot 10^{+26}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z\_m}\\ \end{array}\right) \end{array} \]

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) 3e+26)
     (* (/ 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) <= 3e+26) {
		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) <= 3d+26) 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) <= 3e+26) {
		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) <= 3e+26:
		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) <= 3e+26)
		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) <= 3e+26)
		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], 3e+26], 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)) < 2.99999999999999997e26

   1. Initial program 92.3%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. times-frac 95.3%

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

   6. Applied egg-rr 95.3%

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

   if 2.99999999999999997e26 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 80.1%

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

      1. distribute-rgt-out-- 85.7%

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

   3. Simplified 85.7%

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

      1. *-commutative 85.7%

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

      2. times-frac 96.0%

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

   6. Applied egg-rr 96.0%

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

Alternative 11: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9.5 \cdot 10^{-55}:\\ \;\;\;\;x\_m \cdot \frac{2}{z\_m \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z\_m} \cdot \frac{2}{y - t}\\ \end{array}\right) \end{array} \]

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 9.5e-55)
     (* 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 <= 9.5e-55) {
		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 <= 9.5d-55) 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 <= 9.5e-55) {
		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 <= 9.5e-55:
		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 <= 9.5e-55)
		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 <= 9.5e-55)
		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, 9.5e-55], 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 < 9.5000000000000006e-55

   1. Initial program 91.8%

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

      1. distribute-rgt-out-- 93.5%

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

   3. Simplified 93.5%

      \[\leadsto \color{blue}{\frac{x \cdot 2}{z \cdot \left(y - t\right)}} \]
   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* 91.1%

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

      3. *-commutative 91.1%

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

      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%

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

   if 9.5000000000000006e-55 < z

   1. Initial program 84.9%

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

      1. distribute-rgt-out-- 87.5%

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

   3. Simplified 87.5%

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

      1. times-frac 95.5%

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

   6. Applied egg-rr 95.5%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{-55}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.6 \cdot 10^{-177}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z\_m \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z\_m}}{t}\\ \end{array}\right) \end{array} \]

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 3.6e-177)
     (* -2.0 (/ x_m (* z_m t)))
     (* -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 (z_m <= 3.6e-177) {
		tmp = -2.0 * (x_m / (z_m * t));
	} 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 (z_m <= 3.6d-177) then
        tmp = (-2.0d0) * (x_m / (z_m * t))
    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 (z_m <= 3.6e-177) {
		tmp = -2.0 * (x_m / (z_m * t));
	} 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 z_m <= 3.6e-177:
		tmp = -2.0 * (x_m / (z_m * t))
	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 (z_m <= 3.6e-177)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z_m * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / 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 (z_m <= 3.6e-177)
		tmp = -2.0 * (x_m / (z_m * t));
	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[LessEqual[z$95$m, 3.6e-177], N[(-2.0 * N[(x$95$m / N[(z$95$m * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z$95$m), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.59999999999999983e-177

   1. Initial program 92.5%

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

      1. distribute-rgt-out-- 93.9%

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

   3. Simplified 93.9%

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

   5. Taylor expanded in y around 0 54.5%

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

      1. *-commutative 54.5%

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

   7. Simplified 54.5%

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

   if 3.59999999999999983e-177 < 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-- 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in y around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-/r* 62.2%

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

   7. Simplified 62.2%

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

Alternative 13: 91.5% accurate, 1.2× speedup

Math

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

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 89.8%

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

   1. distribute-rgt-out-- 91.7%

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

3. Simplified 91.7%

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

   1. distribute-rgt-out-- 89.8%

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

   2. associate-/l* 89.2%

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

   3. *-commutative 89.2%

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

   4. distribute-rgt-out-- 91.2%

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

6. Applied egg-rr 91.2%

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

7. Final simplification 91.2%

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

Alternative 14: 54.1% 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 89.8%

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

   1. distribute-rgt-out-- 91.7%

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

3. Simplified 91.7%

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

5. Taylor expanded in y around 0 55.1%

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

   1. *-commutative 55.1%

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

7. Simplified 55.1%

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

Developer Target 1: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{\left(y - t\right) \cdot z} \cdot 2\\ t_2 := \frac{x \cdot 2}{y \cdot z - t \cdot z}\\ \mathbf{if}\;t\_2 < -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x (* (- y t) z)) 2.0))
        (t_2 (/ (* x 2.0) (- (* y z) (* t z)))))
   (if (< t_2 -2.559141628295061e-13)
     t_1
     (if (< t_2 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / ((y - t) * z)) * 2.0;
	double t_2 = (x * 2.0) / ((y * z) - (t * z));
	double tmp;
	if (t_2 < -2.559141628295061e-13) {
		tmp = t_1;
	} else if (t_2 < 1.045027827330126e-269) {
		tmp = ((x / z) * 2.0) / (y - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (x / ((y - t) * z)) * 2.0d0
    t_2 = (x * 2.0d0) / ((y * z) - (t * z))
    if (t_2 < (-2.559141628295061d-13)) then
        tmp = t_1
    else if (t_2 < 1.045027827330126d-269) then
        tmp = ((x / z) * 2.0d0) / (y - t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / ((y - t) * z)) * 2.0;
	double t_2 = (x * 2.0) / ((y * z) - (t * z));
	double tmp;
	if (t_2 < -2.559141628295061e-13) {
		tmp = t_1;
	} else if (t_2 < 1.045027827330126e-269) {
		tmp = ((x / z) * 2.0) / (y - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / ((y - t) * z)) * 2.0
	t_2 = (x * 2.0) / ((y * z) - (t * z))
	tmp = 0
	if t_2 < -2.559141628295061e-13:
		tmp = t_1
	elif t_2 < 1.045027827330126e-269:
		tmp = ((x / z) * 2.0) / (y - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / Float64(Float64(y - t) * z)) * 2.0)
	t_2 = Float64(Float64(x * 2.0) / Float64(Float64(y * z) - Float64(t * z)))
	tmp = 0.0
	if (t_2 < -2.559141628295061e-13)
		tmp = t_1;
	elseif (t_2 < 1.045027827330126e-269)
		tmp = Float64(Float64(Float64(x / z) * 2.0) / Float64(y - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / ((y - t) * z)) * 2.0;
	t_2 = (x * 2.0) / ((y * z) - (t * z));
	tmp = 0.0;
	if (t_2 < -2.559141628295061e-13)
		tmp = t_1;
	elseif (t_2 < 1.045027827330126e-269)
		tmp = ((x / z) * 2.0) / (y - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / N[(N[(y - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 2.0), $MachinePrecision] / N[(N[(y * z), $MachinePrecision] - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -2.559141628295061e-13], t$95$1, If[Less[t$95$2, 1.045027827330126e-269], N[(N[(N[(x / z), $MachinePrecision] * 2.0), $MachinePrecision] / N[(y - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024163 
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (* x 2) (- (* y z) (* t z))) -2559141628295061/10000000000000000000000000000) (* (/ x (* (- y t) z)) 2) (if (< (/ (* x 2) (- (* y z) (* t z))) 522513913665063/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (* (/ x z) 2) (- y t)) (* (/ x (* (- y t) z)) 2))))

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