Linear.Projection:infinitePerspective from linear-1.19.1.3, A

Percentage Accurate: 89.6% → 96.1%

Time: 11.3s

Alternatives: 13

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{+103}:\\ \;\;\;\;\frac{-2 \cdot \frac{x\_m}{z}}{t - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 5e+103)
    (/ (* -2.0 (/ x_m z)) (- t y))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e+103) {
		tmp = (-2.0 * (x_m / z)) / (t - y);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d+103) then
        tmp = ((-2.0d0) * (x_m / z)) / (t - y)
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e+103) {
		tmp = (-2.0 * (x_m / z)) / (t - y);
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 5e+103:
		tmp = (-2.0 * (x_m / z)) / (t - y)
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e+103)
		tmp = Float64(Float64(-2.0 * Float64(x_m / z)) / Float64(t - y));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e+103)
		tmp = (-2.0 * (x_m / z)) / (t - y);
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e+103], N[(N[(-2.0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / N[(t - y), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 5e103

   1. Initial program 86.8%

      \[\frac{x \cdot 2}{y \cdot z - t \cdot z} \]
   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. Taylor expanded in x around 0 92.5%

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

      1. associate-*r/ 92.5%

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

      2. metadata-eval 92.5%

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

      3. distribute-lft-neg-in 92.5%

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

      4. *-commutative 92.5%

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

      5. distribute-neg-frac 92.5%

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

      6. associate-/r* 95.4%

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

      7. *-commutative 95.4%

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

      8. associate-*r/ 95.4%

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

      9. distribute-neg-frac2 95.4%

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

      10. neg-sub0 95.4%

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

      11. sub-neg 95.4%

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

      12. +-commutative 95.4%

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

      13. associate--r+ 95.4%

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

      14. neg-sub0 95.4%

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

      15. remove-double-neg 95.4%

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

   7. Simplified 95.4%

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

   if 5e103 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 90.7%

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

      1. distribute-rgt-out-- 90.7%

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

   3. Simplified 90.7%

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

      1. *-commutative 90.7%

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

      2. times-frac 99.6%

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

   6. Applied egg-rr 99.6%

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

Alternative 2: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+73} \lor \neg \left(y \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -5e+73) (not (<= y 3.5e-52)))
    (* x_m (/ 2.0 (* z y)))
    (* -2.0 (/ (/ x_m z) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -5e+73) || !(y <= 3.5e-52)) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-5d+73)) .or. (.not. (y <= 3.5d-52))) then
        tmp = x_m * (2.0d0 / (z * y))
    else
        tmp = (-2.0d0) * ((x_m / z) / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -5e+73) || !(y <= 3.5e-52)) {
		tmp = x_m * (2.0 / (z * y));
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -5e+73) or not (y <= 3.5e-52):
		tmp = x_m * (2.0 / (z * y))
	else:
		tmp = -2.0 * ((x_m / z) / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -5e+73) || !(y <= 3.5e-52))
		tmp = Float64(x_m * Float64(2.0 / Float64(z * y)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -5e+73) || ~((y <= 3.5e-52)))
		tmp = x_m * (2.0 / (z * y));
	else
		tmp = -2.0 * ((x_m / z) / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -5e+73], N[Not[LessEqual[y, 3.5e-52]], $MachinePrecision]], N[(x$95$m * N[(2.0 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -4.99999999999999976e73 or 3.5e-52 < y

   1. Initial program 81.0%

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

      1. distribute-rgt-out-- 89.6%

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

   3. Simplified 89.6%

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

      1. distribute-rgt-out-- 81.0%

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

      2. associate-/l* 81.0%

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

      3. *-commutative 81.0%

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

      4. distribute-rgt-out-- 89.6%

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

   6. Applied egg-rr 89.6%

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

   7. Taylor expanded in y around inf 75.6%

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

      1. *-commutative 75.6%

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

   9. Simplified 75.6%

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

   if -4.99999999999999976e73 < y < 3.5e-52

   1. Initial program 92.3%

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

      1. distribute-rgt-out-- 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in y around 0 79.5%

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

      1. *-commutative 79.5%

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

      2. associate-/r* 80.7%

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

   7. Simplified 80.7%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+73} \lor \neg \left(y \leq 3.5 \cdot 10^{-52}\right):\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+77}:\\ \;\;\;\;\frac{2 \cdot \frac{x\_m}{z}}{y}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{x\_m \cdot -2}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{z \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -4e+77)
    (/ (* 2.0 (/ x_m z)) y)
    (if (<= y 2e-52) (/ (/ (* x_m -2.0) z) t) (/ (/ x_m y) (* z 0.5))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -4e+77) {
		tmp = (2.0 * (x_m / z)) / y;
	} else if (y <= 2e-52) {
		tmp = ((x_m * -2.0) / z) / t;
	} else {
		tmp = (x_m / y) / (z * 0.5);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-4d+77)) then
        tmp = (2.0d0 * (x_m / z)) / y
    else if (y <= 2d-52) then
        tmp = ((x_m * (-2.0d0)) / z) / t
    else
        tmp = (x_m / y) / (z * 0.5d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -4e+77) {
		tmp = (2.0 * (x_m / z)) / y;
	} else if (y <= 2e-52) {
		tmp = ((x_m * -2.0) / z) / t;
	} else {
		tmp = (x_m / y) / (z * 0.5);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -4e+77:
		tmp = (2.0 * (x_m / z)) / y
	elif y <= 2e-52:
		tmp = ((x_m * -2.0) / z) / t
	else:
		tmp = (x_m / y) / (z * 0.5)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -4e+77)
		tmp = Float64(Float64(2.0 * Float64(x_m / z)) / y);
	elseif (y <= 2e-52)
		tmp = Float64(Float64(Float64(x_m * -2.0) / z) / t);
	else
		tmp = Float64(Float64(x_m / y) / Float64(z * 0.5));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -4e+77)
		tmp = (2.0 * (x_m / z)) / y;
	elseif (y <= 2e-52)
		tmp = ((x_m * -2.0) / z) / t;
	else
		tmp = (x_m / y) / (z * 0.5);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -4e+77], N[(N[(2.0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 2e-52], N[(N[(N[(x$95$m * -2.0), $MachinePrecision] / z), $MachinePrecision] / t), $MachinePrecision], N[(N[(x$95$m / y), $MachinePrecision] / N[(z * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.99999999999999993e77

   1. Initial program 77.2%

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

      1. distribute-rgt-out-- 88.3%

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

   3. Simplified 88.3%

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

      1. *-commutative 88.3%

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

      2. times-frac 86.8%

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

   6. Applied egg-rr 86.8%

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

   7. Taylor expanded in y around inf 71.8%

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

      1. associate-*l/ 82.0%

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

      2. associate-*r/ 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-/l* 82.1%

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

   9. Applied egg-rr 82.1%

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

   if -3.99999999999999993e77 < y < 2e-52

   1. Initial program 92.3%

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

      1. distribute-rgt-out-- 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in y around 0 79.5%

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

      1. *-commutative 79.5%

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

   7. Simplified 79.5%

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

      1. *-commutative 79.5%

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

      2. associate-*l/ 79.5%

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

      3. metadata-eval 79.5%

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

      4. distribute-rgt-neg-in 79.5%

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

      5. associate-/r* 80.7%

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

      6. distribute-rgt-neg-in 80.7%

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

      7. metadata-eval 80.7%

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

   9. Applied egg-rr 80.7%

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

   if 2e-52 < y

   1. Initial program 83.3%

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

      1. distribute-rgt-out-- 90.4%

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

   3. Simplified 90.4%

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

      1. *-commutative 90.4%

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

      2. times-frac 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in y around inf 80.2%

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

      1. clear-num 80.2%

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

      2. un-div-inv 80.2%

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

      3. div-inv 80.2%

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

      4. metadata-eval 80.2%

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

   9. Applied egg-rr 80.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{z \cdot 0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+73}:\\ \;\;\;\;\frac{2 \cdot \frac{x\_m}{z}}{y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{y}}{z \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -5.3e+73)
    (/ (* 2.0 (/ x_m z)) y)
    (if (<= y 1.25e-52) (* -2.0 (/ (/ x_m z) t)) (/ (/ x_m y) (* z 0.5))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -5.3e+73) {
		tmp = (2.0 * (x_m / z)) / y;
	} else if (y <= 1.25e-52) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = (x_m / y) / (z * 0.5);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-5.3d+73)) then
        tmp = (2.0d0 * (x_m / z)) / y
    else if (y <= 1.25d-52) then
        tmp = (-2.0d0) * ((x_m / z) / t)
    else
        tmp = (x_m / y) / (z * 0.5d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -5.3e+73) {
		tmp = (2.0 * (x_m / z)) / y;
	} else if (y <= 1.25e-52) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = (x_m / y) / (z * 0.5);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -5.3e+73:
		tmp = (2.0 * (x_m / z)) / y
	elif y <= 1.25e-52:
		tmp = -2.0 * ((x_m / z) / t)
	else:
		tmp = (x_m / y) / (z * 0.5)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -5.3e+73)
		tmp = Float64(Float64(2.0 * Float64(x_m / z)) / y);
	elseif (y <= 1.25e-52)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	else
		tmp = Float64(Float64(x_m / y) / Float64(z * 0.5));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -5.3e+73)
		tmp = (2.0 * (x_m / z)) / y;
	elseif (y <= 1.25e-52)
		tmp = -2.0 * ((x_m / z) / t);
	else
		tmp = (x_m / y) / (z * 0.5);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -5.3e+73], N[(N[(2.0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.25e-52], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / y), $MachinePrecision] / N[(z * 0.5), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -5.29999999999999996e73

   1. Initial program 77.2%

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

      1. distribute-rgt-out-- 88.3%

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

   3. Simplified 88.3%

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

      1. *-commutative 88.3%

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

      2. times-frac 86.8%

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

   6. Applied egg-rr 86.8%

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

   7. Taylor expanded in y around inf 71.8%

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

      1. associate-*l/ 82.0%

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

      2. associate-*r/ 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-/l* 82.1%

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

   9. Applied egg-rr 82.1%

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

   if -5.29999999999999996e73 < y < 1.25e-52

   1. Initial program 92.3%

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

      1. distribute-rgt-out-- 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in y around 0 79.5%

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

      1. *-commutative 79.5%

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

      2. associate-/r* 80.7%

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

   7. Simplified 80.7%

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

   if 1.25e-52 < y

   1. Initial program 83.3%

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

      1. distribute-rgt-out-- 90.4%

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

   3. Simplified 90.4%

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

      1. *-commutative 90.4%

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

      2. times-frac 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in y around inf 80.2%

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

      1. clear-num 80.2%

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

      2. un-div-inv 80.2%

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

      3. div-inv 80.2%

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

      4. metadata-eval 80.2%

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

   9. Applied egg-rr 80.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{z \cdot 0.5}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{2 \cdot \frac{x\_m}{z}}{y}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -3.6e+78)
    (/ (* 2.0 (/ x_m z)) y)
    (if (<= y 1.55e-52) (* -2.0 (/ (/ x_m z) t)) (* (/ 2.0 z) (/ x_m y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -3.6e+78) {
		tmp = (2.0 * (x_m / z)) / y;
	} else if (y <= 1.55e-52) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-3.6d+78)) then
        tmp = (2.0d0 * (x_m / z)) / y
    else if (y <= 1.55d-52) then
        tmp = (-2.0d0) * ((x_m / z) / t)
    else
        tmp = (2.0d0 / z) * (x_m / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -3.6e+78) {
		tmp = (2.0 * (x_m / z)) / y;
	} else if (y <= 1.55e-52) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -3.6e+78:
		tmp = (2.0 * (x_m / z)) / y
	elif y <= 1.55e-52:
		tmp = -2.0 * ((x_m / z) / t)
	else:
		tmp = (2.0 / z) * (x_m / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -3.6e+78)
		tmp = Float64(Float64(2.0 * Float64(x_m / z)) / y);
	elseif (y <= 1.55e-52)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	else
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -3.6e+78)
		tmp = (2.0 * (x_m / z)) / y;
	elseif (y <= 1.55e-52)
		tmp = -2.0 * ((x_m / z) / t);
	else
		tmp = (2.0 / z) * (x_m / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -3.6e+78], N[(N[(2.0 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 1.55e-52], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -3.6000000000000002e78

   1. Initial program 77.2%

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

      1. distribute-rgt-out-- 88.3%

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

   3. Simplified 88.3%

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

      1. *-commutative 88.3%

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

      2. times-frac 86.8%

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

   6. Applied egg-rr 86.8%

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

   7. Taylor expanded in y around inf 71.8%

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

      1. associate-*l/ 82.0%

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

      2. associate-*r/ 82.1%

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

      3. *-commutative 82.1%

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

      4. associate-/l* 82.1%

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

   9. Applied egg-rr 82.1%

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

   if -3.6000000000000002e78 < y < 1.5499999999999999e-52

   1. Initial program 92.3%

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

      1. distribute-rgt-out-- 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in y around 0 79.5%

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

      1. *-commutative 79.5%

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

      2. associate-/r* 80.7%

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

   7. Simplified 80.7%

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

   if 1.5499999999999999e-52 < y

   1. Initial program 83.3%

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

      1. distribute-rgt-out-- 90.4%

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

   3. Simplified 90.4%

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

      1. *-commutative 90.4%

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

      2. times-frac 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in y around inf 80.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+78}:\\ \;\;\;\;\frac{2 \cdot \frac{x}{z}}{y}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-52}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-52}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -7.8e+73)
    (* (/ x_m z) (/ 2.0 y))
    (if (<= y 4.4e-52) (* -2.0 (/ (/ x_m z) t)) (* (/ 2.0 z) (/ x_m y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -7.8e+73) {
		tmp = (x_m / z) * (2.0 / y);
	} else if (y <= 4.4e-52) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-7.8d+73)) then
        tmp = (x_m / z) * (2.0d0 / y)
    else if (y <= 4.4d-52) then
        tmp = (-2.0d0) * ((x_m / z) / t)
    else
        tmp = (2.0d0 / z) * (x_m / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -7.8e+73) {
		tmp = (x_m / z) * (2.0 / y);
	} else if (y <= 4.4e-52) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -7.8e+73:
		tmp = (x_m / z) * (2.0 / y)
	elif y <= 4.4e-52:
		tmp = -2.0 * ((x_m / z) / t)
	else:
		tmp = (2.0 / z) * (x_m / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -7.8e+73)
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / y));
	elseif (y <= 4.4e-52)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	else
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -7.8e+73)
		tmp = (x_m / z) * (2.0 / y);
	elseif (y <= 4.4e-52)
		tmp = -2.0 * ((x_m / z) / t);
	else
		tmp = (2.0 / z) * (x_m / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -7.8e+73], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e-52], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -7.8000000000000002e73

   1. Initial program 77.2%

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

      1. distribute-rgt-out-- 88.3%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in y around inf 77.0%

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

      1. *-commutative 77.0%

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

   7. Simplified 77.0%

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

      1. times-frac 82.0%

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

   9. Applied egg-rr 82.0%

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

   if -7.8000000000000002e73 < y < 4.40000000000000018e-52

   1. Initial program 92.3%

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

      1. distribute-rgt-out-- 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in y around 0 79.5%

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

      1. *-commutative 79.5%

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

      2. associate-/r* 80.7%

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

   7. Simplified 80.7%

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

   if 4.40000000000000018e-52 < y

   1. Initial program 83.3%

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

      1. distribute-rgt-out-- 90.4%

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

   3. Simplified 90.4%

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

      1. *-commutative 90.4%

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

      2. times-frac 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in y around inf 80.2%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-52}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+73}:\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-52}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -5e+73)
    (* x_m (/ 2.0 (* z y)))
    (if (<= y 6.5e-52) (* -2.0 (/ (/ x_m z) t)) (* (/ 2.0 z) (/ x_m y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -5e+73) {
		tmp = x_m * (2.0 / (z * y));
	} else if (y <= 6.5e-52) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-5d+73)) then
        tmp = x_m * (2.0d0 / (z * y))
    else if (y <= 6.5d-52) then
        tmp = (-2.0d0) * ((x_m / z) / t)
    else
        tmp = (2.0d0 / z) * (x_m / y)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -5e+73) {
		tmp = x_m * (2.0 / (z * y));
	} else if (y <= 6.5e-52) {
		tmp = -2.0 * ((x_m / z) / t);
	} else {
		tmp = (2.0 / z) * (x_m / y);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -5e+73:
		tmp = x_m * (2.0 / (z * y))
	elif y <= 6.5e-52:
		tmp = -2.0 * ((x_m / z) / t)
	else:
		tmp = (2.0 / z) * (x_m / y)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -5e+73)
		tmp = Float64(x_m * Float64(2.0 / Float64(z * y)));
	elseif (y <= 6.5e-52)
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	else
		tmp = Float64(Float64(2.0 / z) * Float64(x_m / y));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -5e+73)
		tmp = x_m * (2.0 / (z * y));
	elseif (y <= 6.5e-52)
		tmp = -2.0 * ((x_m / z) / t);
	else
		tmp = (2.0 / z) * (x_m / y);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -5e+73], N[(x$95$m * N[(2.0 / N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e-52], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / z), $MachinePrecision] * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -4.99999999999999976e73

   1. Initial program 77.2%

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

      1. distribute-rgt-out-- 88.3%

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

   3. Simplified 88.3%

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

      1. distribute-rgt-out-- 77.2%

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

      2. associate-/l* 77.1%

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

      3. *-commutative 77.1%

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

      4. distribute-rgt-out-- 88.2%

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

   6. Applied egg-rr 88.2%

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

   7. Taylor expanded in y around inf 77.0%

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

      1. *-commutative 77.0%

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

   9. Simplified 77.0%

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

   if -4.99999999999999976e73 < y < 6.5e-52

   1. Initial program 92.3%

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

      1. distribute-rgt-out-- 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in y around 0 79.5%

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

      1. *-commutative 79.5%

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

      2. associate-/r* 80.7%

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

   7. Simplified 80.7%

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

   if 6.5e-52 < y

   1. Initial program 83.3%

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

      1. distribute-rgt-out-- 90.4%

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

   3. Simplified 90.4%

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

      1. *-commutative 90.4%

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

      2. times-frac 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in y around inf 80.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+73}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot y}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-52}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{z} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot 2 \leq 5 \cdot 10^{+78}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y - t} \cdot \frac{2}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m 2.0) 5e+78)
    (* (/ x_m z) (/ 2.0 (- y t)))
    (* (/ x_m (- y t)) (/ 2.0 z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e+78) {
		tmp = (x_m / z) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x_m * 2.0d0) <= 5d+78) then
        tmp = (x_m / z) * (2.0d0 / (y - t))
    else
        tmp = (x_m / (y - t)) * (2.0d0 / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((x_m * 2.0) <= 5e+78) {
		tmp = (x_m / z) * (2.0 / (y - t));
	} else {
		tmp = (x_m / (y - t)) * (2.0 / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (x_m * 2.0) <= 5e+78:
		tmp = (x_m / z) * (2.0 / (y - t))
	else:
		tmp = (x_m / (y - t)) * (2.0 / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(x_m * 2.0) <= 5e+78)
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / Float64(y - t)));
	else
		tmp = Float64(Float64(x_m / Float64(y - t)) * Float64(2.0 / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((x_m * 2.0) <= 5e+78)
		tmp = (x_m / z) * (2.0 / (y - t));
	else
		tmp = (x_m / (y - t)) * (2.0 / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(x$95$m * 2.0), $MachinePrecision], 5e+78], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y - t), $MachinePrecision]), $MachinePrecision] * N[(2.0 / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x #s(literal 2 binary64)) < 4.99999999999999984e78

   1. Initial program 86.9%

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

      1. distribute-rgt-out-- 92.8%

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

   3. Simplified 92.8%

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

      1. times-frac 95.2%

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

   6. Applied egg-rr 95.2%

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

   if 4.99999999999999984e78 < (*.f64 x #s(literal 2 binary64))

   1. Initial program 89.5%

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

      1. distribute-rgt-out-- 89.5%

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

   3. Simplified 89.5%

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

      1. *-commutative 89.5%

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

      2. times-frac 99.6%

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

   6. Applied egg-rr 99.6%

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

Alternative 9: 94.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-42}:\\ \;\;\;\;x\_m \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{2}{y - t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z 1.15e-42)
    (* x_m (/ 2.0 (* z (- y t))))
    (* (/ x_m z) (/ 2.0 (- y t))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.15e-42) {
		tmp = x_m * (2.0 / (z * (y - t)));
	} else {
		tmp = (x_m / z) * (2.0 / (y - t));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.15d-42) then
        tmp = x_m * (2.0d0 / (z * (y - t)))
    else
        tmp = (x_m / z) * (2.0d0 / (y - t))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.15e-42) {
		tmp = x_m * (2.0 / (z * (y - t)));
	} else {
		tmp = (x_m / z) * (2.0 / (y - t));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 1.15e-42:
		tmp = x_m * (2.0 / (z * (y - t)))
	else:
		tmp = (x_m / z) * (2.0 / (y - t))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 1.15e-42)
		tmp = Float64(x_m * Float64(2.0 / Float64(z * Float64(y - t))));
	else
		tmp = Float64(Float64(x_m / z) * Float64(2.0 / Float64(y - t)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 1.15e-42)
		tmp = x_m * (2.0 / (z * (y - t)));
	else
		tmp = (x_m / z) * (2.0 / (y - t));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 1.15e-42], N[(x$95$m * N[(2.0 / N[(z * N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(2.0 / N[(y - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.15000000000000002e-42

   1. Initial program 89.3%

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

      1. distribute-rgt-out-- 93.2%

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

   3. Simplified 93.2%

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

      1. distribute-rgt-out-- 89.3%

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

      2. associate-/l* 89.3%

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

      3. *-commutative 89.3%

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

      4. distribute-rgt-out-- 93.2%

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

   6. Applied egg-rr 93.2%

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

   if 1.15000000000000002e-42 < z

   1. Initial program 82.7%

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

      1. distribute-rgt-out-- 90.3%

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

   3. Simplified 90.3%

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

      1. times-frac 98.5%

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

   6. Applied egg-rr 98.5%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{-42}:\\ \;\;\;\;x \cdot \frac{2}{z \cdot \left(y - t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{2}{y - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-218}:\\ \;\;\;\;x\_m \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z 1.05e-218) (* x_m (/ -2.0 (* z t))) (* -2.0 (/ (/ x_m z) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.05e-218) {
		tmp = x_m * (-2.0 / (z * t));
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.05d-218) then
        tmp = x_m * ((-2.0d0) / (z * t))
    else
        tmp = (-2.0d0) * ((x_m / z) / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.05e-218) {
		tmp = x_m * (-2.0 / (z * t));
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 1.05e-218:
		tmp = x_m * (-2.0 / (z * t))
	else:
		tmp = -2.0 * ((x_m / z) / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 1.05e-218)
		tmp = Float64(x_m * Float64(-2.0 / Float64(z * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 1.05e-218)
		tmp = x_m * (-2.0 / (z * t));
	else
		tmp = -2.0 * ((x_m / z) / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 1.05e-218], N[(x$95$m * N[(-2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.04999999999999997e-218

   1. Initial program 88.1%

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

      1. distribute-rgt-out-- 92.0%

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

   3. Simplified 92.0%

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

      1. distribute-rgt-out-- 88.1%

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

      2. associate-/l* 88.0%

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

      3. *-commutative 88.0%

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

      4. distribute-rgt-out-- 92.0%

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

   6. Applied egg-rr 92.0%

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

   7. Taylor expanded in y around 0 62.4%

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

      1. *-commutative 62.4%

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

   9. Simplified 62.4%

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

   if 1.04999999999999997e-218 < z

   1. Initial program 86.1%

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

      1. distribute-rgt-out-- 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in y around 0 60.7%

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

      1. *-commutative 60.7%

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

      2. associate-/r* 63.9%

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

   7. Simplified 63.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-218}:\\ \;\;\;\;x \cdot \frac{-2}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.55 \cdot 10^{-218}:\\ \;\;\;\;-2 \cdot \frac{x\_m}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\frac{x\_m}{z}}{t}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z 1.55e-218) (* -2.0 (/ x_m (* z t))) (* -2.0 (/ (/ x_m z) t)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.55e-218) {
		tmp = -2.0 * (x_m / (z * t));
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 1.55d-218) then
        tmp = (-2.0d0) * (x_m / (z * t))
    else
        tmp = (-2.0d0) * ((x_m / z) / t)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= 1.55e-218) {
		tmp = -2.0 * (x_m / (z * t));
	} else {
		tmp = -2.0 * ((x_m / z) / t);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= 1.55e-218:
		tmp = -2.0 * (x_m / (z * t))
	else:
		tmp = -2.0 * ((x_m / z) / t)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= 1.55e-218)
		tmp = Float64(-2.0 * Float64(x_m / Float64(z * t)));
	else
		tmp = Float64(-2.0 * Float64(Float64(x_m / z) / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= 1.55e-218)
		tmp = -2.0 * (x_m / (z * t));
	else
		tmp = -2.0 * ((x_m / z) / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, 1.55e-218], N[(-2.0 * N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.54999999999999999e-218

   1. Initial program 88.1%

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

      1. distribute-rgt-out-- 92.0%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in y around 0 62.4%

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

      1. *-commutative 62.4%

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

   7. Simplified 62.4%

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

   if 1.54999999999999999e-218 < z

   1. Initial program 86.1%

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

      1. distribute-rgt-out-- 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in y around 0 60.7%

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

      1. *-commutative 60.7%

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

      2. associate-/r* 63.9%

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

   7. Simplified 63.9%

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

Alternative 12: 92.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. distribute-rgt-out-- 92.3%

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

3. Simplified 92.3%

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

   1. distribute-rgt-out-- 87.3%

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

   2. associate-/l* 87.2%

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

   3. *-commutative 87.2%

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

   4. distribute-rgt-out-- 92.3%

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

6. Applied egg-rr 92.3%

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

7. Final simplification 92.3%

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

Alternative 13: 53.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

   1. distribute-rgt-out-- 92.3%

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

3. Simplified 92.3%

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

5. Taylor expanded in y around 0 61.7%

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

   1. *-commutative 61.7%

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

7. Simplified 61.7%

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

Developer Target 1: 96.9% 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 2024132 
(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))))