Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B

Percentage Accurate: 82.0% → 99.6%

Time: 5.1s

Alternatives: 6

Speedup: 1.8×

Specification

Math

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

FPCore

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

C

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

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 * (((y / z) * t) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (((y / z) * t) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t] = \mathsf{sort}([x_m, y_m, z_m, t])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y_m}{z_m} \leq 2 \cdot 10^{+235}:\\ \;\;\;\;\frac{x_m}{\frac{z_m}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z_m}{y_m \cdot x_m}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ y_m z_m) 2e+235)
      (/ x_m (/ z_m y_m))
      (/ 1.0 (/ z_m (* y_m x_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	double tmp;
	if ((y_m / z_m) <= 2e+235) {
		tmp = x_m / (z_m / y_m);
	} else {
		tmp = 1.0 / (z_m / (y_m * x_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y_m / z_m) <= 2d+235) then
        tmp = x_m / (z_m / y_m)
    else
        tmp = 1.0d0 / (z_m / (y_m * x_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	double tmp;
	if ((y_m / z_m) <= 2e+235) {
		tmp = x_m / (z_m / y_m);
	} else {
		tmp = 1.0 / (z_m / (y_m * x_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t] = sort([x_m, y_m, z_m, t])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t):
	tmp = 0
	if (y_m / z_m) <= 2e+235:
		tmp = x_m / (z_m / y_m)
	else:
		tmp = 1.0 / (z_m / (y_m * x_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y_m, z_m, t = sort([x_m, y_m, z_m, t])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	tmp = 0.0
	if (Float64(y_m / z_m) <= 2e+235)
		tmp = Float64(x_m / Float64(z_m / y_m));
	else
		tmp = Float64(1.0 / Float64(z_m / Float64(y_m * x_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t = num2cell(sort([x_m, y_m, z_m, t])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	tmp = 0.0;
	if ((y_m / z_m) <= 2e+235)
		tmp = x_m / (z_m / y_m);
	else
		tmp = 1.0 / (z_m / (y_m * x_m));
	end
	tmp_2 = z_s * (y_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]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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]
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m / z$95$m), $MachinePrecision], 2e+235], N[(x$95$m / N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(z$95$m / N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 y z) < 2.0000000000000001e235

   1. Initial program 81.4%

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

      1. associate-/l* 91.8%

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

      2. associate-*r/ 91.8%

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

      3. *-commutative 91.8%

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

      4. *-inverses 91.8%

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

      5. /-rgt-identity 91.8%

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

      6. *-commutative 91.8%

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

   3. Simplified 91.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 93.4%

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

      1. associate-/l* 91.3%

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

   7. Simplified 91.3%

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

   if 2.0000000000000001e235 < (/.f64 y z)

   1. Initial program 81.1%

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

      1. associate-/l* 90.4%

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

      2. associate-*r/ 90.4%

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

      3. *-commutative 90.4%

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

      4. *-inverses 90.4%

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

      5. /-rgt-identity 90.4%

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

      6. *-commutative 90.4%

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

   3. Simplified 90.4%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.8%

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

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq 2 \cdot 10^{+235}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t] = \mathsf{sort}([x_m, y_m, z_m, t])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y_m}{z_m} \leq 10^{+293}:\\ \;\;\;\;\frac{x_m}{\frac{z_m}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \frac{x_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ y_m z_m) 1e+293) (/ x_m (/ z_m y_m)) (* y_m (/ x_m z_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	double tmp;
	if ((y_m / z_m) <= 1e+293) {
		tmp = x_m / (z_m / y_m);
	} else {
		tmp = y_m * (x_m / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y_m / z_m) <= 1d+293) then
        tmp = x_m / (z_m / y_m)
    else
        tmp = y_m * (x_m / z_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	double tmp;
	if ((y_m / z_m) <= 1e+293) {
		tmp = x_m / (z_m / y_m);
	} else {
		tmp = y_m * (x_m / z_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t] = sort([x_m, y_m, z_m, t])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t):
	tmp = 0
	if (y_m / z_m) <= 1e+293:
		tmp = x_m / (z_m / y_m)
	else:
		tmp = y_m * (x_m / z_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y_m, z_m, t = sort([x_m, y_m, z_m, t])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	tmp = 0.0
	if (Float64(y_m / z_m) <= 1e+293)
		tmp = Float64(x_m / Float64(z_m / y_m));
	else
		tmp = Float64(y_m * Float64(x_m / z_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t = num2cell(sort([x_m, y_m, z_m, t])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	tmp = 0.0;
	if ((y_m / z_m) <= 1e+293)
		tmp = x_m / (z_m / y_m);
	else
		tmp = y_m * (x_m / z_m);
	end
	tmp_2 = z_s * (y_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]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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]
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m / z$95$m), $MachinePrecision], 1e+293], N[(x$95$m / N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 y z) < 9.9999999999999992e292

   1. Initial program 80.9%

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

      1. associate-/l* 91.9%

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

      2. associate-*r/ 91.9%

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

      3. *-commutative 91.9%

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

      4. *-inverses 91.9%

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

      5. /-rgt-identity 91.9%

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

      6. *-commutative 91.9%

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

   3. Simplified 91.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 93.5%

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

      1. associate-/l* 91.5%

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

   7. Simplified 91.5%

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

   if 9.9999999999999992e292 < (/.f64 y z)

   1. Initial program 88.2%

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

      1. associate-/l* 88.2%

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

      2. associate-*r/ 88.2%

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

      3. *-commutative 88.2%

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

      4. *-inverses 88.2%

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

      5. /-rgt-identity 88.2%

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

      6. *-commutative 88.2%

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

   3. Simplified 88.2%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

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

      1. associate-*l/ 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq 10^{+293}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t] = \mathsf{sort}([x_m, y_m, z_m, t])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y_m}{z_m} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\frac{x_m}{\frac{z_m}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{\frac{z_m}{x_m}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ y_m z_m) 2e+286) (/ x_m (/ z_m y_m)) (/ y_m (/ z_m x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	double tmp;
	if ((y_m / z_m) <= 2e+286) {
		tmp = x_m / (z_m / y_m);
	} else {
		tmp = y_m / (z_m / x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y_m / z_m) <= 2d+286) then
        tmp = x_m / (z_m / y_m)
    else
        tmp = y_m / (z_m / x_m)
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	double tmp;
	if ((y_m / z_m) <= 2e+286) {
		tmp = x_m / (z_m / y_m);
	} else {
		tmp = y_m / (z_m / x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t] = sort([x_m, y_m, z_m, t])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t):
	tmp = 0
	if (y_m / z_m) <= 2e+286:
		tmp = x_m / (z_m / y_m)
	else:
		tmp = y_m / (z_m / x_m)
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y_m, z_m, t = sort([x_m, y_m, z_m, t])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	tmp = 0.0
	if (Float64(y_m / z_m) <= 2e+286)
		tmp = Float64(x_m / Float64(z_m / y_m));
	else
		tmp = Float64(y_m / Float64(z_m / x_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t = num2cell(sort([x_m, y_m, z_m, t])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	tmp = 0.0;
	if ((y_m / z_m) <= 2e+286)
		tmp = x_m / (z_m / y_m);
	else
		tmp = y_m / (z_m / x_m);
	end
	tmp_2 = z_s * (y_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]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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]
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m / z$95$m), $MachinePrecision], 2e+286], N[(x$95$m / N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(z$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 y z) < 2.00000000000000007e286

   1. Initial program 80.9%

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

      1. associate-/l* 91.9%

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

      2. associate-*r/ 91.9%

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

      3. *-commutative 91.9%

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

      4. *-inverses 91.9%

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

      5. /-rgt-identity 91.9%

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

      6. *-commutative 91.9%

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

   3. Simplified 91.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 93.5%

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

      1. associate-/l* 91.5%

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

   7. Simplified 91.5%

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

   if 2.00000000000000007e286 < (/.f64 y z)

   1. Initial program 88.2%

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

      1. associate-/l* 88.2%

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

      2. *-inverses 88.2%

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

      3. clear-num 88.2%

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

      4. clear-num 88.2%

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

      5. associate-/r/ 88.2%

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

   4. Applied egg-rr 88.2%

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

      1. *-commutative 88.2%

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

      2. associate-*r* 99.9%

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

      3. div-inv 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/l* 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq 2 \cdot 10^{+286}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t] = \mathsf{sort}([x_m, y_m, z_m, t])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y_m}{z_m} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{x_m}{\frac{z_m}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m \cdot x_m}{z_m}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ y_m z_m) 2e+299) (/ x_m (/ z_m y_m)) (/ (* y_m x_m) z_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	double tmp;
	if ((y_m / z_m) <= 2e+299) {
		tmp = x_m / (z_m / y_m);
	} else {
		tmp = (y_m * x_m) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y_m / z_m) <= 2d+299) then
        tmp = x_m / (z_m / y_m)
    else
        tmp = (y_m * x_m) / z_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	double tmp;
	if ((y_m / z_m) <= 2e+299) {
		tmp = x_m / (z_m / y_m);
	} else {
		tmp = (y_m * x_m) / z_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t] = sort([x_m, y_m, z_m, t])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t):
	tmp = 0
	if (y_m / z_m) <= 2e+299:
		tmp = x_m / (z_m / y_m)
	else:
		tmp = (y_m * x_m) / z_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y_m, z_m, t = sort([x_m, y_m, z_m, t])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	tmp = 0.0
	if (Float64(y_m / z_m) <= 2e+299)
		tmp = Float64(x_m / Float64(z_m / y_m));
	else
		tmp = Float64(Float64(y_m * x_m) / z_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t = num2cell(sort([x_m, y_m, z_m, t])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	tmp = 0.0;
	if ((y_m / z_m) <= 2e+299)
		tmp = x_m / (z_m / y_m);
	else
		tmp = (y_m * x_m) / z_m;
	end
	tmp_2 = z_s * (y_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]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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]
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m / z$95$m), $MachinePrecision], 2e+299], N[(x$95$m / N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 y z) < 2.0000000000000001e299

   1. Initial program 81.1%

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

      1. associate-/l* 92.0%

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

      2. associate-*r/ 92.0%

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

      3. *-commutative 92.0%

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

      4. *-inverses 92.0%

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

      5. /-rgt-identity 92.0%

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

      6. *-commutative 92.0%

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

   3. Simplified 92.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 93.6%

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

      1. associate-/l* 91.6%

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

   7. Simplified 91.6%

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

   if 2.0000000000000001e299 < (/.f64 y z)

   1. Initial program 85.6%

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

      1. associate-/l* 85.6%

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

      2. associate-*r/ 85.6%

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

      3. *-commutative 85.6%

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

      4. *-inverses 85.6%

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

      5. /-rgt-identity 85.6%

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

      6. *-commutative 85.6%

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

   3. Simplified 85.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t] = \mathsf{sort}([x_m, y_m, z_m, t])\\ \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 10^{+32}:\\ \;\;\;\;y_m \cdot \frac{x_m}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{z_m} \cdot x_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t)
 :precision binary64
 (*
  z_s
  (* y_s (* x_s (if (<= z_m 1e+32) (* y_m (/ x_m z_m)) (* (/ y_m z_m) x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	double tmp;
	if (z_m <= 1e+32) {
		tmp = y_m * (x_m / z_m);
	} else {
		tmp = (y_m / z_m) * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z_m <= 1d+32) then
        tmp = y_m * (x_m / z_m)
    else
        tmp = (y_m / z_m) * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	double tmp;
	if (z_m <= 1e+32) {
		tmp = y_m * (x_m / z_m);
	} else {
		tmp = (y_m / z_m) * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t] = sort([x_m, y_m, z_m, t])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t):
	tmp = 0
	if z_m <= 1e+32:
		tmp = y_m * (x_m / z_m)
	else:
		tmp = (y_m / z_m) * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y_m, z_m, t = sort([x_m, y_m, z_m, t])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	tmp = 0.0
	if (z_m <= 1e+32)
		tmp = Float64(y_m * Float64(x_m / z_m));
	else
		tmp = Float64(Float64(y_m / z_m) * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t = num2cell(sort([x_m, y_m, z_m, t])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	tmp = 0.0;
	if (z_m <= 1e+32)
		tmp = y_m * (x_m / z_m);
	else
		tmp = (y_m / z_m) * x_m;
	end
	tmp_2 = z_s * (y_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]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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]
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1e+32], N[(y$95$m * N[(x$95$m / z$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.00000000000000005e32

   1. Initial program 82.1%

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

      1. associate-/l* 92.4%

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

      2. associate-*r/ 92.4%

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

      3. *-commutative 92.4%

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

      4. *-inverses 92.4%

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

      5. /-rgt-identity 92.4%

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

      6. *-commutative 92.4%

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

   3. Simplified 92.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 93.3%

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

      1. associate-*l/ 94.1%

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

   7. Applied egg-rr 94.1%

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

   if 1.00000000000000005e32 < z

   1. Initial program 78.7%

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

      1. associate-/l* 89.2%

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

      2. associate-*r/ 89.2%

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

      3. *-commutative 89.2%

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

      4. *-inverses 89.2%

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

      5. /-rgt-identity 89.2%

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

      6. *-commutative 89.2%

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

   3. Simplified 89.2%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+32}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.0% accurate, 1.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t)
 :precision binary64
 (* z_s (* y_s (* x_s (* (/ y_m z_m) x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	return z_s * (y_s * (x_s * ((y_m / z_m) * x_m)));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    code = z_s * (y_s * (x_s * ((y_m / z_m) * x_m)))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t) {
	return z_s * (y_s * (x_s * ((y_m / z_m) * x_m)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t] = sort([x_m, y_m, z_m, t])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t):
	return z_s * (y_s * (x_s * ((y_m / z_m) * x_m)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y_m, z_m, t = sort([x_m, y_m, z_m, t])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	return Float64(z_s * Float64(y_s * Float64(x_s * Float64(Float64(y_m / z_m) * x_m))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t = num2cell(sort([x_m, y_m, z_m, t])){:}
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t)
	tmp = z_s * (y_s * (x_s * ((y_m / z_m) * x_m)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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]
NOTE: x_m, y_m, z_m, and t should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(N[(y$95$m / z$95$m), $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.4%

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

   1. associate-/l* 91.7%

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

   2. associate-*r/ 91.7%

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

   3. *-commutative 91.7%

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

   4. *-inverses 91.7%

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

   5. /-rgt-identity 91.7%

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

   6. *-commutative 91.7%

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

3. Simplified 91.7%

   \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]
4. Add Preprocessing

5. Final simplification 91.7%

   \[\leadsto \frac{y}{z} \cdot x \]
6. Add Preprocessing

Developer target: 96.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ t_2 := \frac{\frac{y}{z} \cdot t}{t}\\ t_3 := \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;t_2 < -1.20672205123045 \cdot 10^{+245}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 < -5.907522236933906 \cdot 10^{-275}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_2 < 5.658954423153415 \cdot 10^{-65}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_2 < 2.0087180502407133 \cdot 10^{+217}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ y z))) (t_2 (/ (* (/ y z) t) t)) (t_3 (/ y (/ z x))))
   (if (< t_2 -1.20672205123045e+245)
     t_3
     (if (< t_2 -5.907522236933906e-275)
       t_1
       (if (< t_2 5.658954423153415e-65)
         t_3
         (if (< t_2 2.0087180502407133e+217) t_1 (/ (* y x) z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (y / z);
	double t_2 = ((y / z) * t) / t;
	double t_3 = y / (z / x);
	double tmp;
	if (t_2 < -1.20672205123045e+245) {
		tmp = t_3;
	} else if (t_2 < -5.907522236933906e-275) {
		tmp = t_1;
	} else if (t_2 < 5.658954423153415e-65) {
		tmp = t_3;
	} else if (t_2 < 2.0087180502407133e+217) {
		tmp = t_1;
	} else {
		tmp = (y * x) / z;
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = x * (y / z)
    t_2 = ((y / z) * t) / t
    t_3 = y / (z / x)
    if (t_2 < (-1.20672205123045d+245)) then
        tmp = t_3
    else if (t_2 < (-5.907522236933906d-275)) then
        tmp = t_1
    else if (t_2 < 5.658954423153415d-65) then
        tmp = t_3
    else if (t_2 < 2.0087180502407133d+217) then
        tmp = t_1
    else
        tmp = (y * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (y / z);
	double t_2 = ((y / z) * t) / t;
	double t_3 = y / (z / x);
	double tmp;
	if (t_2 < -1.20672205123045e+245) {
		tmp = t_3;
	} else if (t_2 < -5.907522236933906e-275) {
		tmp = t_1;
	} else if (t_2 < 5.658954423153415e-65) {
		tmp = t_3;
	} else if (t_2 < 2.0087180502407133e+217) {
		tmp = t_1;
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (y / z)
	t_2 = ((y / z) * t) / t
	t_3 = y / (z / x)
	tmp = 0
	if t_2 < -1.20672205123045e+245:
		tmp = t_3
	elif t_2 < -5.907522236933906e-275:
		tmp = t_1
	elif t_2 < 5.658954423153415e-65:
		tmp = t_3
	elif t_2 < 2.0087180502407133e+217:
		tmp = t_1
	else:
		tmp = (y * x) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y / z))
	t_2 = Float64(Float64(Float64(y / z) * t) / t)
	t_3 = Float64(y / Float64(z / x))
	tmp = 0.0
	if (t_2 < -1.20672205123045e+245)
		tmp = t_3;
	elseif (t_2 < -5.907522236933906e-275)
		tmp = t_1;
	elseif (t_2 < 5.658954423153415e-65)
		tmp = t_3;
	elseif (t_2 < 2.0087180502407133e+217)
		tmp = t_1;
	else
		tmp = Float64(Float64(y * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (y / z);
	t_2 = ((y / z) * t) / t;
	t_3 = y / (z / x);
	tmp = 0.0;
	if (t_2 < -1.20672205123045e+245)
		tmp = t_3;
	elseif (t_2 < -5.907522236933906e-275)
		tmp = t_1;
	elseif (t_2 < 5.658954423153415e-65)
		tmp = t_3;
	elseif (t_2 < 2.0087180502407133e+217)
		tmp = t_1;
	else
		tmp = (y * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y / z), $MachinePrecision] * t), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$3 = N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -1.20672205123045e+245], t$95$3, If[Less[t$95$2, -5.907522236933906e-275], t$95$1, If[Less[t$95$2, 5.658954423153415e-65], t$95$3, If[Less[t$95$2, 2.0087180502407133e+217], t$95$1, N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]]]]]

Reproduce

herbie shell --seed 2024019 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

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