Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3

Percentage Accurate: 84.8% → 97.9%

Time: 7.1s

Alternatives: 9

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(y - z\right)}{y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(y - z\right)}{y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{+20}:\\ \;\;\;\;x\_m - z \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 2e+20) (- x_m (* z (/ x_m y))) (* x_m (- 1.0 (/ z 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 tmp;
	if (x_m <= 2e+20) {
		tmp = x_m - (z * (x_m / y));
	} else {
		tmp = x_m * (1.0 - (z / 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)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 2d+20) then
        tmp = x_m - (z * (x_m / y))
    else
        tmp = x_m * (1.0d0 - (z / 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 tmp;
	if (x_m <= 2e+20) {
		tmp = x_m - (z * (x_m / y));
	} else {
		tmp = x_m * (1.0 - (z / 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):
	tmp = 0
	if x_m <= 2e+20:
		tmp = x_m - (z * (x_m / y))
	else:
		tmp = x_m * (1.0 - (z / y))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2e+20)
		tmp = Float64(x_m - Float64(z * Float64(x_m / y)));
	else
		tmp = Float64(x_m * Float64(1.0 - Float64(z / 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)
	tmp = 0.0;
	if (x_m <= 2e+20)
		tmp = x_m - (z * (x_m / y));
	else
		tmp = x_m * (1.0 - (z / 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_] := N[(x$95$s * If[LessEqual[x$95$m, 2e+20], N[(x$95$m - N[(z * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e20

   1. Initial program 82.7%

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

      1. remove-double-neg 82.7%

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

      2. distribute-frac-neg2 82.7%

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

      3. distribute-frac-neg 82.7%

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

      4. distribute-rgt-neg-in 82.7%

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

      5. associate-/l* 95.2%

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

      6. distribute-frac-neg 95.2%

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

      7. distribute-frac-neg2 95.2%

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

      8. remove-double-neg 95.2%

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

      9. div-sub 95.2%

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

      10. *-inverses 95.2%

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

   3. Simplified 95.2%

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

      1. sub-neg 95.2%

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

      2. distribute-rgt-in 95.2%

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

      3. *-un-lft-identity 95.2%

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

      4. distribute-neg-frac2 95.2%

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

   6. Applied egg-rr 95.2%

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

      1. *-commutative 95.2%

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

      2. add-sqr-sqrt 28.8%

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

      3. sqrt-unprod 47.2%

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

      4. sqr-neg 47.2%

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

      5. sqrt-unprod 33.0%

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

      6. add-sqr-sqrt 51.8%

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

      7. cancel-sign-sub-inv 51.8%

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

      8. *-commutative 51.8%

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

      9. associate-*l/ 50.4%

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

      10. associate-/l* 50.5%

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

      11. add-sqr-sqrt 25.6%

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

      12. sqrt-unprod 63.7%

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

      13. sqr-neg 63.7%

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

      14. sqrt-unprod 46.7%

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

      15. add-sqr-sqrt 95.1%

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

   8. Applied egg-rr 95.1%

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

   if 2e20 < x

   1. Initial program 76.1%

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

      1. remove-double-neg 76.1%

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

      2. distribute-frac-neg2 76.1%

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

      3. distribute-frac-neg 76.1%

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

      4. distribute-rgt-neg-in 76.1%

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

      5. associate-/l* 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. remove-double-neg 99.9%

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

      9. div-sub 99.9%

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

      10. *-inverses 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{+20}:\\ \;\;\;\;x - z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 71.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+33} \lor \neg \left(z \leq -1.65 \cdot 10^{-6}\right) \land \left(z \leq -5.6 \cdot 10^{-46} \lor \neg \left(z \leq 7.2 \cdot 10^{-51}\right)\right):\\ \;\;\;\;x\_m \cdot \frac{z}{-y}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= z -1.26e+33)
          (and (not (<= z -1.65e-6))
               (or (<= z -5.6e-46) (not (<= z 7.2e-51)))))
    (* x_m (/ z (- y)))
    x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -1.26e+33) || (!(z <= -1.65e-6) && ((z <= -5.6e-46) || !(z <= 7.2e-51)))) {
		tmp = x_m * (z / -y);
	} else {
		tmp = x_m;
	}
	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)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.26d+33)) .or. (.not. (z <= (-1.65d-6))) .and. (z <= (-5.6d-46)) .or. (.not. (z <= 7.2d-51))) then
        tmp = x_m * (z / -y)
    else
        tmp = x_m
    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 tmp;
	if ((z <= -1.26e+33) || (!(z <= -1.65e-6) && ((z <= -5.6e-46) || !(z <= 7.2e-51)))) {
		tmp = x_m * (z / -y);
	} else {
		tmp = x_m;
	}
	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):
	tmp = 0
	if (z <= -1.26e+33) or (not (z <= -1.65e-6) and ((z <= -5.6e-46) or not (z <= 7.2e-51))):
		tmp = x_m * (z / -y)
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -1.26e+33) || (!(z <= -1.65e-6) && ((z <= -5.6e-46) || !(z <= 7.2e-51))))
		tmp = Float64(x_m * Float64(z / Float64(-y)));
	else
		tmp = x_m;
	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)
	tmp = 0.0;
	if ((z <= -1.26e+33) || (~((z <= -1.65e-6)) && ((z <= -5.6e-46) || ~((z <= 7.2e-51)))))
		tmp = x_m * (z / -y);
	else
		tmp = x_m;
	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_] := N[(x$95$s * If[Or[LessEqual[z, -1.26e+33], And[N[Not[LessEqual[z, -1.65e-6]], $MachinePrecision], Or[LessEqual[z, -5.6e-46], N[Not[LessEqual[z, 7.2e-51]], $MachinePrecision]]]], N[(x$95$m * N[(z / (-y)), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.26e33 or -1.65000000000000008e-6 < z < -5.5999999999999997e-46 or 7.2000000000000001e-51 < z

   1. Initial program 88.4%

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

      1. remove-double-neg 88.4%

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

      2. distribute-frac-neg2 88.4%

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

      3. distribute-frac-neg 88.4%

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

      4. distribute-rgt-neg-in 88.4%

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

      5. associate-/l* 92.7%

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

      6. distribute-frac-neg 92.7%

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

      7. distribute-frac-neg2 92.7%

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

      8. remove-double-neg 92.7%

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

      9. div-sub 92.8%

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

      10. *-inverses 92.8%

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

   3. Simplified 92.8%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 69.9%

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

      1. mul-1-neg 69.9%

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

      2. distribute-frac-neg2 69.9%

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

   7. Simplified 69.9%

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

   if -1.26e33 < z < -1.65000000000000008e-6 or -5.5999999999999997e-46 < z < 7.2000000000000001e-51

   1. Initial program 73.8%

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

      1. remove-double-neg 73.8%

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

      2. distribute-frac-neg2 73.8%

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

      3. distribute-frac-neg 73.8%

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

      4. distribute-rgt-neg-in 73.8%

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

      5. associate-/l* 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. remove-double-neg 99.9%

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

      9. div-sub 99.9%

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

      10. *-inverses 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 83.9%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{+33} \lor \neg \left(z \leq -1.65 \cdot 10^{-6}\right) \land \left(z \leq -5.6 \cdot 10^{-46} \lor \neg \left(z \leq 7.2 \cdot 10^{-51}\right)\right):\\ \;\;\;\;x \cdot \frac{z}{-y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8.4 \cdot 10^{+30} \lor \neg \left(z \leq -1.52 \cdot 10^{-6} \lor \neg \left(z \leq -1.15 \cdot 10^{-44}\right) \land z \leq 1.05 \cdot 10^{-50}\right):\\ \;\;\;\;z \cdot \frac{x\_m}{-y}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (or (<= z -8.4e+30)
          (not
           (or (<= z -1.52e-6) (and (not (<= z -1.15e-44)) (<= z 1.05e-50)))))
    (* z (/ x_m (- y)))
    x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((z <= -8.4e+30) || !((z <= -1.52e-6) || (!(z <= -1.15e-44) && (z <= 1.05e-50)))) {
		tmp = z * (x_m / -y);
	} else {
		tmp = x_m;
	}
	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)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-8.4d+30)) .or. (.not. (z <= (-1.52d-6)) .or. (.not. (z <= (-1.15d-44))) .and. (z <= 1.05d-50))) then
        tmp = z * (x_m / -y)
    else
        tmp = x_m
    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 tmp;
	if ((z <= -8.4e+30) || !((z <= -1.52e-6) || (!(z <= -1.15e-44) && (z <= 1.05e-50)))) {
		tmp = z * (x_m / -y);
	} else {
		tmp = x_m;
	}
	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):
	tmp = 0
	if (z <= -8.4e+30) or not ((z <= -1.52e-6) or (not (z <= -1.15e-44) and (z <= 1.05e-50))):
		tmp = z * (x_m / -y)
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if ((z <= -8.4e+30) || !((z <= -1.52e-6) || (!(z <= -1.15e-44) && (z <= 1.05e-50))))
		tmp = Float64(z * Float64(x_m / Float64(-y)));
	else
		tmp = x_m;
	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)
	tmp = 0.0;
	if ((z <= -8.4e+30) || ~(((z <= -1.52e-6) || (~((z <= -1.15e-44)) && (z <= 1.05e-50)))))
		tmp = z * (x_m / -y);
	else
		tmp = x_m;
	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_] := N[(x$95$s * If[Or[LessEqual[z, -8.4e+30], N[Not[Or[LessEqual[z, -1.52e-6], And[N[Not[LessEqual[z, -1.15e-44]], $MachinePrecision], LessEqual[z, 1.05e-50]]]], $MachinePrecision]], N[(z * N[(x$95$m / (-y)), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -8.4000000000000001e30 or -1.52000000000000006e-6 < z < -1.14999999999999999e-44 or 1.05e-50 < z

   1. Initial program 88.4%

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

      1. remove-double-neg 88.4%

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

      2. distribute-frac-neg2 88.4%

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

      3. distribute-frac-neg 88.4%

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

      4. distribute-rgt-neg-in 88.4%

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

      5. associate-/l* 92.7%

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

      6. distribute-frac-neg 92.7%

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

      7. distribute-frac-neg2 92.7%

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

      8. remove-double-neg 92.7%

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

      9. div-sub 92.8%

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

      10. *-inverses 92.8%

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

   3. Simplified 92.8%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 75.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. associate-*l/ 74.1%

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

      2. associate-*l* 74.1%

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

      3. *-commutative 74.1%

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

      4. associate-*r/ 74.1%

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

      5. mul-1-neg 74.1%

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

   7. Simplified 74.1%

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

   if -8.4000000000000001e30 < z < -1.52000000000000006e-6 or -1.14999999999999999e-44 < z < 1.05e-50

   1. Initial program 73.8%

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

      1. remove-double-neg 73.8%

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

      2. distribute-frac-neg2 73.8%

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

      3. distribute-frac-neg 73.8%

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

      4. distribute-rgt-neg-in 73.8%

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

      5. associate-/l* 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. remove-double-neg 99.9%

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

      9. div-sub 99.9%

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

      10. *-inverses 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 83.9%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.4 \cdot 10^{+30} \lor \neg \left(z \leq -1.52 \cdot 10^{-6} \lor \neg \left(z \leq -1.15 \cdot 10^{-44}\right) \land z \leq 1.05 \cdot 10^{-50}\right):\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := z \cdot \frac{x\_m}{-y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-45}:\\ \;\;\;\;\frac{z}{\frac{y}{-x\_m}}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-53}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* z (/ x_m (- y)))))
   (*
    x_s
    (if (<= z -1.16e+36)
      t_0
      (if (<= z -9.5e-8)
        x_m
        (if (<= z -2e-45) (/ z (/ y (- x_m))) (if (<= z 3.9e-53) x_m t_0)))))))

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_0 = z * (x_m / -y);
	double tmp;
	if (z <= -1.16e+36) {
		tmp = t_0;
	} else if (z <= -9.5e-8) {
		tmp = x_m;
	} else if (z <= -2e-45) {
		tmp = z / (y / -x_m);
	} else if (z <= 3.9e-53) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	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)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * (x_m / -y)
    if (z <= (-1.16d+36)) then
        tmp = t_0
    else if (z <= (-9.5d-8)) then
        tmp = x_m
    else if (z <= (-2d-45)) then
        tmp = z / (y / -x_m)
    else if (z <= 3.9d-53) then
        tmp = x_m
    else
        tmp = t_0
    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_0 = z * (x_m / -y);
	double tmp;
	if (z <= -1.16e+36) {
		tmp = t_0;
	} else if (z <= -9.5e-8) {
		tmp = x_m;
	} else if (z <= -2e-45) {
		tmp = z / (y / -x_m);
	} else if (z <= 3.9e-53) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	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_0 = z * (x_m / -y)
	tmp = 0
	if z <= -1.16e+36:
		tmp = t_0
	elif z <= -9.5e-8:
		tmp = x_m
	elif z <= -2e-45:
		tmp = z / (y / -x_m)
	elif z <= 3.9e-53:
		tmp = x_m
	else:
		tmp = t_0
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(z * Float64(x_m / Float64(-y)))
	tmp = 0.0
	if (z <= -1.16e+36)
		tmp = t_0;
	elseif (z <= -9.5e-8)
		tmp = x_m;
	elseif (z <= -2e-45)
		tmp = Float64(z / Float64(y / Float64(-x_m)));
	elseif (z <= 3.9e-53)
		tmp = x_m;
	else
		tmp = t_0;
	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_0 = z * (x_m / -y);
	tmp = 0.0;
	if (z <= -1.16e+36)
		tmp = t_0;
	elseif (z <= -9.5e-8)
		tmp = x_m;
	elseif (z <= -2e-45)
		tmp = z / (y / -x_m);
	elseif (z <= 3.9e-53)
		tmp = x_m;
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(z * N[(x$95$m / (-y)), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.16e+36], t$95$0, If[LessEqual[z, -9.5e-8], x$95$m, If[LessEqual[z, -2e-45], N[(z / N[(y / (-x$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.9e-53], x$95$m, t$95$0]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15999999999999998e36 or 3.9000000000000002e-53 < z

   1. Initial program 87.8%

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

      1. remove-double-neg 87.8%

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

      2. distribute-frac-neg2 87.8%

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

      3. distribute-frac-neg 87.8%

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

      4. distribute-rgt-neg-in 87.8%

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

      5. associate-/l* 92.3%

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

      6. distribute-frac-neg 92.3%

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

      7. distribute-frac-neg2 92.3%

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

      8. remove-double-neg 92.3%

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

      9. div-sub 92.4%

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

      10. *-inverses 92.4%

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

   3. Simplified 92.4%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 75.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. associate-*l/ 73.4%

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

      2. associate-*l* 73.4%

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

      3. *-commutative 73.4%

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

      4. associate-*r/ 73.4%

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

      5. mul-1-neg 73.4%

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

   7. Simplified 73.4%

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

   if -1.15999999999999998e36 < z < -9.50000000000000036e-8 or -1.99999999999999997e-45 < z < 3.9000000000000002e-53

   1. Initial program 73.8%

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

      1. remove-double-neg 73.8%

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

      2. distribute-frac-neg2 73.8%

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

      3. distribute-frac-neg 73.8%

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

      4. distribute-rgt-neg-in 73.8%

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

      5. associate-/l* 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. remove-double-neg 99.9%

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

      9. div-sub 99.9%

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

      10. *-inverses 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 83.9%

      \[\leadsto \color{blue}{x} \]

   if -9.50000000000000036e-8 < z < -1.99999999999999997e-45

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 85.3%

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

      1. associate-*r* 85.3%

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

      2. mul-1-neg 85.3%

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

   5. Simplified 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-/l* 85.6%

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

      3. add-sqr-sqrt 56.2%

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

      4. sqrt-unprod 29.8%

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

      5. sqr-neg 29.8%

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

      6. sqrt-unprod 0.8%

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

      7. add-sqr-sqrt 1.5%

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

   7. Applied egg-rr 1.5%

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

      1. add-sqr-sqrt 0.9%

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

      2. sqrt-unprod 57.7%

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

      3. swap-sqr 57.5%

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

      4. sqr-neg 57.5%

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

      5. distribute-frac-neg 57.5%

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

      6. distribute-frac-neg 57.5%

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

      7. swap-sqr 57.7%

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

      8. sqrt-unprod 57.1%

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

      9. add-sqr-sqrt 85.6%

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

      10. distribute-frac-neg 85.6%

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

      11. distribute-rgt-neg-in 85.6%

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

      12. distribute-lft-neg-in 85.6%

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

      13. clear-num 85.8%

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

      14. un-div-inv 85.8%

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

   9. Applied egg-rr 85.8%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.16 \cdot 10^{+36}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{elif}\;z \leq -9.5 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-45}:\\ \;\;\;\;\frac{z}{\frac{y}{-x}}\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{-53}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{x\_m \cdot \left(-z\right)}{y}\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-7}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-47}:\\ \;\;\;\;\frac{z}{\frac{y}{-x\_m}}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x\_m}{-y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -6.5e+40)
    (/ (* x_m (- z)) y)
    (if (<= z -3.7e-7)
      x_m
      (if (<= z -1.15e-47)
        (/ z (/ y (- x_m)))
        (if (<= z 1.05e-50) x_m (* 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 tmp;
	if (z <= -6.5e+40) {
		tmp = (x_m * -z) / y;
	} else if (z <= -3.7e-7) {
		tmp = x_m;
	} else if (z <= -1.15e-47) {
		tmp = z / (y / -x_m);
	} else if (z <= 1.05e-50) {
		tmp = x_m;
	} else {
		tmp = 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)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-6.5d+40)) then
        tmp = (x_m * -z) / y
    else if (z <= (-3.7d-7)) then
        tmp = x_m
    else if (z <= (-1.15d-47)) then
        tmp = z / (y / -x_m)
    else if (z <= 1.05d-50) then
        tmp = x_m
    else
        tmp = 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 tmp;
	if (z <= -6.5e+40) {
		tmp = (x_m * -z) / y;
	} else if (z <= -3.7e-7) {
		tmp = x_m;
	} else if (z <= -1.15e-47) {
		tmp = z / (y / -x_m);
	} else if (z <= 1.05e-50) {
		tmp = x_m;
	} else {
		tmp = 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):
	tmp = 0
	if z <= -6.5e+40:
		tmp = (x_m * -z) / y
	elif z <= -3.7e-7:
		tmp = x_m
	elif z <= -1.15e-47:
		tmp = z / (y / -x_m)
	elif z <= 1.05e-50:
		tmp = x_m
	else:
		tmp = 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)
	tmp = 0.0
	if (z <= -6.5e+40)
		tmp = Float64(Float64(x_m * Float64(-z)) / y);
	elseif (z <= -3.7e-7)
		tmp = x_m;
	elseif (z <= -1.15e-47)
		tmp = Float64(z / Float64(y / Float64(-x_m)));
	elseif (z <= 1.05e-50)
		tmp = x_m;
	else
		tmp = Float64(z * Float64(x_m / Float64(-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)
	tmp = 0.0;
	if (z <= -6.5e+40)
		tmp = (x_m * -z) / y;
	elseif (z <= -3.7e-7)
		tmp = x_m;
	elseif (z <= -1.15e-47)
		tmp = z / (y / -x_m);
	elseif (z <= 1.05e-50)
		tmp = x_m;
	else
		tmp = 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_] := N[(x$95$s * If[LessEqual[z, -6.5e+40], N[(N[(x$95$m * (-z)), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[z, -3.7e-7], x$95$m, If[LessEqual[z, -1.15e-47], N[(z / N[(y / (-x$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.05e-50], x$95$m, N[(z * N[(x$95$m / (-y)), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < -6.5000000000000001e40

   1. Initial program 90.6%

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

   3. Taylor expanded in y around 0 77.4%

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

      1. associate-*r* 77.4%

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

      2. mul-1-neg 77.4%

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

   5. Simplified 77.4%

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

   if -6.5000000000000001e40 < z < -3.70000000000000004e-7 or -1.14999999999999991e-47 < z < 1.05e-50

   1. Initial program 73.8%

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

      1. remove-double-neg 73.8%

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

      2. distribute-frac-neg2 73.8%

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

      3. distribute-frac-neg 73.8%

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

      4. distribute-rgt-neg-in 73.8%

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

      5. associate-/l* 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. remove-double-neg 99.9%

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

      9. div-sub 99.9%

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

      10. *-inverses 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 83.9%

      \[\leadsto \color{blue}{x} \]

   if -3.70000000000000004e-7 < z < -1.14999999999999991e-47

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 85.3%

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

      1. associate-*r* 85.3%

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

      2. mul-1-neg 85.3%

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

   5. Simplified 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-/l* 85.6%

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

      3. add-sqr-sqrt 56.2%

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

      4. sqrt-unprod 29.8%

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

      5. sqr-neg 29.8%

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

      6. sqrt-unprod 0.8%

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

      7. add-sqr-sqrt 1.5%

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

   7. Applied egg-rr 1.5%

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

      1. add-sqr-sqrt 0.9%

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

      2. sqrt-unprod 57.7%

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

      3. swap-sqr 57.5%

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

      4. sqr-neg 57.5%

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

      5. distribute-frac-neg 57.5%

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

      6. distribute-frac-neg 57.5%

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

      7. swap-sqr 57.7%

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

      8. sqrt-unprod 57.1%

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

      9. add-sqr-sqrt 85.6%

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

      10. distribute-frac-neg 85.6%

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

      11. distribute-rgt-neg-in 85.6%

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

      12. distribute-lft-neg-in 85.6%

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

      13. clear-num 85.8%

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

      14. un-div-inv 85.8%

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

   9. Applied egg-rr 85.8%

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

   if 1.05e-50 < z

   1. Initial program 85.8%

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

      1. remove-double-neg 85.8%

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

      2. distribute-frac-neg2 85.8%

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

      3. distribute-frac-neg 85.8%

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

      4. distribute-rgt-neg-in 85.8%

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

      5. associate-/l* 93.6%

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

      6. distribute-frac-neg 93.6%

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

      7. distribute-frac-neg2 93.6%

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

      8. remove-double-neg 93.6%

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

      9. div-sub 93.6%

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

      10. *-inverses 93.6%

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

   3. Simplified 93.6%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 73.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{y}} \]
   6. Step-by-step derivation

      1. associate-*l/ 74.6%

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

      2. associate-*l* 74.6%

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

      3. *-commutative 74.6%

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

      4. associate-*r/ 74.6%

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

      5. mul-1-neg 74.6%

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

   7. Simplified 74.6%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{x \cdot \left(-z\right)}{y}\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{-47}:\\ \;\;\;\;\frac{z}{\frac{y}{-x}}\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{+24}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 5e+24) x_m (* y (/ 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 tmp;
	if (x_m <= 5e+24) {
		tmp = x_m;
	} else {
		tmp = y * (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)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 5d+24) then
        tmp = x_m
    else
        tmp = y * (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 tmp;
	if (x_m <= 5e+24) {
		tmp = x_m;
	} else {
		tmp = y * (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):
	tmp = 0
	if x_m <= 5e+24:
		tmp = x_m
	else:
		tmp = y * (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)
	tmp = 0.0
	if (x_m <= 5e+24)
		tmp = x_m;
	else
		tmp = Float64(y * 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)
	tmp = 0.0;
	if (x_m <= 5e+24)
		tmp = x_m;
	else
		tmp = y * (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_] := N[(x$95$s * If[LessEqual[x$95$m, 5e+24], x$95$m, N[(y * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000045e24

   1. Initial program 82.7%

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

      1. remove-double-neg 82.7%

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

      2. distribute-frac-neg2 82.7%

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

      3. distribute-frac-neg 82.7%

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

      4. distribute-rgt-neg-in 82.7%

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

      5. associate-/l* 95.2%

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

      6. distribute-frac-neg 95.2%

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

      7. distribute-frac-neg2 95.2%

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

      8. remove-double-neg 95.2%

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

      9. div-sub 95.2%

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

      10. *-inverses 95.2%

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

   3. Simplified 95.2%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 53.0%

      \[\leadsto \color{blue}{x} \]

   if 5.00000000000000045e24 < x

   1. Initial program 76.1%

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

   3. Taylor expanded in y around inf 28.4%

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

      1. *-commutative 28.4%

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

      2. associate-/l* 63.7%

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

   5. Applied egg-rr 63.7%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.1% accurate, 1.0× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (* x_m (- 1.0 (/ z y)))))

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (x_m * (1.0d0 - (z / y)))
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) {
	return x_s * (x_m * (1.0 - (z / y)));
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (x_m * (1.0 - (z / y)));
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_] := N[(x$95$s * N[(x$95$m * N[(1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

   1. remove-double-neg 81.3%

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

   2. distribute-frac-neg2 81.3%

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

   3. distribute-frac-neg 81.3%

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

   4. distribute-rgt-neg-in 81.3%

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

   5. associate-/l* 96.2%

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

   6. distribute-frac-neg 96.2%

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

   7. distribute-frac-neg2 96.2%

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

   8. remove-double-neg 96.2%

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

   9. div-sub 96.2%

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

   10. *-inverses 96.2%

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

3. Simplified 96.2%

   \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
4. Add Preprocessing

5. Final simplification 96.2%

   \[\leadsto x \cdot \left(1 - \frac{z}{y}\right) \]
6. Add Preprocessing

Alternative 8: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * (x_m - (x_m / (y / z)))
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) {
	return x_s * (x_m - (x_m / (y / z)));
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (x_m - (x_m / (y / z)));
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_] := N[(x$95$s * N[(x$95$m - N[(x$95$m / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

   1. remove-double-neg 81.3%

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

   2. distribute-frac-neg2 81.3%

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

   3. distribute-frac-neg 81.3%

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

   4. distribute-rgt-neg-in 81.3%

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

   5. associate-/l* 96.2%

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

   6. distribute-frac-neg 96.2%

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

   7. distribute-frac-neg2 96.2%

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

   8. remove-double-neg 96.2%

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

   9. div-sub 96.2%

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

   10. *-inverses 96.2%

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

3. Simplified 96.2%

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

   1. sub-neg 96.2%

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

   2. distribute-rgt-in 96.2%

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

   3. *-un-lft-identity 96.2%

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

   4. distribute-neg-frac2 96.2%

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

6. Applied egg-rr 96.2%

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

   1. *-commutative 96.2%

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

   2. distribute-frac-neg2 96.2%

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

   3. distribute-rgt-neg-out 96.2%

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

   4. distribute-lft-neg-out 96.2%

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

   5. associate-/l* 95.6%

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

   6. clear-num 95.6%

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

   7. frac-2neg 95.6%

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

   8. metadata-eval 95.6%

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

   9. frac-2neg 95.6%

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

   10. add-sqr-sqrt 50.3%

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

   11. sqrt-unprod 61.7%

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

   12. sqr-neg 61.7%

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

   13. sqrt-unprod 23.0%

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

   14. add-sqr-sqrt 49.3%

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

   15. distribute-lft-neg-out 49.3%

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

   16. remove-double-neg 49.3%

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

   17. distribute-frac-neg2 49.3%

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

   18. distribute-lft-neg-out 49.3%

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

   19. frac-2neg 49.3%

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

   20. add-sqr-sqrt 26.2%

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

   21. sqrt-unprod 60.7%

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

   22. sqr-neg 60.7%

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

   23. sqrt-unprod 45.2%

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

   24. add-sqr-sqrt 95.6%

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

   25. distribute-lft-neg-out 95.6%

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

   26. remove-double-neg 95.6%

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

   27. *-commutative 95.6%

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

8. Applied egg-rr 95.6%

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

9. Taylor expanded in y around 0 95.6%

   \[\leadsto x + \color{blue}{-1 \cdot \frac{x \cdot z}{y}} \]
10. Step-by-step derivation

    1. associate-*r/ 95.6%

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

    2. associate-*r* 95.6%

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

    3. neg-mul-1 95.6%

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

    4. associate-*l/ 92.7%

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

    5. associate-/r/ 97.0%

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

11. Simplified 97.0%

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

12. Final simplification 97.0%

    \[\leadsto x - \frac{x}{\frac{y}{z}} \]
13. Add Preprocessing

Alternative 9: 51.1% accurate, 7.0× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * x_m
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) {
	return x_s * x_m;
}

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * 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]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 81.3%

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

   1. remove-double-neg 81.3%

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

   2. distribute-frac-neg2 81.3%

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

   3. distribute-frac-neg 81.3%

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

   4. distribute-rgt-neg-in 81.3%

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

   5. associate-/l* 96.2%

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

   6. distribute-frac-neg 96.2%

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

   7. distribute-frac-neg2 96.2%

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

   8. remove-double-neg 96.2%

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

   9. div-sub 96.2%

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

   10. *-inverses 96.2%

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

3. Simplified 96.2%

   \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
4. Add Preprocessing

5. Taylor expanded in z around 0 52.9%

   \[\leadsto \color{blue}{x} \]

6. Final simplification 52.9%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z -2.060202331921739e+104)
   (- x (/ (* z x) y))
   (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < -2.060202331921739e+104) {
		tmp = x - ((z * x) / y);
	} else if (z < 1.6939766013828526e+213) {
		tmp = x / (y / (y - z));
	} else {
		tmp = (y - z) * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < (-2.060202331921739d+104)) then
        tmp = x - ((z * x) / y)
    else if (z < 1.6939766013828526d+213) then
        tmp = x / (y / (y - z))
    else
        tmp = (y - z) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < -2.060202331921739e+104) {
		tmp = x - ((z * x) / y);
	} else if (z < 1.6939766013828526e+213) {
		tmp = x / (y / (y - z));
	} else {
		tmp = (y - z) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < -2.060202331921739e+104:
		tmp = x - ((z * x) / y)
	elif z < 1.6939766013828526e+213:
		tmp = x / (y / (y - z))
	else:
		tmp = (y - z) * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < -2.060202331921739e+104)
		tmp = Float64(x - Float64(Float64(z * x) / y));
	elseif (z < 1.6939766013828526e+213)
		tmp = Float64(x / Float64(y / Float64(y - z)));
	else
		tmp = Float64(Float64(y - z) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < -2.060202331921739e+104)
		tmp = x - ((z * x) / y);
	elseif (z < 1.6939766013828526e+213)
		tmp = x / (y / (y - z));
	else
		tmp = (y - z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, -2.060202331921739e+104], N[(x - N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[Less[z, 1.6939766013828526e+213], N[(x / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024059 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :alt
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

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