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

Percentage Accurate: 84.4% → 97.9%

Time: 7.4s

Alternatives: 10

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: 84.4% 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, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{-45}:\\ \;\;\;\;x\_m + \frac{-1}{\frac{y}{x\_m \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x\_m - \frac{x\_m}{\frac{y}{z}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 4e-45)
    (+ x_m (/ -1.0 (/ y (* x_m z))))
    (- 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) {
	double tmp;
	if (x_m <= 4e-45) {
		tmp = x_m + (-1.0 / (y / (x_m * z)));
	} else {
		tmp = x_m - (x_m / (y / z));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    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 <= 4d-45) then
        tmp = x_m + ((-1.0d0) / (y / (x_m * z)))
    else
        tmp = x_m - (x_m / (y / z))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 4e-45) {
		tmp = x_m + (-1.0 / (y / (x_m * z)));
	} else {
		tmp = x_m - (x_m / (y / z));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 4e-45:
		tmp = x_m + (-1.0 / (y / (x_m * z)))
	else:
		tmp = x_m - (x_m / (y / z))
	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 <= 4e-45)
		tmp = Float64(x_m + Float64(-1.0 / Float64(y / Float64(x_m * z))));
	else
		tmp = Float64(x_m - Float64(x_m / Float64(y / z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 4e-45)
		tmp = x_m + (-1.0 / (y / (x_m * z)));
	else
		tmp = x_m - (x_m / (y / z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 4e-45], N[(x$95$m + N[(-1.0 / N[(y / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m - N[(x$95$m / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999994e-45

   1. Initial program 89.4%

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

      1. remove-double-neg 89.4%

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

      2. distribute-frac-neg2 89.4%

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

      3. distribute-frac-neg 89.4%

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

      4. distribute-rgt-neg-in 89.4%

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

      5. associate-/l* 94.5%

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

      6. distribute-frac-neg 94.5%

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

      7. distribute-frac-neg2 94.5%

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

      8. remove-double-neg 94.5%

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

      9. div-sub 94.5%

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

      10. *-inverses 94.5%

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

   3. Simplified 94.5%

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

      1. sub-neg 94.5%

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

      2. distribute-rgt-in 94.5%

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

      3. *-un-lft-identity 94.5%

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

      4. distribute-neg-frac2 94.5%

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

   6. Applied egg-rr 94.5%

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

      1. *-commutative 94.5%

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

      2. distribute-frac-neg2 94.5%

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

      3. distribute-rgt-neg-out 94.5%

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

      4. distribute-lft-neg-out 94.5%

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

      5. associate-/l* 95.4%

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

      6. clear-num 95.4%

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

      7. frac-2neg 95.4%

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

      8. metadata-eval 95.4%

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

      9. frac-2neg 95.4%

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

      10. add-sqr-sqrt 51.1%

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

      11. sqrt-unprod 68.5%

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

      12. sqr-neg 68.5%

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

      13. sqrt-unprod 24.8%

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

      14. add-sqr-sqrt 53.4%

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

      15. distribute-lft-neg-out 53.4%

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

      16. remove-double-neg 53.4%

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

      17. distribute-frac-neg2 53.4%

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

      18. distribute-lft-neg-out 53.4%

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

      19. frac-2neg 53.4%

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

      20. add-sqr-sqrt 28.5%

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

      21. sqrt-unprod 62.9%

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

      22. sqr-neg 62.9%

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

      23. sqrt-unprod 44.2%

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

      24. add-sqr-sqrt 95.4%

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

      25. distribute-lft-neg-out 95.4%

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

      26. remove-double-neg 95.4%

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

      27. *-commutative 95.4%

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

   8. Applied egg-rr 95.4%

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

   if 3.99999999999999994e-45 < x

   1. Initial program 85.5%

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

      1. remove-double-neg 85.5%

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

      2. distribute-frac-neg2 85.5%

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

      3. distribute-frac-neg 85.5%

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

      4. distribute-rgt-neg-in 85.5%

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

      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 99.8%

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

      10. *-inverses 99.8%

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

   3. Simplified 99.8%

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

      1. sub-neg 99.8%

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

      2. distribute-rgt-in 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. distribute-neg-frac2 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in z around 0 92.0%

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

      1. mul-1-neg 92.0%

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

      2. associate-*l/ 91.7%

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

      3. associate-/r/ 100.0%

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

      4. distribute-neg-frac2 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-45}:\\ \;\;\;\;x + \frac{-1}{\frac{y}{x \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+176} \lor \neg \left(z \leq -6 \cdot 10^{+85} \lor \neg \left(z \leq -4.4 \cdot 10^{+54}\right) \land z \leq 2.7 \cdot 10^{+20}\right):\\ \;\;\;\;x\_m \cdot \frac{-z}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -7.8e+176)
          (not (or (<= z -6e+85) (and (not (<= z -4.4e+54)) (<= z 2.7e+20)))))
    (* 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 <= -7.8e+176) || !((z <= -6e+85) || (!(z <= -4.4e+54) && (z <= 2.7e+20)))) {
		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 <= (-7.8d+176)) .or. (.not. (z <= (-6d+85)) .or. (.not. (z <= (-4.4d+54))) .and. (z <= 2.7d+20))) 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 <= -7.8e+176) || !((z <= -6e+85) || (!(z <= -4.4e+54) && (z <= 2.7e+20)))) {
		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 <= -7.8e+176) or not ((z <= -6e+85) or (not (z <= -4.4e+54) and (z <= 2.7e+20))):
		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 <= -7.8e+176) || !((z <= -6e+85) || (!(z <= -4.4e+54) && (z <= 2.7e+20))))
		tmp = Float64(x_m * Float64(Float64(-z) / 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 <= -7.8e+176) || ~(((z <= -6e+85) || (~((z <= -4.4e+54)) && (z <= 2.7e+20)))))
		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, -7.8e+176], N[Not[Or[LessEqual[z, -6e+85], And[N[Not[LessEqual[z, -4.4e+54]], $MachinePrecision], LessEqual[z, 2.7e+20]]]], $MachinePrecision]], N[(x$95$m * N[((-z) / y), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -7.8000000000000003e176 or -6.0000000000000001e85 < z < -4.3999999999999998e54 or 2.7e20 < z

   1. Initial program 95.0%

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

      1. remove-double-neg 95.0%

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

      2. distribute-frac-neg2 95.0%

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

      3. distribute-frac-neg 95.0%

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

      4. distribute-rgt-neg-in 95.0%

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

      5. associate-/l* 90.0%

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

      6. distribute-frac-neg 90.0%

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

      7. distribute-frac-neg2 90.0%

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

      8. remove-double-neg 90.0%

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

      9. div-sub 90.1%

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

      10. *-inverses 90.1%

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

   3. Simplified 90.1%

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

   5. Taylor expanded in z around inf 75.0%

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

      1. mul-1-neg 75.0%

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

      2. distribute-frac-neg2 75.0%

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

   7. Simplified 75.0%

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

   if -7.8000000000000003e176 < z < -6.0000000000000001e85 or -4.3999999999999998e54 < z < 2.7e20

   1. Initial program 84.0%

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

      1. remove-double-neg 84.0%

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

      2. distribute-frac-neg2 84.0%

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

      3. distribute-frac-neg 84.0%

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

      4. distribute-rgt-neg-in 84.0%

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

      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 78.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.8 \cdot 10^{+176} \lor \neg \left(z \leq -6 \cdot 10^{+85} \lor \neg \left(z \leq -4.4 \cdot 10^{+54}\right) \land z \leq 2.7 \cdot 10^{+20}\right):\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+16}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-39} \lor \neg \left(y \leq -4 \cdot 10^{-50}\right) \land y \leq 8.6 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{x\_m}{-y}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (<= y -3.2e+16)
    x_m
    (if (or (<= y -4.8e-39) (and (not (<= y -4e-50)) (<= y 8.6e+37)))
      (* 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 (y <= -3.2e+16) {
		tmp = x_m;
	} else if ((y <= -4.8e-39) || (!(y <= -4e-50) && (y <= 8.6e+37))) {
		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 (y <= (-3.2d+16)) then
        tmp = x_m
    else if ((y <= (-4.8d-39)) .or. (.not. (y <= (-4d-50))) .and. (y <= 8.6d+37)) 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 (y <= -3.2e+16) {
		tmp = x_m;
	} else if ((y <= -4.8e-39) || (!(y <= -4e-50) && (y <= 8.6e+37))) {
		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 y <= -3.2e+16:
		tmp = x_m
	elif (y <= -4.8e-39) or (not (y <= -4e-50) and (y <= 8.6e+37)):
		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 (y <= -3.2e+16)
		tmp = x_m;
	elseif ((y <= -4.8e-39) || (!(y <= -4e-50) && (y <= 8.6e+37)))
		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 (y <= -3.2e+16)
		tmp = x_m;
	elseif ((y <= -4.8e-39) || (~((y <= -4e-50)) && (y <= 8.6e+37)))
		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[LessEqual[y, -3.2e+16], x$95$m, If[Or[LessEqual[y, -4.8e-39], And[N[Not[LessEqual[y, -4e-50]], $MachinePrecision], LessEqual[y, 8.6e+37]]], N[(z * N[(x$95$m / (-y)), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -3.2e16 or -4.80000000000000031e-39 < y < -4.00000000000000003e-50 or 8.5999999999999994e37 < y

   1. Initial program 80.1%

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

      1. remove-double-neg 80.1%

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

      2. distribute-frac-neg2 80.1%

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

      3. distribute-frac-neg 80.1%

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

      4. distribute-rgt-neg-in 80.1%

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

      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

   5. Taylor expanded in z around 0 87.4%

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

   if -3.2e16 < y < -4.80000000000000031e-39 or -4.00000000000000003e-50 < y < 8.5999999999999994e37

   1. Initial program 95.4%

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

      1. remove-double-neg 95.4%

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

      2. distribute-frac-neg2 95.4%

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

      3. distribute-frac-neg 95.4%

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

      4. distribute-rgt-neg-in 95.4%

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

      5. associate-/l* 92.9%

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

      6. distribute-frac-neg 92.9%

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

      7. distribute-frac-neg2 92.9%

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

      8. remove-double-neg 92.9%

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

      9. div-sub 92.9%

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

      10. *-inverses 92.9%

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

   3. Simplified 92.9%

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

   5. Taylor expanded in z around inf 72.0%

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

      1. associate-*l/ 71.0%

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

      2. associate-*l* 71.0%

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

      3. *-commutative 71.0%

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

      4. associate-*r/ 71.0%

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

      5. mul-1-neg 71.0%

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

   7. Simplified 71.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-39} \lor \neg \left(y \leq -4 \cdot 10^{-50}\right) \land y \leq 8.6 \cdot 10^{+37}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot z}{-y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{+87}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{z}{\frac{y}{-x\_m}}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (/ (* x_m z) (- y))))
   (*
    x_s
    (if (<= z -1.45e+176)
      t_0
      (if (<= z -1.85e+87)
        x_m
        (if (<= z -5.8e+54)
          (/ z (/ y (- x_m)))
          (if (<= z 8.5e+18) 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 = (x_m * z) / -y;
	double tmp;
	if (z <= -1.45e+176) {
		tmp = t_0;
	} else if (z <= -1.85e+87) {
		tmp = x_m;
	} else if (z <= -5.8e+54) {
		tmp = z / (y / -x_m);
	} else if (z <= 8.5e+18) {
		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 = (x_m * z) / -y
    if (z <= (-1.45d+176)) then
        tmp = t_0
    else if (z <= (-1.85d+87)) then
        tmp = x_m
    else if (z <= (-5.8d+54)) then
        tmp = z / (y / -x_m)
    else if (z <= 8.5d+18) 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 = (x_m * z) / -y;
	double tmp;
	if (z <= -1.45e+176) {
		tmp = t_0;
	} else if (z <= -1.85e+87) {
		tmp = x_m;
	} else if (z <= -5.8e+54) {
		tmp = z / (y / -x_m);
	} else if (z <= 8.5e+18) {
		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 = (x_m * z) / -y
	tmp = 0
	if z <= -1.45e+176:
		tmp = t_0
	elif z <= -1.85e+87:
		tmp = x_m
	elif z <= -5.8e+54:
		tmp = z / (y / -x_m)
	elif z <= 8.5e+18:
		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(Float64(x_m * z) / Float64(-y))
	tmp = 0.0
	if (z <= -1.45e+176)
		tmp = t_0;
	elseif (z <= -1.85e+87)
		tmp = x_m;
	elseif (z <= -5.8e+54)
		tmp = Float64(z / Float64(y / Float64(-x_m)));
	elseif (z <= 8.5e+18)
		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 = (x_m * z) / -y;
	tmp = 0.0;
	if (z <= -1.45e+176)
		tmp = t_0;
	elseif (z <= -1.85e+87)
		tmp = x_m;
	elseif (z <= -5.8e+54)
		tmp = z / (y / -x_m);
	elseif (z <= 8.5e+18)
		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[(N[(x$95$m * z), $MachinePrecision] / (-y)), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.45e+176], t$95$0, If[LessEqual[z, -1.85e+87], x$95$m, If[LessEqual[z, -5.8e+54], N[(z / N[(y / (-x$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e+18], x$95$m, t$95$0]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4500000000000001e176 or 8.5e18 < z

   1. Initial program 94.5%

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

   3. Taylor expanded in y around 0 83.0%

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

      1. associate-*r* 83.0%

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

      2. mul-1-neg 83.0%

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

   5. Simplified 83.0%

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

   if -1.4500000000000001e176 < z < -1.85000000000000001e87 or -5.7999999999999997e54 < z < 8.5e18

   1. Initial program 84.0%

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

      1. remove-double-neg 84.0%

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

      2. distribute-frac-neg2 84.0%

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

      3. distribute-frac-neg 84.0%

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

      4. distribute-rgt-neg-in 84.0%

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

      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 78.7%

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

   if -1.85000000000000001e87 < z < -5.7999999999999997e54

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. mul-1-neg 100.0%

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

   5. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 99.6%

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

      3. add-sqr-sqrt 25.0%

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

      4. sqrt-unprod 26.5%

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

      5. sqr-neg 26.5%

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

      6. sqrt-unprod 1.1%

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

      7. add-sqr-sqrt 1.3%

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

   7. Applied egg-rr 1.3%

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

      1. *-commutative 1.3%

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

      2. clear-num 1.3%

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

      3. frac-2neg 1.3%

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

      4. metadata-eval 1.3%

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

      5. associate-*l/ 1.3%

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

      6. neg-mul-1 1.3%

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

      7. distribute-frac-neg2 1.3%

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

      8. div-inv 1.3%

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

      9. add-sqr-sqrt 0.2%

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

      10. sqrt-unprod 51.4%

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

      11. sqr-neg 51.4%

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

      12. sqrt-unprod 74.6%

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

      13. add-sqr-sqrt 99.8%

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

      14. *-un-lft-identity 99.8%

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

      15. *-commutative 99.8%

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

      16. *-inverses 99.8%

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

      17. associate-/l* 99.6%

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

      18. clear-num 99.4%

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

      19. *-commutative 99.4%

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

      20. associate-/r* 99.8%

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

      21. *-inverses 99.8%

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

      22. div-inv 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+176}:\\ \;\;\;\;\frac{x \cdot z}{-y}\\ \mathbf{elif}\;z \leq -1.85 \cdot 10^{+87}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{z}{\frac{y}{-x}}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+15}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-39}:\\ \;\;\;\;z \cdot \frac{x\_m}{-y}\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-50}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{\frac{y}{-x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (<= y -7.8e+15)
    x_m
    (if (<= y -6e-39)
      (* z (/ x_m (- y)))
      (if (<= y -3.9e-50) x_m (if (<= y 3.3e+31) (/ z (/ y (- x_m))) 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 (y <= -7.8e+15) {
		tmp = x_m;
	} else if (y <= -6e-39) {
		tmp = z * (x_m / -y);
	} else if (y <= -3.9e-50) {
		tmp = x_m;
	} else if (y <= 3.3e+31) {
		tmp = z / (y / -x_m);
	} 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 (y <= (-7.8d+15)) then
        tmp = x_m
    else if (y <= (-6d-39)) then
        tmp = z * (x_m / -y)
    else if (y <= (-3.9d-50)) then
        tmp = x_m
    else if (y <= 3.3d+31) then
        tmp = z / (y / -x_m)
    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 (y <= -7.8e+15) {
		tmp = x_m;
	} else if (y <= -6e-39) {
		tmp = z * (x_m / -y);
	} else if (y <= -3.9e-50) {
		tmp = x_m;
	} else if (y <= 3.3e+31) {
		tmp = z / (y / -x_m);
	} 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 y <= -7.8e+15:
		tmp = x_m
	elif y <= -6e-39:
		tmp = z * (x_m / -y)
	elif y <= -3.9e-50:
		tmp = x_m
	elif y <= 3.3e+31:
		tmp = z / (y / -x_m)
	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 (y <= -7.8e+15)
		tmp = x_m;
	elseif (y <= -6e-39)
		tmp = Float64(z * Float64(x_m / Float64(-y)));
	elseif (y <= -3.9e-50)
		tmp = x_m;
	elseif (y <= 3.3e+31)
		tmp = Float64(z / Float64(y / Float64(-x_m)));
	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 (y <= -7.8e+15)
		tmp = x_m;
	elseif (y <= -6e-39)
		tmp = z * (x_m / -y);
	elseif (y <= -3.9e-50)
		tmp = x_m;
	elseif (y <= 3.3e+31)
		tmp = z / (y / -x_m);
	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[LessEqual[y, -7.8e+15], x$95$m, If[LessEqual[y, -6e-39], N[(z * N[(x$95$m / (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.9e-50], x$95$m, If[LessEqual[y, 3.3e+31], N[(z / N[(y / (-x$95$m)), $MachinePrecision]), $MachinePrecision], x$95$m]]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -7.8e15 or -6.00000000000000055e-39 < y < -3.90000000000000021e-50 or 3.29999999999999992e31 < y

   1. Initial program 80.2%

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

      1. remove-double-neg 80.2%

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

      2. distribute-frac-neg2 80.2%

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

      3. distribute-frac-neg 80.2%

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

      4. distribute-rgt-neg-in 80.2%

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

      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 86.7%

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

   if -7.8e15 < y < -6.00000000000000055e-39

   1. Initial program 99.8%

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

      1. remove-double-neg 99.8%

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

      2. distribute-frac-neg2 99.8%

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

      3. distribute-frac-neg 99.8%

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

      4. distribute-rgt-neg-in 99.8%

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

      5. associate-/l* 99.7%

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

      6. distribute-frac-neg 99.7%

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

      7. distribute-frac-neg2 99.7%

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

      8. remove-double-neg 99.7%

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

      9. div-sub 99.7%

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

      10. *-inverses 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 78.9%

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

      1. associate-*l/ 79.0%

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

      2. associate-*l* 79.0%

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

      3. *-commutative 79.0%

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

      4. associate-*r/ 79.0%

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

      5. mul-1-neg 79.0%

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

   7. Simplified 79.0%

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

   if -3.90000000000000021e-50 < y < 3.29999999999999992e31

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0 70.7%

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

      1. associate-*r* 70.7%

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

      2. mul-1-neg 70.7%

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

   5. Simplified 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-/l* 70.0%

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

      3. add-sqr-sqrt 28.8%

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

      4. sqrt-unprod 17.5%

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

      5. sqr-neg 17.5%

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

      6. sqrt-unprod 1.0%

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

      7. add-sqr-sqrt 1.8%

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

   7. Applied egg-rr 1.8%

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

      1. *-commutative 1.8%

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

      2. clear-num 1.8%

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

      3. frac-2neg 1.8%

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

      4. metadata-eval 1.8%

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

      5. associate-*l/ 1.8%

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

      6. neg-mul-1 1.8%

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

      7. distribute-frac-neg2 1.8%

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

      8. div-inv 1.8%

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

      9. add-sqr-sqrt 0.9%

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

      10. sqrt-unprod 31.3%

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

      11. sqr-neg 31.3%

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

      12. sqrt-unprod 41.5%

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

      13. add-sqr-sqrt 71.0%

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

      14. *-un-lft-identity 71.0%

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

      15. *-commutative 71.0%

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

      16. *-inverses 71.0%

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

      17. associate-/l* 51.3%

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

      18. clear-num 51.3%

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

      19. *-commutative 51.3%

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

      20. associate-/r* 71.0%

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

      21. *-inverses 71.0%

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

      22. div-inv 71.1%

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

   9. Applied egg-rr 71.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+15}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-39}:\\ \;\;\;\;z \cdot \frac{x}{-y}\\ \mathbf{elif}\;y \leq -3.9 \cdot 10^{-50}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+31}:\\ \;\;\;\;\frac{z}{\frac{y}{-x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+14}:\\ \;\;\;\;y \cdot \frac{x\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 (<= z -1.65e+14) (* y (/ 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 <= -1.65e+14) {
		tmp = y * (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 <= (-1.65d+14)) then
        tmp = y * (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 <= -1.65e+14) {
		tmp = y * (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 <= -1.65e+14:
		tmp = y * (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 <= -1.65e+14)
		tmp = Float64(y * Float64(x_m / 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 <= -1.65e+14)
		tmp = y * (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[LessEqual[z, -1.65e+14], N[(y * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision], x$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.65e14

   1. Initial program 86.6%

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

   3. Taylor expanded in y around inf 17.9%

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

      1. *-commutative 17.9%

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

      2. associate-/l* 38.4%

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

   5. Applied egg-rr 38.4%

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

   if -1.65e14 < z

   1. Initial program 88.6%

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

      1. remove-double-neg 88.6%

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

      2. distribute-frac-neg2 88.6%

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

      3. distribute-frac-neg 88.6%

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

      4. distribute-rgt-neg-in 88.6%

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

      5. associate-/l* 97.0%

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

      6. distribute-frac-neg 97.0%

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

      7. distribute-frac-neg2 97.0%

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

      8. remove-double-neg 97.0%

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

      9. div-sub 97.0%

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

      10. *-inverses 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in z around 0 63.6%

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

Alternative 7: 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 88.2%

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

   1. remove-double-neg 88.2%

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

   2. distribute-frac-neg2 88.2%

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

   3. distribute-frac-neg 88.2%

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

   4. distribute-rgt-neg-in 88.2%

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

   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. Taylor expanded in z around 0 94.3%

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

   1. mul-1-neg 94.3%

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

   2. associate-*l/ 92.2%

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

   3. associate-/r/ 96.3%

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

   4. distribute-neg-frac2 96.3%

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

9. Simplified 96.3%

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

10. Final simplification 96.3%

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

Alternative 8: 95.9% accurate, 1.0× speedup

Math

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

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 (/ 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 - (x_m * (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 - (x_m * (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 - (x_m * (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 - (x_m * (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(x_m * 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 - (x_m * (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[(x$95$m * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.2%

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

   1. remove-double-neg 88.2%

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

   2. distribute-frac-neg2 88.2%

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

   3. distribute-frac-neg 88.2%

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

   4. distribute-rgt-neg-in 88.2%

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

   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. Taylor expanded in z around 0 94.3%

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

   1. mul-1-neg 94.3%

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

   2. associate-*l/ 92.2%

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

   3. associate-/r/ 96.3%

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

   4. distribute-neg-frac2 96.3%

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

9. Simplified 96.3%

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

10. Taylor expanded in x around 0 96.2%

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

    1. mul-1-neg 96.2%

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

    2. unsub-neg 96.2%

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

    3. distribute-lft-out-- 96.2%

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

    4. *-rgt-identity 96.2%

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

12. Simplified 96.2%

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

Alternative 9: 95.9% 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 88.2%

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

   1. remove-double-neg 88.2%

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

   2. distribute-frac-neg2 88.2%

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

   3. distribute-frac-neg 88.2%

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

   4. distribute-rgt-neg-in 88.2%

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

   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. Add Preprocessing

Alternative 10: 49.9% 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 88.2%

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

   1. remove-double-neg 88.2%

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

   2. distribute-frac-neg2 88.2%

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

   3. distribute-frac-neg 88.2%

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

   4. distribute-rgt-neg-in 88.2%

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

   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 55.6%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer target: 95.8% 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 2024100 
(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))