Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A

Percentage Accurate: 69.8% → 99.8%

Time: 12.0s

Alternatives: 10

Speedup: 1.2×

• 42.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

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

C

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

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 * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

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

C

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

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 * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(y + \frac{x - z}{\frac{y}{x + z}}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 (+ y (/ (- x z) (/ y (+ x z))))))

C

double code(double x, double y, double z) {
	return 0.5 * (y + ((x - z) / (y / (x + z))));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * (y + ((x - z) / (y / (x + z))))
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * (y + ((x - z) / (y / (x + z))));
}

Python

def code(x, y, z):
	return 0.5 * (y + ((x - z) / (y / (x + z))))

Julia

function code(x, y, z)
	return Float64(0.5 * Float64(y + Float64(Float64(x - z) / Float64(y / Float64(x + z)))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * (y + ((x - z) / (y / (x + z))));
end

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(y + N[(N[(x - z), $MachinePrecision] / N[(y / N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

   1. remove-double-neg 69.3%

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

   2. distribute-lft-neg-out 69.3%

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

   3. distribute-frac-neg2 69.3%

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

   4. distribute-frac-neg 69.3%

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

   5. neg-mul-1 69.3%

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

   6. distribute-lft-neg-out 69.3%

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

   7. *-commutative 69.3%

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

   8. distribute-lft-neg-in 69.3%

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

   9. times-frac 69.3%

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

   10. metadata-eval 69.3%

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

   11. metadata-eval 69.3%

       \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

   12. associate--l+ 69.3%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

   13. fma-define 72.4%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

3. Simplified 72.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 77.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
6. Step-by-step derivation

   1. associate--l+ 77.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

   2. div-sub 79.2%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

7. Simplified 79.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
8. Step-by-step derivation

   1. unpow2 79.2%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

   2. unpow2 79.2%

      \[\leadsto 0.5 \cdot \left(y + \frac{x \cdot x - \color{blue}{z \cdot z}}{y}\right) \]

   3. difference-of-squares 86.7%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

9. Applied egg-rr 86.7%

   \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]
10. Step-by-step derivation

    1. *-commutative 86.7%

       \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x - z\right) \cdot \left(x + z\right)}}{y}\right) \]

    2. associate-/l* 99.9%

       \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

11. Applied egg-rr 99.9%

    \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]
12. Step-by-step derivation

    1. clear-num 99.9%

       \[\leadsto 0.5 \cdot \left(y + \left(x - z\right) \cdot \color{blue}{\frac{1}{\frac{y}{x + z}}}\right) \]

    2. un-div-inv 99.9%

       \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{x - z}{\frac{y}{x + z}}}\right) \]

13. Applied egg-rr 99.9%

    \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{x - z}{\frac{y}{x + z}}}\right) \]

14. Final simplification 99.9%

    \[\leadsto 0.5 \cdot \left(y + \frac{x - z}{\frac{y}{x + z}}\right) \]
15. Add Preprocessing

Alternative 2: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+62} \lor \neg \left(y \leq 2.2 \cdot 10^{+82}\right) \land y \leq 1.75 \cdot 10^{+183}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y 3.8e+62) (and (not (<= y 2.2e+82)) (<= y 1.75e+183)))
   (* 0.5 (* (+ x z) (/ (- x z) y)))
   (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= 3.8e+62) || (!(y <= 2.2e+82) && (y <= 1.75e+183))) {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	} else {
		tmp = 0.5 * 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 ((y <= 3.8d+62) .or. (.not. (y <= 2.2d+82)) .and. (y <= 1.75d+183)) then
        tmp = 0.5d0 * ((x + z) * ((x - z) / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= 3.8e+62) || (!(y <= 2.2e+82) && (y <= 1.75e+183))) {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= 3.8e+62) or (not (y <= 2.2e+82) and (y <= 1.75e+183)):
		tmp = 0.5 * ((x + z) * ((x - z) / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= 3.8e+62) || (!(y <= 2.2e+82) && (y <= 1.75e+183)))
		tmp = Float64(0.5 * Float64(Float64(x + z) * Float64(Float64(x - z) / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= 3.8e+62) || (~((y <= 2.2e+82)) && (y <= 1.75e+183)))
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, 3.8e+62], And[N[Not[LessEqual[y, 2.2e+82]], $MachinePrecision], LessEqual[y, 1.75e+183]]], N[(0.5 * N[(N[(x + z), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.79999999999999984e62 or 2.2000000000000001e82 < y < 1.74999999999999994e183

   1. Initial program 76.4%

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

      1. remove-double-neg 76.4%

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

      2. distribute-lft-neg-out 76.4%

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

      3. distribute-frac-neg2 76.4%

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

      4. distribute-frac-neg 76.4%

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

      5. neg-mul-1 76.4%

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

      6. distribute-lft-neg-out 76.4%

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

      7. *-commutative 76.4%

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

      8. distribute-lft-neg-in 76.4%

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

      9. times-frac 76.4%

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

      10. metadata-eval 76.4%

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

      11. metadata-eval 76.4%

          \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

      12. associate--l+ 76.4%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

      13. fma-define 80.0%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

   3. Simplified 80.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 81.8%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 81.8%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

      2. div-sub 83.6%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   7. Simplified 83.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   8. Step-by-step derivation

      1. unpow2 83.6%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. unpow2 83.6%

         \[\leadsto 0.5 \cdot \left(y + \frac{x \cdot x - \color{blue}{z \cdot z}}{y}\right) \]

      3. difference-of-squares 91.8%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   9. Applied egg-rr 91.8%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   10. Taylor expanded in y around 0 71.2%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   11. Step-by-step derivation

       1. associate-*r/ 75.9%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(x + z\right) \cdot \frac{x - z}{y}\right)} \]

       2. +-commutative 75.9%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(z + x\right)} \cdot \frac{x - z}{y}\right) \]

   12. Simplified 75.9%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(z + x\right) \cdot \frac{x - z}{y}\right)} \]

   if 3.79999999999999984e62 < y < 2.2000000000000001e82 or 1.74999999999999994e183 < y

   1. Initial program 27.3%

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

   3. Taylor expanded in y around inf 75.6%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 75.6%

         \[\leadsto \color{blue}{y \cdot 0.5} \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{y \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+62} \lor \neg \left(y \leq 2.2 \cdot 10^{+82}\right) \land y \leq 1.75 \cdot 10^{+183}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y}\right)\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{+82} \lor \neg \left(y \leq 2.1 \cdot 10^{+183}\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.5e+62)
   (* 0.5 (* (- x z) (/ (+ x z) y)))
   (if (or (<= y 1.46e+82) (not (<= y 2.1e+183)))
     (* 0.5 y)
     (* 0.5 (* (+ x z) (/ (- x z) y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e+62) {
		tmp = 0.5 * ((x - z) * ((x + z) / y));
	} else if ((y <= 1.46e+82) || !(y <= 2.1e+183)) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * ((x + z) * ((x - z) / 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 (y <= 3.5d+62) then
        tmp = 0.5d0 * ((x - z) * ((x + z) / y))
    else if ((y <= 1.46d+82) .or. (.not. (y <= 2.1d+183))) then
        tmp = 0.5d0 * y
    else
        tmp = 0.5d0 * ((x + z) * ((x - z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e+62) {
		tmp = 0.5 * ((x - z) * ((x + z) / y));
	} else if ((y <= 1.46e+82) || !(y <= 2.1e+183)) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.5e+62:
		tmp = 0.5 * ((x - z) * ((x + z) / y))
	elif (y <= 1.46e+82) or not (y <= 2.1e+183):
		tmp = 0.5 * y
	else:
		tmp = 0.5 * ((x + z) * ((x - z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.5e+62)
		tmp = Float64(0.5 * Float64(Float64(x - z) * Float64(Float64(x + z) / y)));
	elseif ((y <= 1.46e+82) || !(y <= 2.1e+183))
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(0.5 * Float64(Float64(x + z) * Float64(Float64(x - z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.5e+62)
		tmp = 0.5 * ((x - z) * ((x + z) / y));
	elseif ((y <= 1.46e+82) || ~((y <= 2.1e+183)))
		tmp = 0.5 * y;
	else
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.5e+62], N[(0.5 * N[(N[(x - z), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 1.46e+82], N[Not[LessEqual[y, 2.1e+183]], $MachinePrecision]], N[(0.5 * y), $MachinePrecision], N[(0.5 * N[(N[(x + z), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.49999999999999984e62

   1. Initial program 79.0%

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

      1. remove-double-neg 79.0%

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

      2. distribute-lft-neg-out 79.0%

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

      3. distribute-frac-neg2 79.0%

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

      4. distribute-frac-neg 79.0%

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

      5. neg-mul-1 79.0%

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

      6. distribute-lft-neg-out 79.0%

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

      7. *-commutative 79.0%

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

      8. distribute-lft-neg-in 79.0%

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

      9. times-frac 79.0%

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

      10. metadata-eval 79.0%

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

      11. metadata-eval 79.0%

          \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

      12. associate--l+ 79.0%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

      13. fma-define 82.6%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

   3. Simplified 82.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 83.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 83.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

      2. div-sub 85.1%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   7. Simplified 85.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   8. Step-by-step derivation

      1. unpow2 85.1%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. unpow2 85.1%

         \[\leadsto 0.5 \cdot \left(y + \frac{x \cdot x - \color{blue}{z \cdot z}}{y}\right) \]

      3. difference-of-squares 93.2%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   9. Applied egg-rr 93.2%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   10. Taylor expanded in y around 0 74.6%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   11. Step-by-step derivation

       1. *-commutative 93.2%

          \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x - z\right) \cdot \left(x + z\right)}}{y}\right) \]

       2. associate-/l* 99.9%

          \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

   12. Applied egg-rr 77.5%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(x - z\right) \cdot \frac{x + z}{y}\right)} \]

   if 3.49999999999999984e62 < y < 1.4599999999999999e82 or 2.1e183 < y

   1. Initial program 27.3%

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

   3. Taylor expanded in y around inf 75.6%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 75.6%

         \[\leadsto \color{blue}{y \cdot 0.5} \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

   if 1.4599999999999999e82 < y < 2.1e183

   1. Initial program 54.1%

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

      1. remove-double-neg 54.1%

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

      2. distribute-lft-neg-out 54.1%

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

      3. distribute-frac-neg2 54.1%

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

      4. distribute-frac-neg 54.1%

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

      5. neg-mul-1 54.1%

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

      6. distribute-lft-neg-out 54.1%

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

      7. *-commutative 54.1%

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

      8. distribute-lft-neg-in 54.1%

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

      9. times-frac 54.1%

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

      10. metadata-eval 54.1%

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

      11. metadata-eval 54.1%

          \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

      12. associate--l+ 54.1%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

      13. fma-define 58.4%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

   3. Simplified 58.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 71.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 71.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

      2. div-sub 71.1%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   7. Simplified 71.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   8. Step-by-step derivation

      1. unpow2 71.1%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. unpow2 71.1%

         \[\leadsto 0.5 \cdot \left(y + \frac{x \cdot x - \color{blue}{z \cdot z}}{y}\right) \]

      3. difference-of-squares 79.8%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   9. Applied egg-rr 79.8%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   10. Taylor expanded in y around 0 42.3%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   11. Step-by-step derivation

       1. associate-*r/ 62.3%

          \[\leadsto 0.5 \cdot \color{blue}{\left(\left(x + z\right) \cdot \frac{x - z}{y}\right)} \]

       2. +-commutative 62.3%

          \[\leadsto 0.5 \cdot \left(\color{blue}{\left(z + x\right)} \cdot \frac{x - z}{y}\right) \]

   12. Simplified 62.3%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(z + x\right) \cdot \frac{x - z}{y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+62}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y}\right)\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{+82} \lor \neg \left(y \leq 2.1 \cdot 10^{+183}\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0045 \lor \neg \left(x \leq 7.8 \cdot 10^{+77}\right) \land x \leq 1.75 \cdot 10^{+110}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x 0.0045) (and (not (<= x 7.8e+77)) (<= x 1.75e+110)))
   (* 0.5 y)
   (* x (* x (/ 0.5 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= 0.0045) || (!(x <= 7.8e+77) && (x <= 1.75e+110))) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x * (0.5 / 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 ((x <= 0.0045d0) .or. (.not. (x <= 7.8d+77)) .and. (x <= 1.75d+110)) then
        tmp = 0.5d0 * y
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= 0.0045) || (!(x <= 7.8e+77) && (x <= 1.75e+110))) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= 0.0045) or (not (x <= 7.8e+77) and (x <= 1.75e+110)):
		tmp = 0.5 * y
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= 0.0045) || (!(x <= 7.8e+77) && (x <= 1.75e+110)))
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= 0.0045) || (~((x <= 7.8e+77)) && (x <= 1.75e+110)))
		tmp = 0.5 * y;
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, 0.0045], And[N[Not[LessEqual[x, 7.8e+77]], $MachinePrecision], LessEqual[x, 1.75e+110]]], N[(0.5 * y), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00449999999999999966 or 7.7999999999999995e77 < x < 1.75e110

   1. Initial program 69.3%

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

   3. Taylor expanded in y around inf 38.8%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 38.8%

         \[\leadsto \color{blue}{y \cdot 0.5} \]

   5. Simplified 38.8%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

   if 0.00449999999999999966 < x < 7.7999999999999995e77 or 1.75e110 < x

   1. Initial program 69.2%

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

   3. Taylor expanded in x around inf 63.9%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative 63.9%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.5} \]

      2. associate-*l/ 63.9%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.5}{y}} \]

      3. associate-*r/ 63.9%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 63.9%

      \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{y}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 35.6%

         \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot \frac{0.5}{y}} \cdot \sqrt{{x}^{2} \cdot \frac{0.5}{y}}} \]

      2. pow2 35.6%

         \[\leadsto \color{blue}{{\left(\sqrt{{x}^{2} \cdot \frac{0.5}{y}}\right)}^{2}} \]

      3. sqrt-prod 35.6%

         \[\leadsto {\color{blue}{\left(\sqrt{{x}^{2}} \cdot \sqrt{\frac{0.5}{y}}\right)}}^{2} \]

      4. sqrt-pow1 38.8%

         \[\leadsto {\left(\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

      5. metadata-eval 38.8%

         \[\leadsto {\left({x}^{\color{blue}{1}} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

      6. pow1 38.8%

         \[\leadsto {\left(\color{blue}{x} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

   7. Applied egg-rr 38.8%

      \[\leadsto \color{blue}{{\left(x \cdot \sqrt{\frac{0.5}{y}}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 38.8%

         \[\leadsto \color{blue}{\left(x \cdot \sqrt{\frac{0.5}{y}}\right) \cdot \left(x \cdot \sqrt{\frac{0.5}{y}}\right)} \]

      2. swap-sqr 35.6%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\sqrt{\frac{0.5}{y}} \cdot \sqrt{\frac{0.5}{y}}\right)} \]

      3. add-sqr-sqrt 63.9%

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\frac{0.5}{y}} \]

      4. associate-*l* 67.1%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]

   9. Applied egg-rr 67.1%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0045 \lor \neg \left(x \leq 7.8 \cdot 10^{+77}\right) \land x \leq 1.75 \cdot 10^{+110}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.6 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+110}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 8.6e-5)
   (* 0.5 y)
   (if (<= x 2.1e+77)
     (* x (/ (* 0.5 x) y))
     (if (<= x 1.35e+110) (* 0.5 y) (* x (* x (/ 0.5 y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 8.6e-5) {
		tmp = 0.5 * y;
	} else if (x <= 2.1e+77) {
		tmp = x * ((0.5 * x) / y);
	} else if (x <= 1.35e+110) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x * (0.5 / 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 (x <= 8.6d-5) then
        tmp = 0.5d0 * y
    else if (x <= 2.1d+77) then
        tmp = x * ((0.5d0 * x) / y)
    else if (x <= 1.35d+110) then
        tmp = 0.5d0 * y
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 8.6e-5) {
		tmp = 0.5 * y;
	} else if (x <= 2.1e+77) {
		tmp = x * ((0.5 * x) / y);
	} else if (x <= 1.35e+110) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 8.6e-5:
		tmp = 0.5 * y
	elif x <= 2.1e+77:
		tmp = x * ((0.5 * x) / y)
	elif x <= 1.35e+110:
		tmp = 0.5 * y
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 8.6e-5)
		tmp = Float64(0.5 * y);
	elseif (x <= 2.1e+77)
		tmp = Float64(x * Float64(Float64(0.5 * x) / y));
	elseif (x <= 1.35e+110)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 8.6e-5)
		tmp = 0.5 * y;
	elseif (x <= 2.1e+77)
		tmp = x * ((0.5 * x) / y);
	elseif (x <= 1.35e+110)
		tmp = 0.5 * y;
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 8.6e-5], N[(0.5 * y), $MachinePrecision], If[LessEqual[x, 2.1e+77], N[(x * N[(N[(0.5 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+110], N[(0.5 * y), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 8.6000000000000003e-5 or 2.0999999999999999e77 < x < 1.35000000000000005e110

   1. Initial program 69.3%

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

   3. Taylor expanded in y around inf 38.8%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 38.8%

         \[\leadsto \color{blue}{y \cdot 0.5} \]

   5. Simplified 38.8%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

   if 8.6000000000000003e-5 < x < 2.0999999999999999e77

   1. Initial program 84.4%

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

   3. Taylor expanded in x around inf 47.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative 47.8%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.5} \]

      2. associate-*l/ 47.8%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.5}{y}} \]

      3. associate-*r/ 47.7%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{y}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 26.3%

         \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot \frac{0.5}{y}} \cdot \sqrt{{x}^{2} \cdot \frac{0.5}{y}}} \]

      2. pow2 26.3%

         \[\leadsto \color{blue}{{\left(\sqrt{{x}^{2} \cdot \frac{0.5}{y}}\right)}^{2}} \]

      3. sqrt-prod 26.3%

         \[\leadsto {\color{blue}{\left(\sqrt{{x}^{2}} \cdot \sqrt{\frac{0.5}{y}}\right)}}^{2} \]

      4. sqrt-pow1 26.3%

         \[\leadsto {\left(\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

      5. metadata-eval 26.3%

         \[\leadsto {\left({x}^{\color{blue}{1}} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

      6. pow1 26.3%

         \[\leadsto {\left(\color{blue}{x} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

   7. Applied egg-rr 26.3%

      \[\leadsto \color{blue}{{\left(x \cdot \sqrt{\frac{0.5}{y}}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 26.3%

         \[\leadsto \color{blue}{\left(x \cdot \sqrt{\frac{0.5}{y}}\right) \cdot \left(x \cdot \sqrt{\frac{0.5}{y}}\right)} \]

      2. swap-sqr 26.2%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\sqrt{\frac{0.5}{y}} \cdot \sqrt{\frac{0.5}{y}}\right)} \]

      3. add-sqr-sqrt 47.7%

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\frac{0.5}{y}} \]

      4. associate-*l* 47.5%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]

   9. Applied egg-rr 47.5%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]

   10. Taylor expanded in x around 0 47.6%

       \[\leadsto x \cdot \color{blue}{\left(0.5 \cdot \frac{x}{y}\right)} \]
   11. Step-by-step derivation

       1. associate-*r/ 47.6%

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

       2. *-commutative 47.6%

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

   12. Simplified 47.6%

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

   if 1.35000000000000005e110 < x

   1. Initial program 61.6%

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

   3. Taylor expanded in x around inf 72.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative 72.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.5} \]

      2. associate-*l/ 72.0%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.5}{y}} \]

      3. associate-*r/ 72.0%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 72.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{y}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 40.2%

         \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot \frac{0.5}{y}} \cdot \sqrt{{x}^{2} \cdot \frac{0.5}{y}}} \]

      2. pow2 40.2%

         \[\leadsto \color{blue}{{\left(\sqrt{{x}^{2} \cdot \frac{0.5}{y}}\right)}^{2}} \]

      3. sqrt-prod 40.2%

         \[\leadsto {\color{blue}{\left(\sqrt{{x}^{2}} \cdot \sqrt{\frac{0.5}{y}}\right)}}^{2} \]

      4. sqrt-pow1 45.1%

         \[\leadsto {\left(\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

      5. metadata-eval 45.1%

         \[\leadsto {\left({x}^{\color{blue}{1}} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

      6. pow1 45.1%

         \[\leadsto {\left(\color{blue}{x} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

   7. Applied egg-rr 45.1%

      \[\leadsto \color{blue}{{\left(x \cdot \sqrt{\frac{0.5}{y}}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 45.1%

         \[\leadsto \color{blue}{\left(x \cdot \sqrt{\frac{0.5}{y}}\right) \cdot \left(x \cdot \sqrt{\frac{0.5}{y}}\right)} \]

      2. swap-sqr 40.2%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\sqrt{\frac{0.5}{y}} \cdot \sqrt{\frac{0.5}{y}}\right)} \]

      3. add-sqr-sqrt 72.0%

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\frac{0.5}{y}} \]

      4. associate-*l* 76.9%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]

   9. Applied egg-rr 76.9%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8.6 \cdot 10^{-5}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+110}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0225:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot \left(0.5 \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.0225)
   (* 0.5 y)
   (if (<= x 4.1e+77)
     (/ (* x (* 0.5 x)) y)
     (if (<= x 8e+113) (* 0.5 y) (* x (* x (/ 0.5 y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0225) {
		tmp = 0.5 * y;
	} else if (x <= 4.1e+77) {
		tmp = (x * (0.5 * x)) / y;
	} else if (x <= 8e+113) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x * (0.5 / 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 (x <= 0.0225d0) then
        tmp = 0.5d0 * y
    else if (x <= 4.1d+77) then
        tmp = (x * (0.5d0 * x)) / y
    else if (x <= 8d+113) then
        tmp = 0.5d0 * y
    else
        tmp = x * (x * (0.5d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.0225) {
		tmp = 0.5 * y;
	} else if (x <= 4.1e+77) {
		tmp = (x * (0.5 * x)) / y;
	} else if (x <= 8e+113) {
		tmp = 0.5 * y;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.0225:
		tmp = 0.5 * y
	elif x <= 4.1e+77:
		tmp = (x * (0.5 * x)) / y
	elif x <= 8e+113:
		tmp = 0.5 * y
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.0225)
		tmp = Float64(0.5 * y);
	elseif (x <= 4.1e+77)
		tmp = Float64(Float64(x * Float64(0.5 * x)) / y);
	elseif (x <= 8e+113)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.0225)
		tmp = 0.5 * y;
	elseif (x <= 4.1e+77)
		tmp = (x * (0.5 * x)) / y;
	elseif (x <= 8e+113)
		tmp = 0.5 * y;
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.0225], N[(0.5 * y), $MachinePrecision], If[LessEqual[x, 4.1e+77], N[(N[(x * N[(0.5 * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 8e+113], N[(0.5 * y), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.022499999999999999 or 4.1000000000000001e77 < x < 8e113

   1. Initial program 69.3%

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

   3. Taylor expanded in y around inf 38.8%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 38.8%

         \[\leadsto \color{blue}{y \cdot 0.5} \]

   5. Simplified 38.8%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

   if 0.022499999999999999 < x < 4.1000000000000001e77

   1. Initial program 84.4%

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

   3. Taylor expanded in x around inf 47.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative 47.8%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.5} \]

      2. associate-*l/ 47.8%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.5}{y}} \]

      3. associate-*r/ 47.7%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 47.7%

      \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{y}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 26.3%

         \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot \frac{0.5}{y}} \cdot \sqrt{{x}^{2} \cdot \frac{0.5}{y}}} \]

      2. pow2 26.3%

         \[\leadsto \color{blue}{{\left(\sqrt{{x}^{2} \cdot \frac{0.5}{y}}\right)}^{2}} \]

      3. sqrt-prod 26.3%

         \[\leadsto {\color{blue}{\left(\sqrt{{x}^{2}} \cdot \sqrt{\frac{0.5}{y}}\right)}}^{2} \]

      4. sqrt-pow1 26.3%

         \[\leadsto {\left(\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

      5. metadata-eval 26.3%

         \[\leadsto {\left({x}^{\color{blue}{1}} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

      6. pow1 26.3%

         \[\leadsto {\left(\color{blue}{x} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

   7. Applied egg-rr 26.3%

      \[\leadsto \color{blue}{{\left(x \cdot \sqrt{\frac{0.5}{y}}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 26.3%

         \[\leadsto \color{blue}{\left(x \cdot \sqrt{\frac{0.5}{y}}\right) \cdot \left(x \cdot \sqrt{\frac{0.5}{y}}\right)} \]

      2. swap-sqr 26.2%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\sqrt{\frac{0.5}{y}} \cdot \sqrt{\frac{0.5}{y}}\right)} \]

      3. add-sqr-sqrt 47.7%

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\frac{0.5}{y}} \]

      4. associate-*l* 47.5%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]

   9. Applied egg-rr 47.5%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 47.5%

          \[\leadsto \color{blue}{\left(x \cdot \frac{0.5}{y}\right) \cdot x} \]

       2. associate-*r/ 47.6%

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

       3. associate-*l/ 47.8%

          \[\leadsto \color{blue}{\frac{\left(x \cdot 0.5\right) \cdot x}{y}} \]

   11. Applied egg-rr 47.8%

       \[\leadsto \color{blue}{\frac{\left(x \cdot 0.5\right) \cdot x}{y}} \]

   if 8e113 < x

   1. Initial program 61.6%

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

   3. Taylor expanded in x around inf 72.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative 72.0%

         \[\leadsto \color{blue}{\frac{{x}^{2}}{y} \cdot 0.5} \]

      2. associate-*l/ 72.0%

         \[\leadsto \color{blue}{\frac{{x}^{2} \cdot 0.5}{y}} \]

      3. associate-*r/ 72.0%

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{y}} \]

   5. Simplified 72.0%

      \[\leadsto \color{blue}{{x}^{2} \cdot \frac{0.5}{y}} \]
   6. Step-by-step derivation

      1. add-sqr-sqrt 40.2%

         \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot \frac{0.5}{y}} \cdot \sqrt{{x}^{2} \cdot \frac{0.5}{y}}} \]

      2. pow2 40.2%

         \[\leadsto \color{blue}{{\left(\sqrt{{x}^{2} \cdot \frac{0.5}{y}}\right)}^{2}} \]

      3. sqrt-prod 40.2%

         \[\leadsto {\color{blue}{\left(\sqrt{{x}^{2}} \cdot \sqrt{\frac{0.5}{y}}\right)}}^{2} \]

      4. sqrt-pow1 45.1%

         \[\leadsto {\left(\color{blue}{{x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

      5. metadata-eval 45.1%

         \[\leadsto {\left({x}^{\color{blue}{1}} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

      6. pow1 45.1%

         \[\leadsto {\left(\color{blue}{x} \cdot \sqrt{\frac{0.5}{y}}\right)}^{2} \]

   7. Applied egg-rr 45.1%

      \[\leadsto \color{blue}{{\left(x \cdot \sqrt{\frac{0.5}{y}}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 45.1%

         \[\leadsto \color{blue}{\left(x \cdot \sqrt{\frac{0.5}{y}}\right) \cdot \left(x \cdot \sqrt{\frac{0.5}{y}}\right)} \]

      2. swap-sqr 40.2%

         \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\sqrt{\frac{0.5}{y}} \cdot \sqrt{\frac{0.5}{y}}\right)} \]

      3. add-sqr-sqrt 72.0%

         \[\leadsto \left(x \cdot x\right) \cdot \color{blue}{\frac{0.5}{y}} \]

      4. associate-*l* 76.9%

         \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]

   9. Applied egg-rr 76.9%

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \frac{0.5}{y}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0225:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+77}:\\ \;\;\;\;\frac{x \cdot \left(0.5 \cdot x\right)}{y}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00037:\\ \;\;\;\;0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 0.00037)
   (* 0.5 (+ y (* (- x z) (/ z y))))
   (if (<= x 1.06e+77)
     (* 0.5 (* (- x z) (/ (+ x z) y)))
     (* 0.5 (+ y (* (- x z) (/ x y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00037) {
		tmp = 0.5 * (y + ((x - z) * (z / y)));
	} else if (x <= 1.06e+77) {
		tmp = 0.5 * ((x - z) * ((x + z) / y));
	} else {
		tmp = 0.5 * (y + ((x - 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 (x <= 0.00037d0) then
        tmp = 0.5d0 * (y + ((x - z) * (z / y)))
    else if (x <= 1.06d+77) then
        tmp = 0.5d0 * ((x - z) * ((x + z) / y))
    else
        tmp = 0.5d0 * (y + ((x - z) * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 0.00037) {
		tmp = 0.5 * (y + ((x - z) * (z / y)));
	} else if (x <= 1.06e+77) {
		tmp = 0.5 * ((x - z) * ((x + z) / y));
	} else {
		tmp = 0.5 * (y + ((x - z) * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 0.00037:
		tmp = 0.5 * (y + ((x - z) * (z / y)))
	elif x <= 1.06e+77:
		tmp = 0.5 * ((x - z) * ((x + z) / y))
	else:
		tmp = 0.5 * (y + ((x - z) * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 0.00037)
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x - z) * Float64(z / y))));
	elseif (x <= 1.06e+77)
		tmp = Float64(0.5 * Float64(Float64(x - z) * Float64(Float64(x + z) / y)));
	else
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x - z) * Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 0.00037)
		tmp = 0.5 * (y + ((x - z) * (z / y)));
	elseif (x <= 1.06e+77)
		tmp = 0.5 * ((x - z) * ((x + z) / y));
	else
		tmp = 0.5 * (y + ((x - z) * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 0.00037], N[(0.5 * N[(y + N[(N[(x - z), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.06e+77], N[(0.5 * N[(N[(x - z), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(N[(x - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 3.6999999999999999e-4

   1. Initial program 69.1%

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

      1. remove-double-neg 69.1%

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

      2. distribute-lft-neg-out 69.1%

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

      3. distribute-frac-neg2 69.1%

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

      4. distribute-frac-neg 69.1%

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

      5. neg-mul-1 69.1%

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

      6. distribute-lft-neg-out 69.1%

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

      7. *-commutative 69.1%

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

      8. distribute-lft-neg-in 69.1%

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

      9. times-frac 69.1%

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

      10. metadata-eval 69.1%

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

      11. metadata-eval 69.1%

          \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

      12. associate--l+ 69.1%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

      13. fma-define 71.7%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

   3. Simplified 71.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 78.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 78.6%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

      2. div-sub 79.6%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   7. Simplified 79.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   8. Step-by-step derivation

      1. unpow2 79.6%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. unpow2 79.6%

         \[\leadsto 0.5 \cdot \left(y + \frac{x \cdot x - \color{blue}{z \cdot z}}{y}\right) \]

      3. difference-of-squares 85.9%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   9. Applied egg-rr 85.9%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]
   10. Step-by-step derivation

       1. *-commutative 85.9%

          \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x - z\right) \cdot \left(x + z\right)}}{y}\right) \]

       2. associate-/l* 99.9%

          \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

   11. Applied egg-rr 99.9%

       \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

   12. Taylor expanded in x around 0 75.1%

       \[\leadsto 0.5 \cdot \left(y + \left(x - z\right) \cdot \color{blue}{\frac{z}{y}}\right) \]

   if 3.6999999999999999e-4 < x < 1.06000000000000003e77

   1. Initial program 84.4%

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

      1. remove-double-neg 84.4%

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

      2. distribute-lft-neg-out 84.4%

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

      3. distribute-frac-neg2 84.4%

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

      4. distribute-frac-neg 84.4%

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

      5. neg-mul-1 84.4%

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

      6. distribute-lft-neg-out 84.4%

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

      7. *-commutative 84.4%

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

      8. distribute-lft-neg-in 84.4%

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

      9. times-frac 84.4%

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

      10. metadata-eval 84.4%

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

      11. metadata-eval 84.4%

          \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

      12. associate--l+ 84.4%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

      13. fma-define 84.4%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

   3. Simplified 84.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 95.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 95.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

      2. div-sub 95.0%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   7. Simplified 95.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   8. Step-by-step derivation

      1. unpow2 95.0%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. unpow2 95.0%

         \[\leadsto 0.5 \cdot \left(y + \frac{x \cdot x - \color{blue}{z \cdot z}}{y}\right) \]

      3. difference-of-squares 95.0%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   9. Applied egg-rr 95.0%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   10. Taylor expanded in y around 0 89.9%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   11. Step-by-step derivation

       1. *-commutative 95.0%

          \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x - z\right) \cdot \left(x + z\right)}}{y}\right) \]

       2. associate-/l* 99.8%

          \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

   12. Applied egg-rr 94.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(x - z\right) \cdot \frac{x + z}{y}\right)} \]

   if 1.06000000000000003e77 < x

   1. Initial program 63.0%

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

      1. remove-double-neg 63.0%

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

      2. distribute-lft-neg-out 63.0%

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

      3. distribute-frac-neg2 63.0%

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

      4. distribute-frac-neg 63.0%

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

      5. neg-mul-1 63.0%

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

      6. distribute-lft-neg-out 63.0%

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

      7. *-commutative 63.0%

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

      8. distribute-lft-neg-in 63.0%

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

      9. times-frac 63.0%

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

      10. metadata-eval 63.0%

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

      11. metadata-eval 63.0%

          \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

      12. associate--l+ 63.0%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

      13. fma-define 70.2%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

   3. Simplified 70.2%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 65.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 65.3%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

      2. div-sub 70.1%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   7. Simplified 70.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   8. Step-by-step derivation

      1. unpow2 70.1%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. unpow2 70.1%

         \[\leadsto 0.5 \cdot \left(y + \frac{x \cdot x - \color{blue}{z \cdot z}}{y}\right) \]

      3. difference-of-squares 86.8%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   9. Applied egg-rr 86.8%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]
   10. Step-by-step derivation

       1. *-commutative 86.8%

          \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x - z\right) \cdot \left(x + z\right)}}{y}\right) \]

       2. associate-/l* 99.9%

          \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

   11. Applied egg-rr 99.9%

       \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

   12. Taylor expanded in x around inf 92.9%

       \[\leadsto 0.5 \cdot \left(y + \left(x - z\right) \cdot \color{blue}{\frac{x}{y}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00037:\\ \;\;\;\;0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.2e-38)
   (* 0.5 (* (- x z) (/ (+ x z) y)))
   (* 0.5 (+ y (* (- x z) (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.2e-38) {
		tmp = 0.5 * ((x - z) * ((x + z) / y));
	} else {
		tmp = 0.5 * (y + ((x - 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 (y <= 1.2d-38) then
        tmp = 0.5d0 * ((x - z) * ((x + z) / y))
    else
        tmp = 0.5d0 * (y + ((x - z) * (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.2e-38) {
		tmp = 0.5 * ((x - z) * ((x + z) / y));
	} else {
		tmp = 0.5 * (y + ((x - z) * (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.2e-38:
		tmp = 0.5 * ((x - z) * ((x + z) / y))
	else:
		tmp = 0.5 * (y + ((x - z) * (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.2e-38)
		tmp = Float64(0.5 * Float64(Float64(x - z) * Float64(Float64(x + z) / y)));
	else
		tmp = Float64(0.5 * Float64(y + Float64(Float64(x - z) * Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.2e-38)
		tmp = 0.5 * ((x - z) * ((x + z) / y));
	else
		tmp = 0.5 * (y + ((x - z) * (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.2e-38], N[(0.5 * N[(N[(x - z), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y + N[(N[(x - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.20000000000000011e-38

   1. Initial program 76.5%

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

      1. remove-double-neg 76.5%

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

      2. distribute-lft-neg-out 76.5%

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

      3. distribute-frac-neg2 76.5%

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

      4. distribute-frac-neg 76.5%

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

      5. neg-mul-1 76.5%

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

      6. distribute-lft-neg-out 76.5%

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

      7. *-commutative 76.5%

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

      8. distribute-lft-neg-in 76.5%

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

      9. times-frac 76.5%

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

      10. metadata-eval 76.5%

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

      11. metadata-eval 76.5%

          \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

      12. associate--l+ 76.5%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

      13. fma-define 80.5%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

   3. Simplified 80.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 81.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 81.0%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

      2. div-sub 83.3%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   7. Simplified 83.3%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   8. Step-by-step derivation

      1. unpow2 83.3%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. unpow2 83.3%

         \[\leadsto 0.5 \cdot \left(y + \frac{x \cdot x - \color{blue}{z \cdot z}}{y}\right) \]

      3. difference-of-squares 92.4%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   9. Applied egg-rr 92.4%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   10. Taylor expanded in y around 0 75.3%

       \[\leadsto 0.5 \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   11. Step-by-step derivation

       1. *-commutative 92.4%

          \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x - z\right) \cdot \left(x + z\right)}}{y}\right) \]

       2. associate-/l* 99.9%

          \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

   12. Applied egg-rr 78.5%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\left(x - z\right) \cdot \frac{x + z}{y}\right)} \]

   if 1.20000000000000011e-38 < y

   1. Initial program 53.7%

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

      1. remove-double-neg 53.7%

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

      2. distribute-lft-neg-out 53.7%

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

      3. distribute-frac-neg2 53.7%

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

      4. distribute-frac-neg 53.7%

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

      5. neg-mul-1 53.7%

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

      6. distribute-lft-neg-out 53.7%

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

      7. *-commutative 53.7%

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

      8. distribute-lft-neg-in 53.7%

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

      9. times-frac 53.7%

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

      10. metadata-eval 53.7%

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

      11. metadata-eval 53.7%

          \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

      12. associate--l+ 53.7%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

      13. fma-define 55.0%

          \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

   3. Simplified 55.0%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 70.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
   6. Step-by-step derivation

      1. associate--l+ 70.4%

         \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

      2. div-sub 70.4%

         \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

   7. Simplified 70.4%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
   8. Step-by-step derivation

      1. unpow2 70.4%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

      2. unpow2 70.4%

         \[\leadsto 0.5 \cdot \left(y + \frac{x \cdot x - \color{blue}{z \cdot z}}{y}\right) \]

      3. difference-of-squares 74.3%

         \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

   9. Applied egg-rr 74.3%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]
   10. Step-by-step derivation

       1. *-commutative 74.3%

          \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x - z\right) \cdot \left(x + z\right)}}{y}\right) \]

       2. associate-/l* 99.9%

          \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

   11. Applied egg-rr 99.9%

       \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

   12. Taylor expanded in x around inf 83.2%

       \[\leadsto 0.5 \cdot \left(y + \left(x - z\right) \cdot \color{blue}{\frac{x}{y}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-38}:\\ \;\;\;\;0.5 \cdot \left(\left(x - z\right) \cdot \frac{x + z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{x + z}{y}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 (+ y (* (- x z) (/ (+ x z) y)))))

C

double code(double x, double y, double z) {
	return 0.5 * (y + ((x - z) * ((x + 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 = 0.5d0 * (y + ((x - z) * ((x + z) / y)))
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * (y + ((x - z) * ((x + z) / y)));
}

Python

def code(x, y, z):
	return 0.5 * (y + ((x - z) * ((x + z) / y)))

Julia

function code(x, y, z)
	return Float64(0.5 * Float64(y + Float64(Float64(x - z) * Float64(Float64(x + z) / y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * (y + ((x - z) * ((x + z) / y)));
end

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(y + N[(N[(x - z), $MachinePrecision] * N[(N[(x + z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

   1. remove-double-neg 69.3%

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

   2. distribute-lft-neg-out 69.3%

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

   3. distribute-frac-neg2 69.3%

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

   4. distribute-frac-neg 69.3%

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

   5. neg-mul-1 69.3%

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

   6. distribute-lft-neg-out 69.3%

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

   7. *-commutative 69.3%

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

   8. distribute-lft-neg-in 69.3%

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

   9. times-frac 69.3%

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

   10. metadata-eval 69.3%

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

   11. metadata-eval 69.3%

       \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

   12. associate--l+ 69.3%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

   13. fma-define 72.4%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

3. Simplified 72.4%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 77.6%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
6. Step-by-step derivation

   1. associate--l+ 77.6%

      \[\leadsto 0.5 \cdot \color{blue}{\left(y + \left(\frac{{x}^{2}}{y} - \frac{{z}^{2}}{y}\right)\right)} \]

   2. div-sub 79.2%

      \[\leadsto 0.5 \cdot \left(y + \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}\right) \]

7. Simplified 79.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
8. Step-by-step derivation

   1. unpow2 79.2%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{x \cdot x} - {z}^{2}}{y}\right) \]

   2. unpow2 79.2%

      \[\leadsto 0.5 \cdot \left(y + \frac{x \cdot x - \color{blue}{z \cdot z}}{y}\right) \]

   3. difference-of-squares 86.7%

      \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]

9. Applied egg-rr 86.7%

   \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]
10. Step-by-step derivation

    1. *-commutative 86.7%

       \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x - z\right) \cdot \left(x + z\right)}}{y}\right) \]

    2. associate-/l* 99.9%

       \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

11. Applied egg-rr 99.9%

    \[\leadsto 0.5 \cdot \left(y + \color{blue}{\left(x - z\right) \cdot \frac{x + z}{y}}\right) \]

12. Final simplification 99.9%

    \[\leadsto 0.5 \cdot \left(y + \left(x - z\right) \cdot \frac{x + z}{y}\right) \]
13. Add Preprocessing

Alternative 10: 33.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 y))

C

double code(double x, double y, double z) {
	return 0.5 * 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 = 0.5d0 * y
end function

Java

public static double code(double x, double y, double z) {
	return 0.5 * y;
}

Python

def code(x, y, z):
	return 0.5 * y

Julia

function code(x, y, z)
	return Float64(0.5 * y)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.5 * y;
end

Wolfram

code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 69.3%

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

3. Taylor expanded in y around inf 32.6%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
4. Step-by-step derivation

   1. *-commutative 32.6%

      \[\leadsto \color{blue}{y \cdot 0.5} \]

5. Simplified 32.6%

   \[\leadsto \color{blue}{y \cdot 0.5} \]

6. Final simplification 32.6%

   \[\leadsto 0.5 \cdot y \]
7. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Python

def code(x, y, z):
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x))

Julia

function code(x, y, z)
	return Float64(Float64(y * 0.5) - Float64(Float64(Float64(0.5 / y) * Float64(z + x)) * Float64(z - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024115 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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