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

Percentage Accurate: 68.8% → 99.9%

Time: 7.1s

Alternatives: 12

Speedup: 1.2×

• 41.8% 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: 68.8% 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: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. sub-neg 69.7%

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

   2. +-commutative 69.7%

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

   3. neg-sub0 69.7%

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

   4. associate-+l- 69.7%

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

   5. sub0-neg 69.7%

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

   6. neg-mul-1 69.7%

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

   7. *-commutative 69.7%

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

   8. times-frac 69.7%

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

   9. associate--r+ 69.7%

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

   10. div-sub 69.7%

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

   11. difference-of-squares 74.9%

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

   12. +-commutative 74.9%

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

   13. associate-*l/ 76.5%

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

   14. *-commutative 76.5%

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

   15. associate-/l* 99.9%

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

   16. *-inverses 99.9%

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

   17. /-rgt-identity 99.9%

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

   18. metadata-eval 99.9%

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

3. Simplified 99.9%

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

   1. clear-num 99.9%

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

   2. un-div-inv 99.9%

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

   3. +-commutative 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-187} \lor \neg \left(x \cdot x \leq 4 \cdot 10^{-129}\right) \land x \cdot x \leq 10^{+46}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x x) 2e-187)
         (and (not (<= (* x x) 4e-129)) (<= (* x x) 1e+46)))
   (* -0.5 (- (/ (* z z) y) y))
   (* -0.5 (- (* (- z x) (/ x y)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 2e-187) || (!((x * x) <= 4e-129) && ((x * x) <= 1e+46))) {
		tmp = -0.5 * (((z * z) / y) - y);
	} else {
		tmp = -0.5 * (((z - x) * (x / y)) - 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 * x) <= 2d-187) .or. (.not. ((x * x) <= 4d-129)) .and. ((x * x) <= 1d+46)) then
        tmp = (-0.5d0) * (((z * z) / y) - y)
    else
        tmp = (-0.5d0) * (((z - x) * (x / y)) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 2e-187) || (!((x * x) <= 4e-129) && ((x * x) <= 1e+46))) {
		tmp = -0.5 * (((z * z) / y) - y);
	} else {
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * x) <= 2e-187) or (not ((x * x) <= 4e-129) and ((x * x) <= 1e+46)):
		tmp = -0.5 * (((z * z) / y) - y)
	else:
		tmp = -0.5 * (((z - x) * (x / y)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * x) <= 2e-187) || (!(Float64(x * x) <= 4e-129) && (Float64(x * x) <= 1e+46)))
		tmp = Float64(-0.5 * Float64(Float64(Float64(z * z) / y) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(z - x) * Float64(x / y)) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * x) <= 2e-187) || (~(((x * x) <= 4e-129)) && ((x * x) <= 1e+46)))
		tmp = -0.5 * (((z * z) / y) - y);
	else
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 2e-187], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 4e-129]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 1e+46]]], N[(-0.5 * N[(N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[(z - x), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2e-187 or 3.9999999999999997e-129 < (*.f64 x x) < 9.9999999999999999e45

   1. Initial program 75.5%

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

      1. sub-neg 75.5%

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

      2. +-commutative 75.5%

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

      3. neg-sub0 75.5%

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

      4. associate-+l- 75.5%

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

      5. sub0-neg 75.5%

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

      6. neg-mul-1 75.5%

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

      7. *-commutative 75.5%

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

      8. times-frac 75.5%

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

      9. associate--r+ 75.5%

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

      10. div-sub 75.5%

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

      11. difference-of-squares 75.5%

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

      12. +-commutative 75.5%

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

      13. associate-*l/ 77.0%

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

      14. *-commutative 77.0%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 94.5%

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

      1. unpow2 94.5%

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

   6. Simplified 94.5%

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

   if 2e-187 < (*.f64 x x) < 3.9999999999999997e-129 or 9.9999999999999999e45 < (*.f64 x x)

   1. Initial program 64.0%

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

      1. sub-neg 64.0%

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

      2. +-commutative 64.0%

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

      3. neg-sub0 64.0%

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

      4. associate-+l- 64.0%

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

      5. sub0-neg 64.0%

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

      6. neg-mul-1 64.0%

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

      7. *-commutative 64.0%

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

      8. times-frac 64.0%

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

      9. associate--r+ 64.0%

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

      10. div-sub 64.0%

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

      11. difference-of-squares 74.4%

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

      12. +-commutative 74.4%

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

      13. associate-*l/ 76.0%

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

      14. *-commutative 76.0%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 84.3%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-187} \lor \neg \left(x \cdot x \leq 4 \cdot 10^{-129}\right) \land x \cdot x \leq 10^{+46}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \]

Alternative 3: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-187} \lor \neg \left(x \cdot x \leq 4 \cdot 10^{-129}\right) \land x \cdot x \leq 10^{+46}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z - x}{\frac{y}{x}} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x x) 2e-187)
         (and (not (<= (* x x) 4e-129)) (<= (* x x) 1e+46)))
   (* -0.5 (- (/ (* z z) y) y))
   (* -0.5 (- (/ (- z x) (/ y x)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 2e-187) || (!((x * x) <= 4e-129) && ((x * x) <= 1e+46))) {
		tmp = -0.5 * (((z * z) / y) - y);
	} else {
		tmp = -0.5 * (((z - x) / (y / 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 * x) <= 2d-187) .or. (.not. ((x * x) <= 4d-129)) .and. ((x * x) <= 1d+46)) then
        tmp = (-0.5d0) * (((z * z) / y) - y)
    else
        tmp = (-0.5d0) * (((z - x) / (y / x)) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 2e-187) || (!((x * x) <= 4e-129) && ((x * x) <= 1e+46))) {
		tmp = -0.5 * (((z * z) / y) - y);
	} else {
		tmp = -0.5 * (((z - x) / (y / x)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * x) <= 2e-187) or (not ((x * x) <= 4e-129) and ((x * x) <= 1e+46)):
		tmp = -0.5 * (((z * z) / y) - y)
	else:
		tmp = -0.5 * (((z - x) / (y / x)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * x) <= 2e-187) || (!(Float64(x * x) <= 4e-129) && (Float64(x * x) <= 1e+46)))
		tmp = Float64(-0.5 * Float64(Float64(Float64(z * z) / y) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(z - x) / Float64(y / x)) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * x) <= 2e-187) || (~(((x * x) <= 4e-129)) && ((x * x) <= 1e+46)))
		tmp = -0.5 * (((z * z) / y) - y);
	else
		tmp = -0.5 * (((z - x) / (y / x)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 2e-187], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 4e-129]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 1e+46]]], N[(-0.5 * N[(N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[(z - x), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2e-187 or 3.9999999999999997e-129 < (*.f64 x x) < 9.9999999999999999e45

   1. Initial program 75.5%

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

      1. sub-neg 75.5%

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

      2. +-commutative 75.5%

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

      3. neg-sub0 75.5%

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

      4. associate-+l- 75.5%

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

      5. sub0-neg 75.5%

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

      6. neg-mul-1 75.5%

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

      7. *-commutative 75.5%

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

      8. times-frac 75.5%

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

      9. associate--r+ 75.5%

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

      10. div-sub 75.5%

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

      11. difference-of-squares 75.5%

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

      12. +-commutative 75.5%

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

      13. associate-*l/ 77.0%

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

      14. *-commutative 77.0%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 94.5%

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

      1. unpow2 94.5%

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

   6. Simplified 94.5%

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

   if 2e-187 < (*.f64 x x) < 3.9999999999999997e-129 or 9.9999999999999999e45 < (*.f64 x x)

   1. Initial program 64.0%

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

      1. sub-neg 64.0%

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

      2. +-commutative 64.0%

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

      3. neg-sub0 64.0%

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

      4. associate-+l- 64.0%

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

      5. sub0-neg 64.0%

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

      6. neg-mul-1 64.0%

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

      7. *-commutative 64.0%

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

      8. times-frac 64.0%

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

      9. associate--r+ 64.0%

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

      10. div-sub 64.0%

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

      11. difference-of-squares 74.4%

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

      12. +-commutative 74.4%

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

      13. associate-*l/ 76.0%

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

      14. *-commutative 76.0%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

      3. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in z around 0 84.3%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-187} \lor \neg \left(x \cdot x \leq 4 \cdot 10^{-129}\right) \land x \cdot x \leq 10^{+46}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z - x}{\frac{y}{x}} - y\right)\\ \end{array} \]

Alternative 4: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-187} \lor \neg \left(x \cdot x \leq 4 \cdot 10^{-129}\right) \land x \cdot x \leq 4 \cdot 10^{+55}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{x}{y} \cdot \left(-x\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= (* x x) 2e-187)
         (and (not (<= (* x x) 4e-129)) (<= (* x x) 4e+55)))
   (* -0.5 (- (/ (* z z) y) y))
   (* -0.5 (- (* (/ x y) (- x)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 2e-187) || (!((x * x) <= 4e-129) && ((x * x) <= 4e+55))) {
		tmp = -0.5 * (((z * z) / y) - y);
	} else {
		tmp = -0.5 * (((x / y) * -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 * x) <= 2d-187) .or. (.not. ((x * x) <= 4d-129)) .and. ((x * x) <= 4d+55)) then
        tmp = (-0.5d0) * (((z * z) / y) - y)
    else
        tmp = (-0.5d0) * (((x / y) * -x) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (((x * x) <= 2e-187) || (!((x * x) <= 4e-129) && ((x * x) <= 4e+55))) {
		tmp = -0.5 * (((z * z) / y) - y);
	} else {
		tmp = -0.5 * (((x / y) * -x) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if ((x * x) <= 2e-187) or (not ((x * x) <= 4e-129) and ((x * x) <= 4e+55)):
		tmp = -0.5 * (((z * z) / y) - y)
	else:
		tmp = -0.5 * (((x / y) * -x) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((Float64(x * x) <= 2e-187) || (!(Float64(x * x) <= 4e-129) && (Float64(x * x) <= 4e+55)))
		tmp = Float64(-0.5 * Float64(Float64(Float64(z * z) / y) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(x / y) * Float64(-x)) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (((x * x) <= 2e-187) || (~(((x * x) <= 4e-129)) && ((x * x) <= 4e+55)))
		tmp = -0.5 * (((z * z) / y) - y);
	else
		tmp = -0.5 * (((x / y) * -x) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[N[(x * x), $MachinePrecision], 2e-187], And[N[Not[LessEqual[N[(x * x), $MachinePrecision], 4e-129]], $MachinePrecision], LessEqual[N[(x * x), $MachinePrecision], 4e+55]]], N[(-0.5 * N[(N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(N[(x / y), $MachinePrecision] * (-x)), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2e-187 or 3.9999999999999997e-129 < (*.f64 x x) < 4.00000000000000004e55

   1. Initial program 75.6%

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

      1. sub-neg 75.6%

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

      2. +-commutative 75.6%

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

      3. neg-sub0 75.6%

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

      4. associate-+l- 75.6%

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

      5. sub0-neg 75.6%

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

      6. neg-mul-1 75.6%

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

      7. *-commutative 75.6%

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

      8. times-frac 75.6%

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

      9. associate--r+ 75.6%

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

      10. div-sub 75.6%

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

      11. difference-of-squares 75.6%

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

      12. +-commutative 75.6%

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

      13. associate-*l/ 77.1%

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

      14. *-commutative 77.1%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 94.5%

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

      1. unpow2 94.5%

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

   6. Simplified 94.5%

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

   if 2e-187 < (*.f64 x x) < 3.9999999999999997e-129 or 4.00000000000000004e55 < (*.f64 x x)

   1. Initial program 63.8%

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

      1. sub-neg 63.8%

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

      2. +-commutative 63.8%

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

      3. neg-sub0 63.8%

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

      4. associate-+l- 63.8%

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

      5. sub0-neg 63.8%

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

      6. neg-mul-1 63.8%

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

      7. *-commutative 63.8%

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

      8. times-frac 63.8%

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

      9. associate--r+ 63.8%

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

      10. div-sub 63.8%

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

      11. difference-of-squares 74.2%

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

      12. +-commutative 74.2%

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

      13. associate-*l/ 75.8%

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

      14. *-commutative 75.8%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

      3. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in z around 0 69.0%

      \[\leadsto \left(\color{blue}{-1 \cdot \frac{{x}^{2}}{y}} - y\right) \cdot -0.5 \]
   7. Step-by-step derivation

      1. unpow2 69.0%

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

      2. associate-*r/ 69.0%

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

      3. mul-1-neg 69.0%

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

      4. distribute-rgt-neg-out 69.0%

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

      5. associate-*l/ 80.4%

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

   8. Simplified 80.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-187} \lor \neg \left(x \cdot x \leq 4 \cdot 10^{-129}\right) \land x \cdot x \leq 4 \cdot 10^{+55}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{x}{y} \cdot \left(-x\right) - y\right)\\ \end{array} \]

Alternative 5: 80.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-187}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{-129}:\\ \;\;\;\;-0.5 \cdot \left(\left(-y\right) - \frac{x \cdot x}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{x}{y} \cdot \left(-x\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (- (/ (* z z) y) y))))
   (if (<= (* x x) 2e-187)
     t_0
     (if (<= (* x x) 4e-129)
       (* -0.5 (- (- y) (/ (* x x) y)))
       (if (<= (* x x) 4e+55) t_0 (* -0.5 (- (* (/ x y) (- x)) y)))))))

C

double code(double x, double y, double z) {
	double t_0 = -0.5 * (((z * z) / y) - y);
	double tmp;
	if ((x * x) <= 2e-187) {
		tmp = t_0;
	} else if ((x * x) <= 4e-129) {
		tmp = -0.5 * (-y - ((x * x) / y));
	} else if ((x * x) <= 4e+55) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (((x / y) * -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) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (((z * z) / y) - y)
    if ((x * x) <= 2d-187) then
        tmp = t_0
    else if ((x * x) <= 4d-129) then
        tmp = (-0.5d0) * (-y - ((x * x) / y))
    else if ((x * x) <= 4d+55) then
        tmp = t_0
    else
        tmp = (-0.5d0) * (((x / y) * -x) - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.5 * (((z * z) / y) - y);
	double tmp;
	if ((x * x) <= 2e-187) {
		tmp = t_0;
	} else if ((x * x) <= 4e-129) {
		tmp = -0.5 * (-y - ((x * x) / y));
	} else if ((x * x) <= 4e+55) {
		tmp = t_0;
	} else {
		tmp = -0.5 * (((x / y) * -x) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.5 * (((z * z) / y) - y)
	tmp = 0
	if (x * x) <= 2e-187:
		tmp = t_0
	elif (x * x) <= 4e-129:
		tmp = -0.5 * (-y - ((x * x) / y))
	elif (x * x) <= 4e+55:
		tmp = t_0
	else:
		tmp = -0.5 * (((x / y) * -x) - y)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(Float64(Float64(z * z) / y) - y))
	tmp = 0.0
	if (Float64(x * x) <= 2e-187)
		tmp = t_0;
	elseif (Float64(x * x) <= 4e-129)
		tmp = Float64(-0.5 * Float64(Float64(-y) - Float64(Float64(x * x) / y)));
	elseif (Float64(x * x) <= 4e+55)
		tmp = t_0;
	else
		tmp = Float64(-0.5 * Float64(Float64(Float64(x / y) * Float64(-x)) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (((z * z) / y) - y);
	tmp = 0.0;
	if ((x * x) <= 2e-187)
		tmp = t_0;
	elseif ((x * x) <= 4e-129)
		tmp = -0.5 * (-y - ((x * x) / y));
	elseif ((x * x) <= 4e+55)
		tmp = t_0;
	else
		tmp = -0.5 * (((x / y) * -x) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.5 * N[(N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e-187], t$95$0, If[LessEqual[N[(x * x), $MachinePrecision], 4e-129], N[(-0.5 * N[((-y) - N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 4e+55], t$95$0, N[(-0.5 * N[(N[(N[(x / y), $MachinePrecision] * (-x)), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2e-187 or 3.9999999999999997e-129 < (*.f64 x x) < 4.00000000000000004e55

   1. Initial program 75.6%

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

      1. sub-neg 75.6%

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

      2. +-commutative 75.6%

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

      3. neg-sub0 75.6%

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

      4. associate-+l- 75.6%

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

      5. sub0-neg 75.6%

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

      6. neg-mul-1 75.6%

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

      7. *-commutative 75.6%

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

      8. times-frac 75.6%

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

      9. associate--r+ 75.6%

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

      10. div-sub 75.6%

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

      11. difference-of-squares 75.6%

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

      12. +-commutative 75.6%

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

      13. associate-*l/ 77.1%

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

      14. *-commutative 77.1%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 94.5%

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

      1. unpow2 94.5%

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

   6. Simplified 94.5%

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

   if 2e-187 < (*.f64 x x) < 3.9999999999999997e-129

   1. Initial program 74.6%

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

      1. sub-neg 74.6%

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

      2. +-commutative 74.6%

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

      3. neg-sub0 74.6%

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

      4. associate-+l- 74.6%

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

      5. sub0-neg 74.6%

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

      6. neg-mul-1 74.6%

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

      7. *-commutative 74.6%

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

      8. times-frac 74.6%

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

      9. associate--r+ 74.6%

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

      10. div-sub 74.6%

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

      11. difference-of-squares 74.6%

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

      12. +-commutative 74.6%

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

      13. associate-*l/ 74.8%

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

      14. *-commutative 74.8%

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

      15. associate-/l* 99.8%

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

      16. *-inverses 99.8%

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

      17. /-rgt-identity 99.8%

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

      18. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 84.3%

      \[\leadsto \left(\color{blue}{-1 \cdot \frac{{x}^{2}}{y}} - y\right) \cdot -0.5 \]
   5. Step-by-step derivation

      1. mul-1-neg 84.3%

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

      2. unpow2 84.3%

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

      3. distribute-neg-frac 84.3%

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

   6. Simplified 84.3%

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

   if 4.00000000000000004e55 < (*.f64 x x)

   1. Initial program 61.9%

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

      1. sub-neg 61.9%

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

      2. +-commutative 61.9%

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

      3. neg-sub0 61.9%

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

      4. associate-+l- 61.9%

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

      5. sub0-neg 61.9%

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

      6. neg-mul-1 61.9%

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

      7. *-commutative 61.9%

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

      8. times-frac 61.9%

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

      9. associate--r+ 61.9%

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

      10. div-sub 61.9%

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

      11. difference-of-squares 74.1%

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

      12. +-commutative 74.1%

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

      13. associate-*l/ 75.9%

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

      14. *-commutative 75.9%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

      3. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in z around 0 66.3%

      \[\leadsto \left(\color{blue}{-1 \cdot \frac{{x}^{2}}{y}} - y\right) \cdot -0.5 \]
   7. Step-by-step derivation

      1. unpow2 66.3%

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

      2. associate-*r/ 66.3%

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

      3. mul-1-neg 66.3%

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

      4. distribute-rgt-neg-out 66.3%

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

      5. associate-*l/ 79.7%

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

   8. Simplified 79.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-187}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{-129}:\\ \;\;\;\;-0.5 \cdot \left(\left(-y\right) - \frac{x \cdot x}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+55}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{x}{y} \cdot \left(-x\right) - y\right)\\ \end{array} \]

Alternative 6: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{+42}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* z (/ z y)))))
   (if (<= y -5.6e+42)
     (* y 0.5)
     (if (<= y 1.08e-67)
       t_0
       (if (<= y 1.8e-22)
         (* x (* x (/ 0.5 y)))
         (if (<= y 3.4e+43) t_0 (* y 0.5)))))))

C

double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double tmp;
	if (y <= -5.6e+42) {
		tmp = y * 0.5;
	} else if (y <= 1.08e-67) {
		tmp = t_0;
	} else if (y <= 1.8e-22) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 3.4e+43) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (z * (z / y))
    if (y <= (-5.6d+42)) then
        tmp = y * 0.5d0
    else if (y <= 1.08d-67) then
        tmp = t_0
    else if (y <= 1.8d-22) then
        tmp = x * (x * (0.5d0 / y))
    else if (y <= 3.4d+43) then
        tmp = t_0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double tmp;
	if (y <= -5.6e+42) {
		tmp = y * 0.5;
	} else if (y <= 1.08e-67) {
		tmp = t_0;
	} else if (y <= 1.8e-22) {
		tmp = x * (x * (0.5 / y));
	} else if (y <= 3.4e+43) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.5 * (z * (z / y))
	tmp = 0
	if y <= -5.6e+42:
		tmp = y * 0.5
	elif y <= 1.08e-67:
		tmp = t_0
	elif y <= 1.8e-22:
		tmp = x * (x * (0.5 / y))
	elif y <= 3.4e+43:
		tmp = t_0
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(z * Float64(z / y)))
	tmp = 0.0
	if (y <= -5.6e+42)
		tmp = Float64(y * 0.5);
	elseif (y <= 1.08e-67)
		tmp = t_0;
	elseif (y <= 1.8e-22)
		tmp = Float64(x * Float64(x * Float64(0.5 / y)));
	elseif (y <= 3.4e+43)
		tmp = t_0;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (z * (z / y));
	tmp = 0.0;
	if (y <= -5.6e+42)
		tmp = y * 0.5;
	elseif (y <= 1.08e-67)
		tmp = t_0;
	elseif (y <= 1.8e-22)
		tmp = x * (x * (0.5 / y));
	elseif (y <= 3.4e+43)
		tmp = t_0;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.6e+42], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, 1.08e-67], t$95$0, If[LessEqual[y, 1.8e-22], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+43], t$95$0, N[(y * 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.5999999999999999e42 or 3.40000000000000012e43 < y

   1. Initial program 44.2%

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

   2. Taylor expanded in y around inf 74.3%

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

      1. *-commutative 74.3%

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

   4. Simplified 74.3%

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

   if -5.5999999999999999e42 < y < 1.0800000000000001e-67 or 1.7999999999999999e-22 < y < 3.40000000000000012e43

   1. Initial program 88.8%

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

   2. Taylor expanded in z around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. unpow2 57.4%

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

      3. associate-/l* 58.1%

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

   4. Simplified 58.1%

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

      1. associate-/r/ 58.1%

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

   6. Applied egg-rr 58.1%

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

   if 1.0800000000000001e-67 < y < 1.7999999999999999e-22

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 71.0%

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

      1. unpow2 71.0%

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

   4. Simplified 71.0%

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

      1. div-inv 70.9%

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

      2. associate-*l* 70.7%

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

      3. *-commutative 70.7%

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

      4. associate-/r* 70.7%

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

      5. metadata-eval 70.7%

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

   6. Applied egg-rr 70.7%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+42}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-67}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+43}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7: 53.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -3.3 \cdot 10^{+42}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* z (/ z y)))))
   (if (<= y -3.3e+42)
     (* y 0.5)
     (if (<= y 6.8e-71)
       t_0
       (if (<= y 9.5e-22)
         (* x (/ x (* y 2.0)))
         (if (<= y 2.8e+43) t_0 (* y 0.5)))))))

C

double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double tmp;
	if (y <= -3.3e+42) {
		tmp = y * 0.5;
	} else if (y <= 6.8e-71) {
		tmp = t_0;
	} else if (y <= 9.5e-22) {
		tmp = x * (x / (y * 2.0));
	} else if (y <= 2.8e+43) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (z * (z / y))
    if (y <= (-3.3d+42)) then
        tmp = y * 0.5d0
    else if (y <= 6.8d-71) then
        tmp = t_0
    else if (y <= 9.5d-22) then
        tmp = x * (x / (y * 2.0d0))
    else if (y <= 2.8d+43) then
        tmp = t_0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double tmp;
	if (y <= -3.3e+42) {
		tmp = y * 0.5;
	} else if (y <= 6.8e-71) {
		tmp = t_0;
	} else if (y <= 9.5e-22) {
		tmp = x * (x / (y * 2.0));
	} else if (y <= 2.8e+43) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.5 * (z * (z / y))
	tmp = 0
	if y <= -3.3e+42:
		tmp = y * 0.5
	elif y <= 6.8e-71:
		tmp = t_0
	elif y <= 9.5e-22:
		tmp = x * (x / (y * 2.0))
	elif y <= 2.8e+43:
		tmp = t_0
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(z * Float64(z / y)))
	tmp = 0.0
	if (y <= -3.3e+42)
		tmp = Float64(y * 0.5);
	elseif (y <= 6.8e-71)
		tmp = t_0;
	elseif (y <= 9.5e-22)
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	elseif (y <= 2.8e+43)
		tmp = t_0;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (z * (z / y));
	tmp = 0.0;
	if (y <= -3.3e+42)
		tmp = y * 0.5;
	elseif (y <= 6.8e-71)
		tmp = t_0;
	elseif (y <= 9.5e-22)
		tmp = x * (x / (y * 2.0));
	elseif (y <= 2.8e+43)
		tmp = t_0;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.3e+42], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, 6.8e-71], t$95$0, If[LessEqual[y, 9.5e-22], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e+43], t$95$0, N[(y * 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2999999999999999e42 or 2.80000000000000019e43 < y

   1. Initial program 44.2%

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

   2. Taylor expanded in y around inf 74.3%

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

      1. *-commutative 74.3%

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

   4. Simplified 74.3%

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

   if -3.2999999999999999e42 < y < 6.80000000000000007e-71 or 9.4999999999999994e-22 < y < 2.80000000000000019e43

   1. Initial program 88.8%

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

   2. Taylor expanded in z around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. unpow2 57.4%

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

      3. associate-/l* 58.1%

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

   4. Simplified 58.1%

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

      1. associate-/r/ 58.1%

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

   6. Applied egg-rr 58.1%

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

   if 6.80000000000000007e-71 < y < 9.4999999999999994e-22

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 71.0%

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

      1. unpow2 71.0%

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

   4. Simplified 71.0%

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

      1. associate-/l* 71.0%

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

      2. associate-/r/ 71.0%

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

   6. Applied egg-rr 71.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+42}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-71}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+43}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 8: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* z (/ z y)))))
   (if (<= y -1.55e+43)
     (* y 0.5)
     (if (<= y 1.25e-69)
       t_0
       (if (<= y 1.35e-22)
         (/ (* x x) (* y 2.0))
         (if (<= y 1.3e+44) t_0 (* y 0.5)))))))

C

double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double tmp;
	if (y <= -1.55e+43) {
		tmp = y * 0.5;
	} else if (y <= 1.25e-69) {
		tmp = t_0;
	} else if (y <= 1.35e-22) {
		tmp = (x * x) / (y * 2.0);
	} else if (y <= 1.3e+44) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (-0.5d0) * (z * (z / y))
    if (y <= (-1.55d+43)) then
        tmp = y * 0.5d0
    else if (y <= 1.25d-69) then
        tmp = t_0
    else if (y <= 1.35d-22) then
        tmp = (x * x) / (y * 2.0d0)
    else if (y <= 1.3d+44) then
        tmp = t_0
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double tmp;
	if (y <= -1.55e+43) {
		tmp = y * 0.5;
	} else if (y <= 1.25e-69) {
		tmp = t_0;
	} else if (y <= 1.35e-22) {
		tmp = (x * x) / (y * 2.0);
	} else if (y <= 1.3e+44) {
		tmp = t_0;
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.5 * (z * (z / y))
	tmp = 0
	if y <= -1.55e+43:
		tmp = y * 0.5
	elif y <= 1.25e-69:
		tmp = t_0
	elif y <= 1.35e-22:
		tmp = (x * x) / (y * 2.0)
	elif y <= 1.3e+44:
		tmp = t_0
	else:
		tmp = y * 0.5
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(z * Float64(z / y)))
	tmp = 0.0
	if (y <= -1.55e+43)
		tmp = Float64(y * 0.5);
	elseif (y <= 1.25e-69)
		tmp = t_0;
	elseif (y <= 1.35e-22)
		tmp = Float64(Float64(x * x) / Float64(y * 2.0));
	elseif (y <= 1.3e+44)
		tmp = t_0;
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (z * (z / y));
	tmp = 0.0;
	if (y <= -1.55e+43)
		tmp = y * 0.5;
	elseif (y <= 1.25e-69)
		tmp = t_0;
	elseif (y <= 1.35e-22)
		tmp = (x * x) / (y * 2.0);
	elseif (y <= 1.3e+44)
		tmp = t_0;
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.55e+43], N[(y * 0.5), $MachinePrecision], If[LessEqual[y, 1.25e-69], t$95$0, If[LessEqual[y, 1.35e-22], N[(N[(x * x), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e+44], t$95$0, N[(y * 0.5), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.5500000000000001e43 or 1.3e44 < y

   1. Initial program 44.2%

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

   2. Taylor expanded in y around inf 74.3%

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

      1. *-commutative 74.3%

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

   4. Simplified 74.3%

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

   if -1.5500000000000001e43 < y < 1.25000000000000008e-69 or 1.3500000000000001e-22 < y < 1.3e44

   1. Initial program 88.8%

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

   2. Taylor expanded in z around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. unpow2 57.4%

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

      3. associate-/l* 58.1%

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

   4. Simplified 58.1%

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

      1. associate-/r/ 58.1%

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

   6. Applied egg-rr 58.1%

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

   if 1.25000000000000008e-69 < y < 1.3500000000000001e-22

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 71.0%

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

      1. unpow2 71.0%

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

   4. Simplified 71.0%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+43}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-69}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-22}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot 2}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+44}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 9: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+269}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e+269) (* -0.5 (- (/ (* z z) y) y)) (* x (* x (/ 0.5 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+269) {
		tmp = -0.5 * (((z * z) / y) - 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 * x) <= 1d+269) then
        tmp = (-0.5d0) * (((z * z) / y) - 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 * x) <= 1e+269) {
		tmp = -0.5 * (((z * z) / y) - y);
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e+269:
		tmp = -0.5 * (((z * z) / y) - y)
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+269)
		tmp = Float64(-0.5 * Float64(Float64(Float64(z * z) / y) - 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 * x) <= 1e+269)
		tmp = -0.5 * (((z * z) / y) - y);
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+269], N[(-0.5 * N[(N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e269

   1. Initial program 75.8%

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

      1. sub-neg 75.8%

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

      2. +-commutative 75.8%

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

      3. neg-sub0 75.8%

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

      4. associate-+l- 75.8%

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

      5. sub0-neg 75.8%

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

      6. neg-mul-1 75.8%

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

      7. *-commutative 75.8%

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

      8. times-frac 75.8%

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

      9. associate--r+ 75.8%

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

      10. div-sub 75.8%

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

      11. difference-of-squares 75.8%

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

      12. +-commutative 75.8%

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

      13. associate-*l/ 76.9%

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

      14. *-commutative 76.9%

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

      15. associate-/l* 99.9%

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

      16. *-inverses 99.9%

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

      17. /-rgt-identity 99.9%

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

      18. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 84.4%

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

      1. unpow2 84.4%

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

   6. Simplified 84.4%

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

   if 1e269 < (*.f64 x x)

   1. Initial program 52.8%

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

   2. Taylor expanded in x around inf 57.9%

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

      1. unpow2 57.9%

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

   4. Simplified 57.9%

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

      1. div-inv 57.9%

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

      2. associate-*l* 67.3%

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

      3. *-commutative 67.3%

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

      4. associate-/r* 67.3%

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

      5. metadata-eval 67.3%

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

   6. Applied egg-rr 67.3%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+269}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z \cdot z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 10: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

   1. sub-neg 69.7%

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

   2. +-commutative 69.7%

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

   3. neg-sub0 69.7%

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

   4. associate-+l- 69.7%

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

   5. sub0-neg 69.7%

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

   6. neg-mul-1 69.7%

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

   7. *-commutative 69.7%

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

   8. times-frac 69.7%

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

   9. associate--r+ 69.7%

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

   10. div-sub 69.7%

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

   11. difference-of-squares 74.9%

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

   12. +-commutative 74.9%

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

   13. associate-*l/ 76.5%

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

   14. *-commutative 76.5%

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

   15. associate-/l* 99.9%

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

   16. *-inverses 99.9%

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

   17. /-rgt-identity 99.9%

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

   18. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 11: 43.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+96}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3.7e+96) (* y 0.5) (* x (* x (/ 0.5 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3.7e+96) {
		tmp = y * 0.5;
	} 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 <= 3.7d+96) then
        tmp = y * 0.5d0
    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 <= 3.7e+96) {
		tmp = y * 0.5;
	} else {
		tmp = x * (x * (0.5 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3.7e+96:
		tmp = y * 0.5
	else:
		tmp = x * (x * (0.5 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3.7e+96)
		tmp = Float64(y * 0.5);
	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 <= 3.7e+96)
		tmp = y * 0.5;
	else
		tmp = x * (x * (0.5 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3.7e+96], N[(y * 0.5), $MachinePrecision], N[(x * N[(x * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.69999999999999991e96

   1. Initial program 72.1%

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

   2. Taylor expanded in y around inf 43.4%

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

      1. *-commutative 43.4%

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

   4. Simplified 43.4%

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

   if 3.69999999999999991e96 < x

   1. Initial program 57.1%

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

   2. Taylor expanded in x around inf 53.0%

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

      1. unpow2 53.0%

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

   4. Simplified 53.0%

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

      1. div-inv 53.0%

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

      2. associate-*l* 61.9%

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

      3. *-commutative 61.9%

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

      4. associate-/r* 61.9%

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

      5. metadata-eval 61.9%

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

   6. Applied egg-rr 61.9%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+96}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 12: 34.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.7%

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

2. Taylor expanded in y around inf 39.0%

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

   1. *-commutative 39.0%

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

4. Simplified 39.0%

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

5. Final simplification 39.0%

   \[\leadsto y \cdot 0.5 \]

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 2023257 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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