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

Percentage Accurate: 68.8% → 99.9%

Time: 7.0s

Alternatives: 10

Speedup: 1.2×

• 41.6% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(z - x), $MachinePrecision] * N[(N[(z + x), $MachinePrecision] / y), $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 75.2%

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

   12. +-commutative 75.2%

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

   13. associate-*l/ 77.6%

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

   14. *-commutative 77.6%

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

   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%

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

Alternative 2: 53.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{y \cdot 2}\\ t_1 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -0.0001:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-85}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-256}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ x (* y 2.0)))) (t_1 (* -0.5 (* z (/ z y)))))
   (if (<= x -1.02e+85)
     t_0
     (if (<= x -0.0001)
       t_1
       (if (<= x -1.05e-85)
         (* y 0.5)
         (if (<= x -2.6e-256) t_1 (if (<= x 2.6e+48) (* y 0.5) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (x / (y * 2.0));
	double t_1 = -0.5 * (z * (z / y));
	double tmp;
	if (x <= -1.02e+85) {
		tmp = t_0;
	} else if (x <= -0.0001) {
		tmp = t_1;
	} else if (x <= -1.05e-85) {
		tmp = y * 0.5;
	} else if (x <= -2.6e-256) {
		tmp = t_1;
	} else if (x <= 2.6e+48) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (x / (y * 2.0d0))
    t_1 = (-0.5d0) * (z * (z / y))
    if (x <= (-1.02d+85)) then
        tmp = t_0
    else if (x <= (-0.0001d0)) then
        tmp = t_1
    else if (x <= (-1.05d-85)) then
        tmp = y * 0.5d0
    else if (x <= (-2.6d-256)) then
        tmp = t_1
    else if (x <= 2.6d+48) then
        tmp = y * 0.5d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (x / (y * 2.0));
	double t_1 = -0.5 * (z * (z / y));
	double tmp;
	if (x <= -1.02e+85) {
		tmp = t_0;
	} else if (x <= -0.0001) {
		tmp = t_1;
	} else if (x <= -1.05e-85) {
		tmp = y * 0.5;
	} else if (x <= -2.6e-256) {
		tmp = t_1;
	} else if (x <= 2.6e+48) {
		tmp = y * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (x / (y * 2.0))
	t_1 = -0.5 * (z * (z / y))
	tmp = 0
	if x <= -1.02e+85:
		tmp = t_0
	elif x <= -0.0001:
		tmp = t_1
	elif x <= -1.05e-85:
		tmp = y * 0.5
	elif x <= -2.6e-256:
		tmp = t_1
	elif x <= 2.6e+48:
		tmp = y * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(x / Float64(y * 2.0)))
	t_1 = Float64(-0.5 * Float64(z * Float64(z / y)))
	tmp = 0.0
	if (x <= -1.02e+85)
		tmp = t_0;
	elseif (x <= -0.0001)
		tmp = t_1;
	elseif (x <= -1.05e-85)
		tmp = Float64(y * 0.5);
	elseif (x <= -2.6e-256)
		tmp = t_1;
	elseif (x <= 2.6e+48)
		tmp = Float64(y * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (x / (y * 2.0));
	t_1 = -0.5 * (z * (z / y));
	tmp = 0.0;
	if (x <= -1.02e+85)
		tmp = t_0;
	elseif (x <= -0.0001)
		tmp = t_1;
	elseif (x <= -1.05e-85)
		tmp = y * 0.5;
	elseif (x <= -2.6e-256)
		tmp = t_1;
	elseif (x <= 2.6e+48)
		tmp = y * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.02e+85], t$95$0, If[LessEqual[x, -0.0001], t$95$1, If[LessEqual[x, -1.05e-85], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, -2.6e-256], t$95$1, If[LessEqual[x, 2.6e+48], N[(y * 0.5), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.02e85 or 2.59999999999999995e48 < x

   1. Initial program 63.9%

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

   2. Taylor expanded in x around inf 63.8%

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

      1. unpow2 63.8%

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

   4. Simplified 63.8%

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

      1. associate-/l* 67.6%

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

      2. associate-/r/ 67.6%

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

   6. Applied egg-rr 67.6%

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

   if -1.02e85 < x < -1.00000000000000005e-4 or -1.05e-85 < x < -2.6000000000000001e-256

   1. Initial program 81.1%

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

   2. Taylor expanded in z around inf 59.9%

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

      1. *-commutative 59.9%

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

      2. unpow2 59.9%

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

      3. associate-/l* 61.2%

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

   4. Simplified 61.2%

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

      1. associate-/r/ 61.2%

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

   6. Applied egg-rr 61.2%

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

   if -1.00000000000000005e-4 < x < -1.05e-85 or -2.6000000000000001e-256 < x < 2.59999999999999995e48

   1. Initial program 67.5%

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

   2. Taylor expanded in y around inf 55.3%

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

      1. *-commutative 55.3%

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

   4. Simplified 55.3%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+85}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;x \leq -0.0001:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-85}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-256}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \]

Alternative 3: 53.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{+85}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.08 \cdot 10^{-84}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-257}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* z (/ z y)))))
   (if (<= x -1.02e+85)
     (* x (/ x (* y 2.0)))
     (if (<= x -7e-5)
       t_0
       (if (<= x -1.08e-84)
         (* y 0.5)
         (if (<= x -7.5e-257)
           t_0
           (if (<= x 4.5e+48) (* y 0.5) (* 0.5 (/ x (/ y x))))))))))

C

double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double tmp;
	if (x <= -1.02e+85) {
		tmp = x * (x / (y * 2.0));
	} else if (x <= -7e-5) {
		tmp = t_0;
	} else if (x <= -1.08e-84) {
		tmp = y * 0.5;
	} else if (x <= -7.5e-257) {
		tmp = t_0;
	} else if (x <= 4.5e+48) {
		tmp = y * 0.5;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	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 (x <= (-1.02d+85)) then
        tmp = x * (x / (y * 2.0d0))
    else if (x <= (-7d-5)) then
        tmp = t_0
    else if (x <= (-1.08d-84)) then
        tmp = y * 0.5d0
    else if (x <= (-7.5d-257)) then
        tmp = t_0
    else if (x <= 4.5d+48) then
        tmp = y * 0.5d0
    else
        tmp = 0.5d0 * (x / (y / x))
    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 (x <= -1.02e+85) {
		tmp = x * (x / (y * 2.0));
	} else if (x <= -7e-5) {
		tmp = t_0;
	} else if (x <= -1.08e-84) {
		tmp = y * 0.5;
	} else if (x <= -7.5e-257) {
		tmp = t_0;
	} else if (x <= 4.5e+48) {
		tmp = y * 0.5;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.5 * (z * (z / y))
	tmp = 0
	if x <= -1.02e+85:
		tmp = x * (x / (y * 2.0))
	elif x <= -7e-5:
		tmp = t_0
	elif x <= -1.08e-84:
		tmp = y * 0.5
	elif x <= -7.5e-257:
		tmp = t_0
	elif x <= 4.5e+48:
		tmp = y * 0.5
	else:
		tmp = 0.5 * (x / (y / x))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(z * Float64(z / y)))
	tmp = 0.0
	if (x <= -1.02e+85)
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	elseif (x <= -7e-5)
		tmp = t_0;
	elseif (x <= -1.08e-84)
		tmp = Float64(y * 0.5);
	elseif (x <= -7.5e-257)
		tmp = t_0;
	elseif (x <= 4.5e+48)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (z * (z / y));
	tmp = 0.0;
	if (x <= -1.02e+85)
		tmp = x * (x / (y * 2.0));
	elseif (x <= -7e-5)
		tmp = t_0;
	elseif (x <= -1.08e-84)
		tmp = y * 0.5;
	elseif (x <= -7.5e-257)
		tmp = t_0;
	elseif (x <= 4.5e+48)
		tmp = y * 0.5;
	else
		tmp = 0.5 * (x / (y / x));
	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[x, -1.02e+85], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7e-5], t$95$0, If[LessEqual[x, -1.08e-84], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, -7.5e-257], t$95$0, If[LessEqual[x, 4.5e+48], N[(y * 0.5), $MachinePrecision], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.02e85

   1. Initial program 60.4%

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

   2. Taylor expanded in x around inf 62.7%

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

      1. unpow2 62.7%

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

   4. Simplified 62.7%

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

      1. associate-/l* 66.8%

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

      2. associate-/r/ 66.8%

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

   6. Applied egg-rr 66.8%

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

   if -1.02e85 < x < -6.9999999999999994e-5 or -1.0800000000000001e-84 < x < -7.4999999999999995e-257

   1. Initial program 81.1%

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

   2. Taylor expanded in z around inf 59.9%

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

      1. *-commutative 59.9%

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

      2. unpow2 59.9%

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

      3. associate-/l* 61.2%

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

   4. Simplified 61.2%

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

      1. associate-/r/ 61.2%

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

   6. Applied egg-rr 61.2%

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

   if -6.9999999999999994e-5 < x < -1.0800000000000001e-84 or -7.4999999999999995e-257 < x < 4.49999999999999995e48

   1. Initial program 67.5%

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

   2. Taylor expanded in y around inf 55.3%

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

      1. *-commutative 55.3%

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

   4. Simplified 55.3%

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

   if 4.49999999999999995e48 < x

   1. Initial program 67.0%

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

   2. Taylor expanded in x around inf 64.9%

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

      1. *-commutative 64.9%

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

      2. unpow2 64.9%

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

      3. associate-/l* 68.3%

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

   4. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{x}} \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+85}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-5}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq -1.08 \cdot 10^{-84}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-257}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 4: 53.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+84}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-5}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-84}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-257}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9.5e+84)
   (* x (/ x (* y 2.0)))
   (if (<= x -9.5e-5)
     (* -0.5 (/ (* z z) y))
     (if (<= x -1.45e-84)
       (* y 0.5)
       (if (<= x -8.8e-257)
         (* -0.5 (* z (/ z y)))
         (if (<= x 5.2e+48) (* y 0.5) (* 0.5 (/ x (/ y x)))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e+84) {
		tmp = x * (x / (y * 2.0));
	} else if (x <= -9.5e-5) {
		tmp = -0.5 * ((z * z) / y);
	} else if (x <= -1.45e-84) {
		tmp = y * 0.5;
	} else if (x <= -8.8e-257) {
		tmp = -0.5 * (z * (z / y));
	} else if (x <= 5.2e+48) {
		tmp = y * 0.5;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	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 <= (-9.5d+84)) then
        tmp = x * (x / (y * 2.0d0))
    else if (x <= (-9.5d-5)) then
        tmp = (-0.5d0) * ((z * z) / y)
    else if (x <= (-1.45d-84)) then
        tmp = y * 0.5d0
    else if (x <= (-8.8d-257)) then
        tmp = (-0.5d0) * (z * (z / y))
    else if (x <= 5.2d+48) then
        tmp = y * 0.5d0
    else
        tmp = 0.5d0 * (x / (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -9.5e+84) {
		tmp = x * (x / (y * 2.0));
	} else if (x <= -9.5e-5) {
		tmp = -0.5 * ((z * z) / y);
	} else if (x <= -1.45e-84) {
		tmp = y * 0.5;
	} else if (x <= -8.8e-257) {
		tmp = -0.5 * (z * (z / y));
	} else if (x <= 5.2e+48) {
		tmp = y * 0.5;
	} else {
		tmp = 0.5 * (x / (y / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9.5e+84:
		tmp = x * (x / (y * 2.0))
	elif x <= -9.5e-5:
		tmp = -0.5 * ((z * z) / y)
	elif x <= -1.45e-84:
		tmp = y * 0.5
	elif x <= -8.8e-257:
		tmp = -0.5 * (z * (z / y))
	elif x <= 5.2e+48:
		tmp = y * 0.5
	else:
		tmp = 0.5 * (x / (y / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9.5e+84)
		tmp = Float64(x * Float64(x / Float64(y * 2.0)));
	elseif (x <= -9.5e-5)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y));
	elseif (x <= -1.45e-84)
		tmp = Float64(y * 0.5);
	elseif (x <= -8.8e-257)
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	elseif (x <= 5.2e+48)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(0.5 * Float64(x / Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9.5e+84)
		tmp = x * (x / (y * 2.0));
	elseif (x <= -9.5e-5)
		tmp = -0.5 * ((z * z) / y);
	elseif (x <= -1.45e-84)
		tmp = y * 0.5;
	elseif (x <= -8.8e-257)
		tmp = -0.5 * (z * (z / y));
	elseif (x <= 5.2e+48)
		tmp = y * 0.5;
	else
		tmp = 0.5 * (x / (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9.5e+84], N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.5e-5], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.45e-84], N[(y * 0.5), $MachinePrecision], If[LessEqual[x, -8.8e-257], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.2e+48], N[(y * 0.5), $MachinePrecision], N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < -9.49999999999999979e84

   1. Initial program 60.4%

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

   2. Taylor expanded in x around inf 62.7%

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

      1. unpow2 62.7%

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

   4. Simplified 62.7%

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

      1. associate-/l* 66.8%

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

      2. associate-/r/ 66.8%

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

   6. Applied egg-rr 66.8%

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

   if -9.49999999999999979e84 < x < -9.5000000000000005e-5

   1. Initial program 79.3%

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

   2. Taylor expanded in z around inf 58.5%

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

      1. *-commutative 58.5%

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

      2. unpow2 58.5%

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

   4. Simplified 58.5%

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

   if -9.5000000000000005e-5 < x < -1.4500000000000001e-84 or -8.7999999999999995e-257 < x < 5.1999999999999999e48

   1. Initial program 67.5%

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

   2. Taylor expanded in y around inf 55.3%

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

      1. *-commutative 55.3%

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

   4. Simplified 55.3%

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

   if -1.4500000000000001e-84 < x < -8.7999999999999995e-257

   1. Initial program 81.8%

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

   2. Taylor expanded in z around inf 60.5%

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

      1. *-commutative 60.5%

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

      2. unpow2 60.5%

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

      3. associate-/l* 62.4%

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

   4. Simplified 62.4%

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

      1. associate-/r/ 62.4%

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

   6. Applied egg-rr 62.4%

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

   if 5.1999999999999999e48 < x

   1. Initial program 67.0%

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

   2. Taylor expanded in x around inf 64.9%

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

      1. *-commutative 64.9%

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

      2. unpow2 64.9%

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

      3. associate-/l* 68.3%

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

   4. Simplified 68.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{x}} \cdot 0.5} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+84}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-5}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-84}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-257}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 5: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{x}{y \cdot 2}\\ t_1 := -0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot 2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+175}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ x (* y 2.0)))) (t_1 (* -0.5 (- (* z (/ z y)) y))))
   (if (<= x -1.02e+86)
     t_0
     (if (<= x 6.2e+48)
       t_1
       (if (<= x 5.4e+100)
         (/ (* x x) (* y 2.0))
         (if (<= x 7.5e+175) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (x / (y * 2.0));
	double t_1 = -0.5 * ((z * (z / y)) - y);
	double tmp;
	if (x <= -1.02e+86) {
		tmp = t_0;
	} else if (x <= 6.2e+48) {
		tmp = t_1;
	} else if (x <= 5.4e+100) {
		tmp = (x * x) / (y * 2.0);
	} else if (x <= 7.5e+175) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (x / (y * 2.0d0))
    t_1 = (-0.5d0) * ((z * (z / y)) - y)
    if (x <= (-1.02d+86)) then
        tmp = t_0
    else if (x <= 6.2d+48) then
        tmp = t_1
    else if (x <= 5.4d+100) then
        tmp = (x * x) / (y * 2.0d0)
    else if (x <= 7.5d+175) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (x / (y * 2.0));
	double t_1 = -0.5 * ((z * (z / y)) - y);
	double tmp;
	if (x <= -1.02e+86) {
		tmp = t_0;
	} else if (x <= 6.2e+48) {
		tmp = t_1;
	} else if (x <= 5.4e+100) {
		tmp = (x * x) / (y * 2.0);
	} else if (x <= 7.5e+175) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (x / (y * 2.0))
	t_1 = -0.5 * ((z * (z / y)) - y)
	tmp = 0
	if x <= -1.02e+86:
		tmp = t_0
	elif x <= 6.2e+48:
		tmp = t_1
	elif x <= 5.4e+100:
		tmp = (x * x) / (y * 2.0)
	elif x <= 7.5e+175:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(x / Float64(y * 2.0)))
	t_1 = Float64(-0.5 * Float64(Float64(z * Float64(z / y)) - y))
	tmp = 0.0
	if (x <= -1.02e+86)
		tmp = t_0;
	elseif (x <= 6.2e+48)
		tmp = t_1;
	elseif (x <= 5.4e+100)
		tmp = Float64(Float64(x * x) / Float64(y * 2.0));
	elseif (x <= 7.5e+175)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (x / (y * 2.0));
	t_1 = -0.5 * ((z * (z / y)) - y);
	tmp = 0.0;
	if (x <= -1.02e+86)
		tmp = t_0;
	elseif (x <= 6.2e+48)
		tmp = t_1;
	elseif (x <= 5.4e+100)
		tmp = (x * x) / (y * 2.0);
	elseif (x <= 7.5e+175)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(x / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.5 * N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.02e+86], t$95$0, If[LessEqual[x, 6.2e+48], t$95$1, If[LessEqual[x, 5.4e+100], N[(N[(x * x), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e+175], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.01999999999999996e86 or 7.5000000000000001e175 < x

   1. Initial program 61.4%

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

   2. Taylor expanded in x around inf 68.9%

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

      1. unpow2 68.9%

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

   4. Simplified 68.9%

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

      1. associate-/l* 74.3%

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

      2. associate-/r/ 74.3%

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

   6. Applied egg-rr 74.3%

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

   if -1.01999999999999996e86 < x < 6.20000000000000011e48 or 5.39999999999999997e100 < x < 7.5000000000000001e175

   1. Initial program 71.5%

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

      1. sub-neg 71.5%

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

      2. +-commutative 71.5%

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

      3. neg-sub0 71.5%

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

      4. associate-+l- 71.5%

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

      5. sub0-neg 71.5%

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

      6. neg-mul-1 71.5%

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

      7. *-commutative 71.5%

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

      8. times-frac 71.5%

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

      9. associate--r+ 71.5%

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

      10. div-sub 71.5%

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

      11. difference-of-squares 71.5%

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

      12. +-commutative 71.5%

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

      13. associate-*l/ 74.0%

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

      14. *-commutative 74.0%

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

      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. associate-*r/ 91.2%

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

      2. clear-num 91.2%

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

      3. *-commutative 91.2%

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

      4. +-commutative 91.2%

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

      5. difference-of-squares 91.2%

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

   5. Applied egg-rr 91.2%

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

   6. Taylor expanded in z around inf 81.3%

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

      1. unpow2 81.3%

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

      2. associate-*r/ 87.4%

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

   8. Simplified 87.4%

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

   if 6.20000000000000011e48 < x < 5.39999999999999997e100

   1. Initial program 91.4%

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

   2. Taylor expanded in x around inf 82.0%

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

      1. unpow2 82.0%

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

   4. Simplified 82.0%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+48}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{+100}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot 2}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+175}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \end{array} \]

Alternative 6: 50.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+123} \lor \neg \left(y \leq 7.6 \cdot 10^{-81}\right) \land \left(y \leq 1.35 \cdot 10^{+30} \lor \neg \left(y \leq 7.5 \cdot 10^{+69}\right)\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.2e+123)
         (and (not (<= y 7.6e-81)) (or (<= y 1.35e+30) (not (<= y 7.5e+69)))))
   (* y 0.5)
   (* -0.5 (* z (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e+123) || (!(y <= 7.6e-81) && ((y <= 1.35e+30) || !(y <= 7.5e+69)))) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (z * (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 <= (-2.2d+123)) .or. (.not. (y <= 7.6d-81)) .and. (y <= 1.35d+30) .or. (.not. (y <= 7.5d+69))) then
        tmp = y * 0.5d0
    else
        tmp = (-0.5d0) * (z * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.2e+123) || (!(y <= 7.6e-81) && ((y <= 1.35e+30) || !(y <= 7.5e+69)))) {
		tmp = y * 0.5;
	} else {
		tmp = -0.5 * (z * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.2e+123) or (not (y <= 7.6e-81) and ((y <= 1.35e+30) or not (y <= 7.5e+69))):
		tmp = y * 0.5
	else:
		tmp = -0.5 * (z * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.2e+123) || (!(y <= 7.6e-81) && ((y <= 1.35e+30) || !(y <= 7.5e+69))))
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.2e+123) || (~((y <= 7.6e-81)) && ((y <= 1.35e+30) || ~((y <= 7.5e+69)))))
		tmp = y * 0.5;
	else
		tmp = -0.5 * (z * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.2e+123], And[N[Not[LessEqual[y, 7.6e-81]], $MachinePrecision], Or[LessEqual[y, 1.35e+30], N[Not[LessEqual[y, 7.5e+69]], $MachinePrecision]]]], N[(y * 0.5), $MachinePrecision], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.19999999999999992e123 or 7.5999999999999997e-81 < y < 1.3499999999999999e30 or 7.49999999999999939e69 < y

   1. Initial program 41.2%

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

   2. Taylor expanded in y around inf 68.1%

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

      1. *-commutative 68.1%

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

   4. Simplified 68.1%

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

   if -2.19999999999999992e123 < y < 7.5999999999999997e-81 or 1.3499999999999999e30 < y < 7.49999999999999939e69

   1. Initial program 89.5%

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

   2. Taylor expanded in z around inf 49.7%

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

      1. *-commutative 49.7%

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

      2. unpow2 49.7%

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

      3. associate-/l* 50.9%

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

   4. Simplified 50.9%

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

      1. associate-/r/ 50.9%

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

   6. Applied egg-rr 50.9%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+123} \lor \neg \left(y \leq 7.6 \cdot 10^{-81}\right) \land \left(y \leq 1.35 \cdot 10^{+30} \lor \neg \left(y \leq 7.5 \cdot 10^{+69}\right)\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \]

Alternative 7: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+43} \lor \neg \left(z \leq 1.8 \cdot 10^{+15}\right):\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -3.2e+43) (not (<= z 1.8e+15)))
   (* -0.5 (- (* (- z x) (/ z y)) y))
   (* -0.5 (- (* (- z x) (/ x y)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -3.2e+43) || !(z <= 1.8e+15)) {
		tmp = -0.5 * (((z - x) * (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 ((z <= (-3.2d+43)) .or. (.not. (z <= 1.8d+15))) then
        tmp = (-0.5d0) * (((z - x) * (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 ((z <= -3.2e+43) || !(z <= 1.8e+15)) {
		tmp = -0.5 * (((z - x) * (z / y)) - y);
	} else {
		tmp = -0.5 * (((z - x) * (x / y)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -3.2e+43) or not (z <= 1.8e+15):
		tmp = -0.5 * (((z - x) * (z / y)) - y)
	else:
		tmp = -0.5 * (((z - x) * (x / y)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -3.2e+43) || !(z <= 1.8e+15))
		tmp = Float64(-0.5 * Float64(Float64(Float64(z - x) * Float64(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 ((z <= -3.2e+43) || ~((z <= 1.8e+15)))
		tmp = -0.5 * (((z - x) * (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[z, -3.2e+43], N[Not[LessEqual[z, 1.8e+15]], $MachinePrecision]], N[(-0.5 * N[(N[(N[(z - x), $MachinePrecision] * N[(z / y), $MachinePrecision]), $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 z < -3.20000000000000014e43 or 1.8e15 < z

   1. Initial program 68.9%

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

      1. sub-neg 68.9%

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

      2. +-commutative 68.9%

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

      3. neg-sub0 68.9%

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

      4. associate-+l- 68.9%

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

      5. sub0-neg 68.9%

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

      6. neg-mul-1 68.9%

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

      7. *-commutative 68.9%

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

      8. times-frac 68.9%

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

      9. associate--r+ 68.9%

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

      10. div-sub 68.9%

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

      11. difference-of-squares 80.7%

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

      12. +-commutative 80.7%

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

      13. associate-*l/ 81.1%

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

      14. *-commutative 81.1%

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

      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 x around 0 86.8%

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

   if -3.20000000000000014e43 < z < 1.8e15

   1. Initial program 70.4%

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

      1. sub-neg 70.4%

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

      2. +-commutative 70.4%

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

      3. neg-sub0 70.4%

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

      4. associate-+l- 70.4%

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

      5. sub0-neg 70.4%

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

      6. neg-mul-1 70.4%

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

      7. *-commutative 70.4%

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

      8. times-frac 70.4%

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

      9. associate--r+ 70.4%

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

      10. div-sub 70.4%

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

      11. difference-of-squares 70.4%

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

      12. +-commutative 70.4%

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

      13. associate-*l/ 74.4%

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

      14. *-commutative 74.4%

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

      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 x around inf 91.5%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+43} \lor \neg \left(z \leq 1.8 \cdot 10^{+15}\right):\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \]

Alternative 8: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+95}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+95)
   (* -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+95) {
		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+95) 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+95) {
		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+95:
		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+95)
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(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+95)
		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[LessEqual[N[(x * x), $MachinePrecision], 2e+95], N[(-0.5 * N[(N[(z * N[(z / y), $MachinePrecision]), $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) < 2.00000000000000004e95

   1. Initial program 72.6%

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

      1. sub-neg 72.6%

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

      2. +-commutative 72.6%

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

      3. neg-sub0 72.6%

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

      4. associate-+l- 72.6%

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

      5. sub0-neg 72.6%

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

      6. neg-mul-1 72.6%

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

      7. *-commutative 72.6%

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

      8. times-frac 72.6%

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

      9. associate--r+ 72.6%

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

      10. div-sub 72.6%

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

      11. difference-of-squares 72.6%

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

      12. +-commutative 72.6%

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

      13. associate-*l/ 75.3%

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

      14. *-commutative 75.3%

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

      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. associate-*r/ 91.8%

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

      2. clear-num 91.8%

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

      3. *-commutative 91.8%

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

      4. +-commutative 91.8%

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

      5. difference-of-squares 91.8%

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

   5. Applied egg-rr 91.8%

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

   6. Taylor expanded in z around inf 82.0%

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

      1. unpow2 82.0%

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

      2. associate-*r/ 88.8%

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

   8. Simplified 88.8%

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

   if 2.00000000000000004e95 < (*.f64 x x)

   1. Initial program 65.1%

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

      1. sub-neg 65.1%

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

      2. +-commutative 65.1%

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

      3. neg-sub0 65.1%

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

      4. associate-+l- 65.1%

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

      5. sub0-neg 65.1%

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

      6. neg-mul-1 65.1%

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

      7. *-commutative 65.1%

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

      8. times-frac 65.1%

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

      9. associate--r+ 65.1%

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

      10. div-sub 65.1%

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

      11. difference-of-squares 79.4%

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

      12. +-commutative 79.4%

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

      13. associate-*l/ 81.1%

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

      14. *-commutative 81.1%

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

      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 x around inf 85.2%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+95}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y} - y\right)\\ \end{array} \]

Alternative 9: 85.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+95}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \left(-\frac{x}{y}\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+95)
   (* -0.5 (- (* z (/ z y)) y))
   (* -0.5 (- (* x (- (/ x y))) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+95) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = -0.5 * ((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+95) then
        tmp = (-0.5d0) * ((z * (z / y)) - y)
    else
        tmp = (-0.5d0) * ((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+95) {
		tmp = -0.5 * ((z * (z / y)) - y);
	} else {
		tmp = -0.5 * ((x * -(x / y)) - y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+95:
		tmp = -0.5 * ((z * (z / y)) - y)
	else:
		tmp = -0.5 * ((x * -(x / y)) - y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+95)
		tmp = Float64(-0.5 * Float64(Float64(z * Float64(z / y)) - y));
	else
		tmp = Float64(-0.5 * Float64(Float64(x * Float64(-Float64(x / y))) - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+95)
		tmp = -0.5 * ((z * (z / y)) - y);
	else
		tmp = -0.5 * ((x * -(x / y)) - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+95], N[(-0.5 * N[(N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(N[(x * (-N[(x / y), $MachinePrecision])), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000004e95

   1. Initial program 72.6%

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

      1. sub-neg 72.6%

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

      2. +-commutative 72.6%

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

      3. neg-sub0 72.6%

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

      4. associate-+l- 72.6%

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

      5. sub0-neg 72.6%

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

      6. neg-mul-1 72.6%

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

      7. *-commutative 72.6%

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

      8. times-frac 72.6%

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

      9. associate--r+ 72.6%

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

      10. div-sub 72.6%

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

      11. difference-of-squares 72.6%

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

      12. +-commutative 72.6%

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

      13. associate-*l/ 75.3%

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

      14. *-commutative 75.3%

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

      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. associate-*r/ 91.8%

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

      2. clear-num 91.8%

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

      3. *-commutative 91.8%

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

      4. +-commutative 91.8%

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

      5. difference-of-squares 91.8%

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

   5. Applied egg-rr 91.8%

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

   6. Taylor expanded in z around inf 82.0%

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

      1. unpow2 82.0%

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

      2. associate-*r/ 88.8%

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

   8. Simplified 88.8%

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

   if 2.00000000000000004e95 < (*.f64 x x)

   1. Initial program 65.1%

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

      1. sub-neg 65.1%

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

      2. +-commutative 65.1%

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

      3. neg-sub0 65.1%

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

      4. associate-+l- 65.1%

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

      5. sub0-neg 65.1%

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

      6. neg-mul-1 65.1%

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

      7. *-commutative 65.1%

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

      8. times-frac 65.1%

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

      9. associate--r+ 65.1%

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

      10. div-sub 65.1%

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

      11. difference-of-squares 79.4%

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

      12. +-commutative 79.4%

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

      13. associate-*l/ 81.1%

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

      14. *-commutative 81.1%

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

      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. Step-by-step derivation

      1. associate-*r/ 88.9%

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

      2. clear-num 88.9%

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

      3. *-commutative 88.9%

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

      4. +-commutative 88.9%

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

      5. difference-of-squares 73.7%

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

   5. Applied egg-rr 73.7%

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

   6. Taylor expanded in z around 0 71.7%

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

      1. unpow2 71.7%

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

      2. associate-*r/ 71.7%

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

      3. mul-1-neg 71.7%

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

      4. distribute-rgt-neg-out 71.7%

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

      5. associate-*l/ 82.8%

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

      6. *-commutative 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+95}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(x \cdot \left(-\frac{x}{y}\right) - y\right)\\ \end{array} \]

Alternative 10: 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 35.4%

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

   1. *-commutative 35.4%

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

4. Simplified 35.4%

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

5. Final simplification 35.4%

   \[\leadsto y \cdot 0.5 \]

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 2023192 
(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)))