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

Percentage Accurate: 68.5% → 91.0%

Time: 9.4s

Alternatives: 10

Speedup: 0.9×

• 41.0% 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: 68.5% 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: 91.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+271}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{1}{y \cdot \frac{1}{{x}^{2} - {z}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+271)
   (* 0.5 (+ y (/ 1.0 (* y (/ 1.0 (- (pow x 2.0) (pow z 2.0)))))))
   (* 0.5 (- y (* z (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+271) {
		tmp = 0.5 * (y + (1.0 / (y * (1.0 / (pow(x, 2.0) - pow(z, 2.0))))));
	} else {
		tmp = 0.5 * (y - (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 ((z * z) <= 1d+271) then
        tmp = 0.5d0 * (y + (1.0d0 / (y * (1.0d0 / ((x ** 2.0d0) - (z ** 2.0d0))))))
    else
        tmp = 0.5d0 * (y - (z * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+271) {
		tmp = 0.5 * (y + (1.0 / (y * (1.0 / (Math.pow(x, 2.0) - Math.pow(z, 2.0))))));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+271:
		tmp = 0.5 * (y + (1.0 / (y * (1.0 / (math.pow(x, 2.0) - math.pow(z, 2.0))))))
	else:
		tmp = 0.5 * (y - (z * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+271)
		tmp = Float64(0.5 * Float64(y + Float64(1.0 / Float64(y * Float64(1.0 / Float64((x ^ 2.0) - (z ^ 2.0)))))));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e+271)
		tmp = 0.5 * (y + (1.0 / (y * (1.0 / ((x ^ 2.0) - (z ^ 2.0))))));
	else
		tmp = 0.5 * (y - (z * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+271], N[(0.5 * N[(y + N[(1.0 / N[(y * N[(1.0 / N[(N[Power[x, 2.0], $MachinePrecision] - N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999953e270

   1. Initial program 71.8%

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

      1. remove-double-neg 71.8%

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

      2. distribute-lft-neg-out 71.8%

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

      3. distribute-frac-neg2 71.8%

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

      4. distribute-frac-neg 71.8%

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

      5. neg-mul-1 71.8%

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

      6. distribute-lft-neg-out 71.8%

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

      7. *-commutative 71.8%

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

      8. distribute-lft-neg-in 71.8%

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

      9. times-frac 71.8%

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

      10. metadata-eval 71.8%

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

      11. metadata-eval 71.8%

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

      12. associate--l+ 71.8%

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

      13. fma-define 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in x around 0 89.2%

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

      1. associate--l+ 89.2%

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

      2. div-sub 92.9%

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

   7. Simplified 92.9%

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

      1. clear-num 92.9%

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

      2. inv-pow 92.9%

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

   9. Applied egg-rr 92.9%

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

       1. unpow-1 92.9%

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

   11. Simplified 92.9%

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

       1. div-inv 92.9%

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

   13. Applied egg-rr 92.9%

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

   if 9.99999999999999953e270 < (*.f64 z z)

   1. Initial program 51.9%

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

      1. remove-double-neg 51.9%

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

      2. distribute-lft-neg-out 51.9%

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

      3. distribute-frac-neg2 51.9%

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

      4. distribute-frac-neg 51.9%

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

      5. neg-mul-1 51.9%

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

      6. distribute-lft-neg-out 51.9%

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

      7. *-commutative 51.9%

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

      8. distribute-lft-neg-in 51.9%

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

      9. times-frac 51.9%

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

      10. metadata-eval 51.9%

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

      11. metadata-eval 51.9%

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

      12. associate--l+ 51.9%

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

      13. fma-define 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--l+ 57.8%

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

      2. div-sub 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in x around 0 74.9%

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

      1. pow2 74.9%

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

      2. *-un-lft-identity 74.9%

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

      3. times-frac 88.7%

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

   10. Applied egg-rr 88.7%

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

   11. Taylor expanded in z around 0 88.7%

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

Alternative 2: 91.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+271}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{1}{\frac{y}{{x}^{2} - {z}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+271)
   (* 0.5 (+ y (/ 1.0 (/ y (- (pow x 2.0) (pow z 2.0))))))
   (* 0.5 (- y (* z (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+271) {
		tmp = 0.5 * (y + (1.0 / (y / (pow(x, 2.0) - pow(z, 2.0)))));
	} else {
		tmp = 0.5 * (y - (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 ((z * z) <= 1d+271) then
        tmp = 0.5d0 * (y + (1.0d0 / (y / ((x ** 2.0d0) - (z ** 2.0d0)))))
    else
        tmp = 0.5d0 * (y - (z * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+271) {
		tmp = 0.5 * (y + (1.0 / (y / (Math.pow(x, 2.0) - Math.pow(z, 2.0)))));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+271:
		tmp = 0.5 * (y + (1.0 / (y / (math.pow(x, 2.0) - math.pow(z, 2.0)))))
	else:
		tmp = 0.5 * (y - (z * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+271)
		tmp = Float64(0.5 * Float64(y + Float64(1.0 / Float64(y / Float64((x ^ 2.0) - (z ^ 2.0))))));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e+271)
		tmp = 0.5 * (y + (1.0 / (y / ((x ^ 2.0) - (z ^ 2.0)))));
	else
		tmp = 0.5 * (y - (z * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+271], N[(0.5 * N[(y + N[(1.0 / N[(y / N[(N[Power[x, 2.0], $MachinePrecision] - N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999953e270

   1. Initial program 71.8%

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

      1. remove-double-neg 71.8%

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

      2. distribute-lft-neg-out 71.8%

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

      3. distribute-frac-neg2 71.8%

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

      4. distribute-frac-neg 71.8%

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

      5. neg-mul-1 71.8%

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

      6. distribute-lft-neg-out 71.8%

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

      7. *-commutative 71.8%

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

      8. distribute-lft-neg-in 71.8%

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

      9. times-frac 71.8%

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

      10. metadata-eval 71.8%

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

      11. metadata-eval 71.8%

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

      12. associate--l+ 71.8%

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

      13. fma-define 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in x around 0 89.2%

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

      1. associate--l+ 89.2%

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

      2. div-sub 92.9%

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

   7. Simplified 92.9%

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

      1. clear-num 92.9%

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

      2. inv-pow 92.9%

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

   9. Applied egg-rr 92.9%

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

       1. unpow-1 92.9%

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

   11. Simplified 92.9%

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

   if 9.99999999999999953e270 < (*.f64 z z)

   1. Initial program 51.9%

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

      1. remove-double-neg 51.9%

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

      2. distribute-lft-neg-out 51.9%

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

      3. distribute-frac-neg2 51.9%

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

      4. distribute-frac-neg 51.9%

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

      5. neg-mul-1 51.9%

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

      6. distribute-lft-neg-out 51.9%

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

      7. *-commutative 51.9%

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

      8. distribute-lft-neg-in 51.9%

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

      9. times-frac 51.9%

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

      10. metadata-eval 51.9%

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

      11. metadata-eval 51.9%

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

      12. associate--l+ 51.9%

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

      13. fma-define 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--l+ 57.8%

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

      2. div-sub 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in x around 0 74.9%

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

      1. pow2 74.9%

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

      2. *-un-lft-identity 74.9%

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

      3. times-frac 88.7%

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

   10. Applied egg-rr 88.7%

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

   11. Taylor expanded in z around 0 88.7%

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

Alternative 3: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+271}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* z z) 1e+271)
   (* 0.5 (+ y (/ (- (pow x 2.0) (pow z 2.0)) y)))
   (* 0.5 (- y (* z (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+271) {
		tmp = 0.5 * (y + ((pow(x, 2.0) - pow(z, 2.0)) / y));
	} else {
		tmp = 0.5 * (y - (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 ((z * z) <= 1d+271) then
        tmp = 0.5d0 * (y + (((x ** 2.0d0) - (z ** 2.0d0)) / y))
    else
        tmp = 0.5d0 * (y - (z * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z * z) <= 1e+271) {
		tmp = 0.5 * (y + ((Math.pow(x, 2.0) - Math.pow(z, 2.0)) / y));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z * z) <= 1e+271:
		tmp = 0.5 * (y + ((math.pow(x, 2.0) - math.pow(z, 2.0)) / y))
	else:
		tmp = 0.5 * (y - (z * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(z * z) <= 1e+271)
		tmp = Float64(0.5 * Float64(y + Float64(Float64((x ^ 2.0) - (z ^ 2.0)) / y)));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z * z) <= 1e+271)
		tmp = 0.5 * (y + (((x ^ 2.0) - (z ^ 2.0)) / y));
	else
		tmp = 0.5 * (y - (z * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(z * z), $MachinePrecision], 1e+271], N[(0.5 * N[(y + N[(N[(N[Power[x, 2.0], $MachinePrecision] - N[Power[z, 2.0], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z z) < 9.99999999999999953e270

   1. Initial program 71.8%

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

      1. remove-double-neg 71.8%

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

      2. distribute-lft-neg-out 71.8%

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

      3. distribute-frac-neg2 71.8%

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

      4. distribute-frac-neg 71.8%

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

      5. neg-mul-1 71.8%

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

      6. distribute-lft-neg-out 71.8%

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

      7. *-commutative 71.8%

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

      8. distribute-lft-neg-in 71.8%

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

      9. times-frac 71.8%

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

      10. metadata-eval 71.8%

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

      11. metadata-eval 71.8%

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

      12. associate--l+ 71.8%

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

      13. fma-define 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in x around 0 89.2%

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

      1. associate--l+ 89.2%

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

      2. div-sub 92.9%

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

   7. Simplified 92.9%

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

   if 9.99999999999999953e270 < (*.f64 z z)

   1. Initial program 51.9%

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

      1. remove-double-neg 51.9%

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

      2. distribute-lft-neg-out 51.9%

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

      3. distribute-frac-neg2 51.9%

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

      4. distribute-frac-neg 51.9%

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

      5. neg-mul-1 51.9%

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

      6. distribute-lft-neg-out 51.9%

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

      7. *-commutative 51.9%

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

      8. distribute-lft-neg-in 51.9%

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

      9. times-frac 51.9%

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

      10. metadata-eval 51.9%

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

      11. metadata-eval 51.9%

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

      12. associate--l+ 51.9%

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

      13. fma-define 61.9%

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

   3. Simplified 61.9%

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

   5. Taylor expanded in x around 0 57.8%

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

      1. associate--l+ 57.8%

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

      2. div-sub 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in x around 0 74.9%

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

      1. pow2 74.9%

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

      2. *-un-lft-identity 74.9%

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

      3. times-frac 88.7%

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

   10. Applied egg-rr 88.7%

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

   11. Taylor expanded in z around 0 88.7%

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

Alternative 4: 66.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* y y) (* x x)) (* z z)) (* y 2.0))))
   (if (<= t_0 0.0)
     (* 0.5 (- y (/ z (/ y z))))
     (if (<= t_0 INFINITY)
       (* 0.5 (+ y (/ (pow x 2.0) y)))
       (* 0.5 (- y (* z (/ z y))))))))

C

double code(double x, double y, double z) {
	double t_0 = (((y * y) + (x * x)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * (y + (pow(x, 2.0) / y));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (((y * y) + (x * x)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * (y - (z / (y / z)));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (y + (Math.pow(x, 2.0) / y));
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((y * y) + (x * x)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 0.5 * (y - (z / (y / z)))
	elif t_0 <= math.inf:
		tmp = 0.5 * (y + (math.pow(x, 2.0) / y))
	else:
		tmp = 0.5 * (y - (z * (z / y)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(y * y) + Float64(x * x)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(0.5 * Float64(y - Float64(z / Float64(y / z))));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 * Float64(y + Float64((x ^ 2.0) / y)));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((y * y) + (x * x)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 0.5 * (y - (z / (y / z)));
	elseif (t_0 <= Inf)
		tmp = 0.5 * (y + ((x ^ 2.0) / y));
	else
		tmp = 0.5 * (y - (z * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(0.5 * N[(y - N[(z / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(y + N[(N[Power[x, 2.0], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 0.0

   1. Initial program 70.7%

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

      1. remove-double-neg 70.7%

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

      2. distribute-lft-neg-out 70.7%

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

      3. distribute-frac-neg2 70.7%

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

      4. distribute-frac-neg 70.7%

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

      5. neg-mul-1 70.7%

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

      6. distribute-lft-neg-out 70.7%

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

      7. *-commutative 70.7%

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

      8. distribute-lft-neg-in 70.7%

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

      9. times-frac 70.7%

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

      10. metadata-eval 70.7%

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

      11. metadata-eval 70.7%

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

      12. associate--l+ 70.7%

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

      13. fma-define 70.7%

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

   3. Simplified 70.7%

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

   5. Taylor expanded in x around 0 86.8%

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

      1. associate--l+ 86.8%

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

      2. div-sub 90.3%

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

   7. Simplified 90.3%

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

   8. Taylor expanded in x around 0 61.4%

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

      1. pow2 61.4%

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

      2. *-un-lft-identity 61.4%

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

      3. times-frac 63.7%

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

   10. Applied egg-rr 63.7%

       \[\leadsto 0.5 \cdot \left(y - \color{blue}{\frac{z}{1} \cdot \frac{z}{y}}\right) \]
   11. Step-by-step derivation

       1. /-rgt-identity 63.7%

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

       2. clear-num 63.7%

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

       3. un-div-inv 63.7%

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

   12. Applied egg-rr 63.7%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 79.0%

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

      1. remove-double-neg 79.0%

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

      2. distribute-lft-neg-out 79.0%

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

      3. distribute-frac-neg2 79.0%

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

      4. distribute-frac-neg 79.0%

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

      5. neg-mul-1 79.0%

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

      6. distribute-lft-neg-out 79.0%

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

      7. *-commutative 79.0%

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

      8. distribute-lft-neg-in 79.0%

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

      9. times-frac 79.0%

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

      10. metadata-eval 79.0%

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

      11. metadata-eval 79.0%

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

      12. associate--l+ 79.0%

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

      13. fma-define 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in x around 0 90.3%

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

      1. associate--l+ 90.3%

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

      2. div-sub 96.6%

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

   7. Simplified 96.6%

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

   8. Taylor expanded in z around 0 63.5%

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

   if +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 0.0%

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

      1. remove-double-neg 0.0%

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

      2. distribute-lft-neg-out 0.0%

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

      3. distribute-frac-neg2 0.0%

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

      4. distribute-frac-neg 0.0%

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

      5. neg-mul-1 0.0%

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

      6. distribute-lft-neg-out 0.0%

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

      7. *-commutative 0.0%

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

      8. distribute-lft-neg-in 0.0%

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

      9. times-frac 0.0%

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

      10. metadata-eval 0.0%

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

      11. metadata-eval 0.0%

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

      12. associate--l+ 0.0%

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

      13. fma-define 24.1%

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

   3. Simplified 24.1%

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

   5. Taylor expanded in x around 0 18.2%

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

      1. associate--l+ 18.2%

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

      2. div-sub 18.2%

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

   7. Simplified 18.2%

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

   8. Taylor expanded in x around 0 45.8%

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

      1. pow2 45.8%

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

      2. *-un-lft-identity 45.8%

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

      3. times-frac 72.7%

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

   10. Applied egg-rr 72.7%

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

   11. Taylor expanded in z around 0 72.7%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2} \leq 0:\\ \;\;\;\;0.5 \cdot \left(y - \frac{z}{\frac{y}{z}}\right)\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \left(y + \frac{{x}^{2}}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+154}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1e+154)
   (* 0.5 (/ (fma x x (- (* y y) (* z z))) y))
   (* 0.5 (- y (* z (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1e+154) {
		tmp = 0.5 * (fma(x, x, ((y * y) - (z * z))) / y);
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1e+154)
		tmp = Float64(0.5 * Float64(fma(x, x, Float64(Float64(y * y) - Float64(z * z))) / y));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1e+154], N[(0.5 * N[(N[(x * x + N[(N[(y * y), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000004e154

   1. Initial program 75.3%

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

      1. remove-double-neg 75.3%

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

      2. distribute-lft-neg-out 75.3%

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

      3. distribute-frac-neg2 75.3%

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

      4. distribute-frac-neg 75.3%

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

      5. neg-mul-1 75.3%

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

      6. distribute-lft-neg-out 75.3%

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

      7. *-commutative 75.3%

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

      8. distribute-lft-neg-in 75.3%

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

      9. times-frac 75.3%

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

      10. metadata-eval 75.3%

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

      11. metadata-eval 75.3%

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

      12. associate--l+ 75.3%

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

      13. fma-define 78.5%

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

   3. Simplified 78.5%

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

   if 1.00000000000000004e154 < y

   1. Initial program 7.6%

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

      1. remove-double-neg 7.6%

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

      2. distribute-lft-neg-out 7.6%

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

      3. distribute-frac-neg2 7.6%

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

      4. distribute-frac-neg 7.6%

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

      5. neg-mul-1 7.6%

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

      6. distribute-lft-neg-out 7.6%

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

      7. *-commutative 7.6%

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

      8. distribute-lft-neg-in 7.6%

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

      9. times-frac 7.6%

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

      10. metadata-eval 7.6%

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

      11. metadata-eval 7.6%

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

      12. associate--l+ 7.6%

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

      13. fma-define 7.6%

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

   3. Simplified 7.6%

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

   5. Taylor expanded in x around 0 71.3%

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

      1. associate--l+ 71.3%

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

      2. div-sub 71.3%

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

   7. Simplified 71.3%

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

   8. Taylor expanded in x around 0 74.0%

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

      1. pow2 74.0%

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

      2. *-un-lft-identity 74.0%

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

      3. times-frac 88.9%

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

   10. Applied egg-rr 88.9%

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

   11. Taylor expanded in z around 0 88.9%

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

Alternative 6: 79.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.3e+99)
   (/ (- (+ (* y y) (* x x)) (* z z)) (* y 2.0))
   (* 0.5 (- y (* z (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3e+99) {
		tmp = (((y * y) + (x * x)) - (z * z)) / (y * 2.0);
	} else {
		tmp = 0.5 * (y - (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 <= 3.3d+99) then
        tmp = (((y * y) + (x * x)) - (z * z)) / (y * 2.0d0)
    else
        tmp = 0.5d0 * (y - (z * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.3e+99) {
		tmp = (((y * y) + (x * x)) - (z * z)) / (y * 2.0);
	} else {
		tmp = 0.5 * (y - (z * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.3e+99:
		tmp = (((y * y) + (x * x)) - (z * z)) / (y * 2.0)
	else:
		tmp = 0.5 * (y - (z * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.3e+99)
		tmp = Float64(Float64(Float64(Float64(y * y) + Float64(x * x)) - Float64(z * z)) / Float64(y * 2.0));
	else
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.3e+99)
		tmp = (((y * y) + (x * x)) - (z * z)) / (y * 2.0);
	else
		tmp = 0.5 * (y - (z * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.3e+99], N[(N[(N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2999999999999999e99

   1. Initial program 75.1%

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

   if 3.2999999999999999e99 < y

   1. Initial program 27.1%

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

      1. remove-double-neg 27.1%

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

      2. distribute-lft-neg-out 27.1%

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

      3. distribute-frac-neg2 27.1%

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

      4. distribute-frac-neg 27.1%

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

      5. neg-mul-1 27.1%

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

      6. distribute-lft-neg-out 27.1%

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

      7. *-commutative 27.1%

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

      8. distribute-lft-neg-in 27.1%

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

      9. times-frac 27.1%

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

      10. metadata-eval 27.1%

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

      11. metadata-eval 27.1%

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

      12. associate--l+ 27.1%

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

      13. fma-define 29.3%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in x around 0 73.2%

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

      1. associate--l+ 73.2%

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

      2. div-sub 73.2%

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

   7. Simplified 73.2%

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

   8. Taylor expanded in x around 0 73.1%

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

      1. pow2 73.1%

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

      2. *-un-lft-identity 73.1%

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

      3. times-frac 85.9%

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

   10. Applied egg-rr 85.9%

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

   11. Taylor expanded in z around 0 85.9%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+249}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 1e+249) (* 0.5 (- y (* z (/ z y)))) (* x (/ (* 0.5 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+249) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else {
		tmp = x * ((0.5 * 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) <= 1d+249) then
        tmp = 0.5d0 * (y - (z * (z / y)))
    else
        tmp = x * ((0.5d0 * x) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 1e+249) {
		tmp = 0.5 * (y - (z * (z / y)));
	} else {
		tmp = x * ((0.5 * x) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 1e+249:
		tmp = 0.5 * (y - (z * (z / y)))
	else:
		tmp = x * ((0.5 * x) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 1e+249)
		tmp = Float64(0.5 * Float64(y - Float64(z * Float64(z / y))));
	else
		tmp = Float64(x * Float64(Float64(0.5 * x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 1e+249)
		tmp = 0.5 * (y - (z * (z / y)));
	else
		tmp = x * ((0.5 * x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+249], N[(0.5 * N[(y - N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(0.5 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.9999999999999992e248

   1. Initial program 69.2%

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

      1. remove-double-neg 69.2%

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

      2. distribute-lft-neg-out 69.2%

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

      3. distribute-frac-neg2 69.2%

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

      4. distribute-frac-neg 69.2%

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

      5. neg-mul-1 69.2%

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

      6. distribute-lft-neg-out 69.2%

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

      7. *-commutative 69.2%

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

      8. distribute-lft-neg-in 69.2%

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

      9. times-frac 69.2%

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

      10. metadata-eval 69.2%

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

      11. metadata-eval 69.2%

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

      12. associate--l+ 69.2%

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

      13. fma-define 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in x around 0 91.5%

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

      1. associate--l+ 91.5%

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

      2. div-sub 95.0%

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

   7. Simplified 95.0%

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

   8. Taylor expanded in x around 0 82.5%

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

      1. pow2 82.5%

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

      2. *-un-lft-identity 82.5%

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

      3. times-frac 86.2%

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

   10. Applied egg-rr 86.2%

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

   11. Taylor expanded in z around 0 86.2%

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

   if 9.9999999999999992e248 < (*.f64 x x)

   1. Initial program 60.3%

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

      1. remove-double-neg 60.3%

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

      2. distribute-lft-neg-out 60.3%

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

      3. distribute-frac-neg2 60.3%

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

      4. distribute-frac-neg 60.3%

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

      5. neg-mul-1 60.3%

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

      6. distribute-lft-neg-out 60.3%

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

      7. *-commutative 60.3%

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

      8. distribute-lft-neg-in 60.3%

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

      9. times-frac 60.3%

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

      10. metadata-eval 60.3%

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

      11. metadata-eval 60.3%

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

      12. associate--l+ 60.3%

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

      13. fma-define 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in x around inf 69.3%

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

      1. *-commutative 69.3%

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

      2. associate-*l/ 69.3%

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

      3. associate-*r/ 69.3%

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

   7. Simplified 69.3%

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

      1. associate-*r/ 69.3%

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

      2. clear-num 69.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{{x}^{2} \cdot 0.5}}} \]

   9. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{{x}^{2} \cdot 0.5}}} \]
   10. Step-by-step derivation

       1. clear-num 69.3%

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

       2. associate-*r/ 69.3%

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

       3. pow2 69.3%

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

       4. associate-*l* 74.8%

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

       5. *-commutative 74.8%

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

       6. associate-*r/ 74.8%

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

   11. Applied egg-rr 74.8%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+249}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.42e+104) (* x (/ (* 0.5 x) y)) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.42e+104) {
		tmp = x * ((0.5 * x) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.42d+104) then
        tmp = x * ((0.5d0 * x) / y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.42e+104) {
		tmp = x * ((0.5 * x) / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.42e+104:
		tmp = x * ((0.5 * x) / y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.42e+104)
		tmp = Float64(x * Float64(Float64(0.5 * x) / y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.42e+104)
		tmp = x * ((0.5 * x) / y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.42e+104], N[(x * N[(N[(0.5 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.42e104

   1. Initial program 74.9%

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

      1. remove-double-neg 74.9%

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

      2. distribute-lft-neg-out 74.9%

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

      3. distribute-frac-neg2 74.9%

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

      4. distribute-frac-neg 74.9%

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

      5. neg-mul-1 74.9%

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

      6. distribute-lft-neg-out 74.9%

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

      7. *-commutative 74.9%

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

      8. distribute-lft-neg-in 74.9%

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

      9. times-frac 74.9%

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

      10. metadata-eval 74.9%

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

      11. metadata-eval 74.9%

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

      12. associate--l+ 74.9%

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

      13. fma-define 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in x around inf 36.0%

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

      1. *-commutative 36.0%

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

      2. associate-*l/ 36.0%

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

      3. associate-*r/ 36.0%

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

   7. Simplified 36.0%

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

      1. associate-*r/ 36.0%

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

      2. clear-num 36.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{{x}^{2} \cdot 0.5}}} \]

   9. Applied egg-rr 36.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{{x}^{2} \cdot 0.5}}} \]
   10. Step-by-step derivation

       1. clear-num 36.0%

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

       2. associate-*r/ 36.0%

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

       3. pow2 36.0%

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

       4. associate-*l* 38.1%

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

       5. *-commutative 38.1%

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

       6. associate-*r/ 38.1%

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

   11. Applied egg-rr 38.1%

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

   if 1.42e104 < y

   1. Initial program 26.1%

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

      1. remove-double-neg 26.1%

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

      2. distribute-lft-neg-out 26.1%

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

      3. distribute-frac-neg2 26.1%

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

      4. distribute-frac-neg 26.1%

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

      5. neg-mul-1 26.1%

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

      6. distribute-lft-neg-out 26.1%

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

      7. *-commutative 26.1%

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

      8. distribute-lft-neg-in 26.1%

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

      9. times-frac 26.1%

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

      10. metadata-eval 26.1%

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

      11. metadata-eval 26.1%

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

      12. associate--l+ 26.1%

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

      13. fma-define 26.1%

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

   3. Simplified 26.1%

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

   5. Taylor expanded in y around inf 77.5%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \frac{0.5 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{+104}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.42e+104) (* (* x x) (/ 0.5 y)) (* 0.5 y)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.42e+104) {
		tmp = (x * x) * (0.5 / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.42d+104) then
        tmp = (x * x) * (0.5d0 / y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.42e+104) {
		tmp = (x * x) * (0.5 / y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.42e+104:
		tmp = (x * x) * (0.5 / y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.42e+104)
		tmp = Float64(Float64(x * x) * Float64(0.5 / y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.42e+104)
		tmp = (x * x) * (0.5 / y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.42e+104], N[(N[(x * x), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.42e104

   1. Initial program 74.9%

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

      1. remove-double-neg 74.9%

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

      2. distribute-lft-neg-out 74.9%

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

      3. distribute-frac-neg2 74.9%

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

      4. distribute-frac-neg 74.9%

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

      5. neg-mul-1 74.9%

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

      6. distribute-lft-neg-out 74.9%

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

      7. *-commutative 74.9%

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

      8. distribute-lft-neg-in 74.9%

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

      9. times-frac 74.9%

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

      10. metadata-eval 74.9%

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

      11. metadata-eval 74.9%

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

      12. associate--l+ 74.9%

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

      13. fma-define 78.2%

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

   3. Simplified 78.2%

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

   5. Taylor expanded in x around inf 36.0%

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

      1. *-commutative 36.0%

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

      2. associate-*l/ 36.0%

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

      3. associate-*r/ 36.0%

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

   7. Simplified 36.0%

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

      1. pow2 36.0%

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

   9. Applied egg-rr 36.0%

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

   if 1.42e104 < y

   1. Initial program 26.1%

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

      1. remove-double-neg 26.1%

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

      2. distribute-lft-neg-out 26.1%

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

      3. distribute-frac-neg2 26.1%

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

      4. distribute-frac-neg 26.1%

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

      5. neg-mul-1 26.1%

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

      6. distribute-lft-neg-out 26.1%

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

      7. *-commutative 26.1%

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

      8. distribute-lft-neg-in 26.1%

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

      9. times-frac 26.1%

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

      10. metadata-eval 26.1%

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

      11. metadata-eval 26.1%

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

      12. associate--l+ 26.1%

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

      13. fma-define 26.1%

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

   3. Simplified 26.1%

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

   5. Taylor expanded in y around inf 77.5%

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

Alternative 10: 35.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.3%

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

   1. remove-double-neg 66.3%

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

   2. distribute-lft-neg-out 66.3%

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

   3. distribute-frac-neg2 66.3%

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

   4. distribute-frac-neg 66.3%

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

   5. neg-mul-1 66.3%

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

   6. distribute-lft-neg-out 66.3%

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

   7. *-commutative 66.3%

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

   8. distribute-lft-neg-in 66.3%

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

   9. times-frac 66.3%

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

   10. metadata-eval 66.3%

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

   11. metadata-eval 66.3%

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

   12. associate--l+ 66.3%

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

   13. fma-define 69.1%

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

3. Simplified 69.1%

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

5. Taylor expanded in y around inf 32.8%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
6. Add Preprocessing

Developer Target 1: 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 2024137 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y 1/2) (* (* (/ 1/2 y) (+ z x)) (- z x))))

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