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

Percentage Accurate: 68.3% → 99.8%

Time: 12.2s

Alternatives: 14

Speedup: 0.9×

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

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z\_m}}{\mathsf{hypot}\left(y, x\right) + z\_m}} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (/ 0.5 (/ (/ y (- (hypot y x) z_m)) (+ (hypot y x) z_m))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	return 0.5 / ((y / (hypot(y, x) - z_m)) / (hypot(y, x) + z_m));
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	return 0.5 / ((y / (Math.hypot(y, x) - z_m)) / (Math.hypot(y, x) + z_m));
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	return 0.5 / ((y / (math.hypot(y, x) - z_m)) / (math.hypot(y, x) + z_m))

Julia

z_m = abs(z)
function code(x, y, z_m)
	return Float64(0.5 / Float64(Float64(y / Float64(hypot(y, x) - z_m)) / Float64(hypot(y, x) + z_m)))
end

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(0.5 / N[(N[(y / N[(N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision] - z$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.9%

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

   1. remove-double-neg 71.9%

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

   2. distribute-lft-neg-out 71.9%

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

   3. distribute-frac-neg2 71.9%

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

   4. distribute-frac-neg 71.9%

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

   5. neg-mul-1 71.9%

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

   6. distribute-lft-neg-out 71.9%

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

   7. *-commutative 71.9%

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

   8. distribute-lft-neg-in 71.9%

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

   9. times-frac 71.5%

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

   10. metadata-eval 71.5%

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

   11. metadata-eval 71.5%

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

   12. associate--l+ 71.5%

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

   13. fma-define 74.7%

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

3. Simplified 74.7%

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

   1. clear-num 74.5%

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

   2. un-div-inv 74.9%

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

   3. fma-undefine 71.7%

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

   4. associate--l+ 71.7%

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

   5. add-sqr-sqrt 71.7%

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

   6. pow2 71.7%

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

   7. hypot-define 71.7%

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

   8. pow2 71.7%

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

6. Applied egg-rr 71.7%

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

   1. *-un-lft-identity 71.7%

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

   2. unpow2 71.7%

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

   3. unpow2 71.7%

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

   4. difference-of-squares 79.4%

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

   5. times-frac 99.7%

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

8. Applied egg-rr 99.7%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{\mathsf{hypot}\left(x, y\right) + z} \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}} \]
9. Step-by-step derivation

   1. associate-*l/ 99.8%

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

   2. *-lft-identity 99.8%

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

   3. hypot-undefine 79.2%

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

   4. unpow2 79.2%

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

   5. unpow2 79.2%

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

   6. +-commutative 79.2%

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

   7. unpow2 79.2%

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

   8. unpow2 79.2%

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

   9. hypot-define 99.8%

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

   10. +-commutative 99.8%

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

   11. hypot-undefine 78.5%

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

   12. unpow2 78.5%

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

   13. unpow2 78.5%

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

   14. +-commutative 78.5%

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

   15. unpow2 78.5%

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

   16. unpow2 78.5%

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

   17. hypot-define 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

    \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{\mathsf{hypot}\left(y, x\right) + z}} \]
12. Add Preprocessing

Alternative 2: 67.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 10^{-24}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\_m\right) \cdot \frac{y - z\_m}{y}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right)}}{\mathsf{hypot}\left(y, x\right) + z\_m}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\_m\right) \cdot \frac{x - z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 1e-24)
     (* 0.5 (* (+ y z_m) (/ (- y z_m) y)))
     (if (<= t_0 INFINITY)
       (/ 0.5 (/ (/ y (hypot y x)) (+ (hypot y x) z_m)))
       (* 0.5 (* (+ x z_m) (/ (- x z_m) y)))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= 1e-24) {
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 / ((y / hypot(y, x)) / (hypot(y, x) + z_m));
	} else {
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y));
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= 1e-24) {
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 / ((y / Math.hypot(y, x)) / (Math.hypot(y, x) + z_m));
	} else {
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_0 <= 1e-24:
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y))
	elif t_0 <= math.inf:
		tmp = 0.5 / ((y / math.hypot(y, x)) / (math.hypot(y, x) + z_m))
	else:
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 1e-24)
		tmp = Float64(0.5 * Float64(Float64(y + z_m) * Float64(Float64(y - z_m) / y)));
	elseif (t_0 <= Inf)
		tmp = Float64(0.5 / Float64(Float64(y / hypot(y, x)) / Float64(hypot(y, x) + z_m)));
	else
		tmp = Float64(0.5 * Float64(Float64(x + z_m) * Float64(Float64(x - z_m) / y)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= 1e-24)
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	elseif (t_0 <= Inf)
		tmp = 0.5 / ((y / hypot(y, x)) / (hypot(y, x) + z_m));
	else
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-24], N[(0.5 * N[(N[(y + z$95$m), $MachinePrecision] * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 / N[(N[(y / N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x + z$95$m), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / y), $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))) < 9.99999999999999924e-25

   1. Initial program 82.2%

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

      1. remove-double-neg 82.2%

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

      2. distribute-lft-neg-out 82.2%

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

      3. distribute-frac-neg2 82.2%

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

      4. distribute-frac-neg 82.2%

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

      5. neg-mul-1 82.2%

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

      6. distribute-lft-neg-out 82.2%

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

      7. *-commutative 82.2%

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

      8. distribute-lft-neg-in 82.2%

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

      9. times-frac 82.2%

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

      10. metadata-eval 82.2%

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

      11. metadata-eval 82.2%

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

      12. associate--l+ 82.2%

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

      13. fma-define 82.2%

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

   3. Simplified 82.2%

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

      1. fma-undefine 82.2%

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

      2. associate--l+ 82.2%

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

      3. add-sqr-sqrt 82.2%

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

      4. difference-of-squares 82.2%

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

      5. hypot-define 82.2%

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

      6. hypot-define 82.2%

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

   6. Applied egg-rr 82.2%

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

   7. Taylor expanded in x around 0 60.2%

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

      1. associate-/l* 73.9%

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

   9. Simplified 73.9%

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

   if 9.99999999999999924e-25 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 78.4%

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

      1. remove-double-neg 78.4%

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

      2. distribute-lft-neg-out 78.4%

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

      3. distribute-frac-neg2 78.4%

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

      4. distribute-frac-neg 78.4%

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

      5. neg-mul-1 78.4%

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

      6. distribute-lft-neg-out 78.4%

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

      7. *-commutative 78.4%

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

      8. distribute-lft-neg-in 78.4%

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

      9. times-frac 77.7%

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

      10. metadata-eval 77.7%

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

      11. metadata-eval 77.7%

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

      12. associate--l+ 77.7%

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

      13. fma-define 77.7%

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

   3. Simplified 77.7%

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

      1. clear-num 77.6%

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

      2. un-div-inv 78.3%

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

      3. fma-undefine 78.3%

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

      4. associate--l+ 78.3%

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

      5. add-sqr-sqrt 78.3%

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

      6. pow2 78.3%

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

      7. hypot-define 78.3%

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

      8. pow2 78.3%

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

   6. Applied egg-rr 78.3%

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

      1. *-un-lft-identity 78.3%

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

      2. unpow2 78.3%

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

      3. unpow2 78.3%

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

      4. difference-of-squares 78.3%

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

      5. times-frac 99.7%

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

   8. Applied egg-rr 99.7%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{\mathsf{hypot}\left(x, y\right) + z} \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. hypot-undefine 79.1%

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

      4. unpow2 79.1%

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

      5. unpow2 79.1%

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

      6. +-commutative 79.1%

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

      7. unpow2 79.1%

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

      8. unpow2 79.1%

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

      9. hypot-define 99.8%

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

      10. +-commutative 99.8%

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

      11. hypot-undefine 79.1%

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

      12. unpow2 79.1%

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

      13. unpow2 79.1%

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

      14. +-commutative 79.1%

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

      15. unpow2 79.1%

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

      16. unpow2 79.1%

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

      17. hypot-define 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in z around 0 47.4%

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

       1. +-commutative 47.4%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{\color{blue}{{y}^{2} + {x}^{2}}}}}{z + \mathsf{hypot}\left(y, x\right)}} \]

       2. unpow2 47.4%

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

       3. unpow2 47.4%

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

       4. hypot-undefine 69.2%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\color{blue}{\mathsf{hypot}\left(y, x\right)}}}{z + \mathsf{hypot}\left(y, x\right)}} \]

   13. Simplified 69.2%

       \[\leadsto \frac{0.5}{\frac{\frac{y}{\color{blue}{\mathsf{hypot}\left(y, x\right)}}}{z + \mathsf{hypot}\left(y, x\right)}} \]

   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 29.6%

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

   3. Simplified 29.6%

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

      1. fma-undefine 0.0%

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

      2. associate--l+ 0.0%

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

      3. add-sqr-sqrt 0.0%

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

      4. difference-of-squares 35.6%

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

      5. hypot-define 57.8%

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

      6. hypot-define 72.6%

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

   6. Applied egg-rr 72.6%

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

   7. Taylor expanded in y around 0 71.4%

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

      1. associate-/l* 78.1%

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

      2. +-commutative 78.1%

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

   9. Simplified 78.1%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 10^{-24}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;\frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right)}}{\mathsf{hypot}\left(y, x\right) + z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \frac{\left(z\_m + \mathsf{hypot}\left(x, y\right)\right) \cdot \left(\mathsf{hypot}\left(x, y\right) - z\_m\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right)}}{\mathsf{hypot}\left(y, x\right) + z\_m}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= y 3.4e+108)
   (* 0.5 (/ (* (+ z_m (hypot x y)) (- (hypot x y) z_m)) y))
   (/ 0.5 (/ (/ y (hypot y x)) (+ (hypot y x) z_m)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (y <= 3.4e+108) {
		tmp = 0.5 * (((z_m + hypot(x, y)) * (hypot(x, y) - z_m)) / y);
	} else {
		tmp = 0.5 / ((y / hypot(y, x)) / (hypot(y, x) + z_m));
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (y <= 3.4e+108) {
		tmp = 0.5 * (((z_m + Math.hypot(x, y)) * (Math.hypot(x, y) - z_m)) / y);
	} else {
		tmp = 0.5 / ((y / Math.hypot(y, x)) / (Math.hypot(y, x) + z_m));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if y <= 3.4e+108:
		tmp = 0.5 * (((z_m + math.hypot(x, y)) * (math.hypot(x, y) - z_m)) / y)
	else:
		tmp = 0.5 / ((y / math.hypot(y, x)) / (math.hypot(y, x) + z_m))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (y <= 3.4e+108)
		tmp = Float64(0.5 * Float64(Float64(Float64(z_m + hypot(x, y)) * Float64(hypot(x, y) - z_m)) / y));
	else
		tmp = Float64(0.5 / Float64(Float64(y / hypot(y, x)) / Float64(hypot(y, x) + z_m)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (y <= 3.4e+108)
		tmp = 0.5 * (((z_m + hypot(x, y)) * (hypot(x, y) - z_m)) / y);
	else
		tmp = 0.5 / ((y / hypot(y, x)) / (hypot(y, x) + z_m));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[y, 3.4e+108], N[(0.5 * N[(N[(N[(z$95$m + N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] - z$95$m), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(y / N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y ^ 2 + x ^ 2], $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.39999999999999996e108

   1. Initial program 81.0%

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

      1. remove-double-neg 81.0%

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

      2. distribute-lft-neg-out 81.0%

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

      3. distribute-frac-neg2 81.0%

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

      4. distribute-frac-neg 81.0%

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

      5. neg-mul-1 81.0%

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

      6. distribute-lft-neg-out 81.0%

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

      7. *-commutative 81.0%

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

      8. distribute-lft-neg-in 81.0%

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

      9. times-frac 80.6%

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

      10. metadata-eval 80.6%

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

      11. metadata-eval 80.6%

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

      12. associate--l+ 80.6%

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

      13. fma-define 84.3%

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

   3. Simplified 84.3%

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

      1. fma-undefine 80.6%

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

      2. associate--l+ 80.6%

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

      3. add-sqr-sqrt 80.6%

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

      4. difference-of-squares 84.4%

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

      5. hypot-define 86.8%

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

      6. hypot-define 88.6%

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

   6. Applied egg-rr 88.6%

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

   if 3.39999999999999996e108 < y

   1. Initial program 25.4%

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

      1. remove-double-neg 25.4%

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

      2. distribute-lft-neg-out 25.4%

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

      3. distribute-frac-neg2 25.4%

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

      4. distribute-frac-neg 25.4%

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

      5. neg-mul-1 25.4%

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

      6. distribute-lft-neg-out 25.4%

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

      7. *-commutative 25.4%

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

      8. distribute-lft-neg-in 25.4%

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

      9. times-frac 25.4%

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

      10. metadata-eval 25.4%

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

      11. metadata-eval 25.4%

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

      12. associate--l+ 25.4%

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

      13. fma-define 25.4%

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

   3. Simplified 25.4%

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

      1. clear-num 25.3%

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

      2. un-div-inv 25.3%

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

      3. fma-undefine 25.3%

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

      4. associate--l+ 25.3%

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

      5. add-sqr-sqrt 25.3%

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

      6. pow2 25.3%

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

      7. hypot-define 25.3%

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

      8. pow2 25.3%

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

   6. Applied egg-rr 25.3%

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

      1. *-un-lft-identity 25.3%

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

      2. unpow2 25.3%

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

      3. unpow2 25.3%

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

      4. difference-of-squares 31.0%

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

      5. times-frac 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{\mathsf{hypot}\left(x, y\right) + z} \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 99.9%

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

      2. *-lft-identity 99.9%

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

      3. hypot-undefine 33.3%

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

      4. unpow2 33.3%

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

      5. unpow2 33.3%

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

      6. +-commutative 33.3%

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

      7. unpow2 33.3%

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

      8. unpow2 33.3%

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

      9. hypot-define 99.9%

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

      10. +-commutative 99.9%

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

      11. hypot-undefine 30.9%

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

      12. unpow2 30.9%

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

      13. unpow2 30.9%

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

      14. +-commutative 30.9%

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

      15. unpow2 30.9%

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

      16. unpow2 30.9%

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

      17. hypot-define 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in z around 0 31.1%

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

       1. +-commutative 31.1%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{\color{blue}{{y}^{2} + {x}^{2}}}}}{z + \mathsf{hypot}\left(y, x\right)}} \]

       2. unpow2 31.1%

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

       3. unpow2 31.1%

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

       4. hypot-undefine 94.2%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\color{blue}{\mathsf{hypot}\left(y, x\right)}}}{z + \mathsf{hypot}\left(y, x\right)}} \]

   13. Simplified 94.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.4 \cdot 10^{+108}:\\ \;\;\;\;0.5 \cdot \frac{\left(z + \mathsf{hypot}\left(x, y\right)\right) \cdot \left(\mathsf{hypot}\left(x, y\right) - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right)}}{\mathsf{hypot}\left(y, x\right) + z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{-130}:\\ \;\;\;\;\frac{0.5}{\frac{y}{\left(x + z\_m\right) \cdot \left(x - z\_m\right)}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, \left(y + z\_m\right) \cdot \left(y - z\_m\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\_m\right) \cdot \frac{y - z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= y 1.52e-130)
   (/ 0.5 (/ y (* (+ x z_m) (- x z_m))))
   (if (<= y 4.2e+131)
     (* 0.5 (/ (fma x x (* (+ y z_m) (- y z_m))) y))
     (* 0.5 (* (+ y z_m) (/ (- y z_m) y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (y <= 1.52e-130) {
		tmp = 0.5 / (y / ((x + z_m) * (x - z_m)));
	} else if (y <= 4.2e+131) {
		tmp = 0.5 * (fma(x, x, ((y + z_m) * (y - z_m))) / y);
	} else {
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (y <= 1.52e-130)
		tmp = Float64(0.5 / Float64(y / Float64(Float64(x + z_m) * Float64(x - z_m))));
	elseif (y <= 4.2e+131)
		tmp = Float64(0.5 * Float64(fma(x, x, Float64(Float64(y + z_m) * Float64(y - z_m))) / y));
	else
		tmp = Float64(0.5 * Float64(Float64(y + z_m) * Float64(Float64(y - z_m) / y)));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[y, 1.52e-130], N[(0.5 / N[(y / N[(N[(x + z$95$m), $MachinePrecision] * N[(x - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+131], N[(0.5 * N[(N[(x * x + N[(N[(y + z$95$m), $MachinePrecision] * N[(y - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y + z$95$m), $MachinePrecision] * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.51999999999999998e-130

   1. Initial program 77.4%

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

      1. remove-double-neg 77.4%

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

      2. distribute-lft-neg-out 77.4%

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

      3. distribute-frac-neg2 77.4%

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

      4. distribute-frac-neg 77.4%

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

      5. neg-mul-1 77.4%

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

      6. distribute-lft-neg-out 77.4%

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

      7. *-commutative 77.4%

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

      8. distribute-lft-neg-in 77.4%

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

      9. times-frac 76.8%

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

      10. metadata-eval 76.8%

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

      11. metadata-eval 76.8%

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

      12. associate--l+ 76.8%

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

      13. fma-define 80.7%

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

   3. Simplified 80.7%

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

      1. clear-num 80.6%

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

      2. un-div-inv 81.1%

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

      3. fma-undefine 77.2%

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

      4. associate--l+ 77.2%

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

      5. add-sqr-sqrt 77.2%

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

      6. pow2 77.2%

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

      7. hypot-define 77.2%

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

      8. pow2 77.2%

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

   6. Applied egg-rr 77.2%

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

      1. *-un-lft-identity 77.2%

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

      2. unpow2 77.2%

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

      3. unpow2 77.2%

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

      4. difference-of-squares 85.8%

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

      5. times-frac 99.7%

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

   8. Applied egg-rr 99.7%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{\mathsf{hypot}\left(x, y\right) + z} \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. hypot-undefine 84.4%

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

      4. unpow2 84.4%

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

      5. unpow2 84.4%

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

      6. +-commutative 84.4%

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

      7. unpow2 84.4%

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

      8. unpow2 84.4%

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

      9. hypot-define 99.8%

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

      10. +-commutative 99.8%

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

      11. hypot-undefine 83.8%

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

      12. unpow2 83.8%

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

      13. unpow2 83.8%

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

      14. +-commutative 83.8%

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

      15. unpow2 83.8%

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

      16. unpow2 83.8%

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

      17. hypot-define 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in y around 0 73.0%

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

   if 1.51999999999999998e-130 < y < 4.19999999999999971e131

   1. Initial program 86.7%

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

      1. remove-double-neg 86.7%

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

      2. distribute-lft-neg-out 86.7%

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

      3. distribute-frac-neg2 86.7%

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

      4. distribute-frac-neg 86.7%

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

      5. neg-mul-1 86.7%

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

      6. distribute-lft-neg-out 86.7%

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

      7. *-commutative 86.7%

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

      8. distribute-lft-neg-in 86.7%

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

      9. times-frac 86.7%

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

      10. metadata-eval 86.7%

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

      11. metadata-eval 86.7%

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

      12. associate--l+ 86.7%

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

      13. fma-define 89.8%

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

   3. Simplified 89.8%

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

      1. difference-of-squares 89.8%

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

      2. *-commutative 89.8%

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

   6. Applied egg-rr 89.8%

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

   if 4.19999999999999971e131 < y

   1. Initial program 21.0%

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

      1. remove-double-neg 21.0%

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

      2. distribute-lft-neg-out 21.0%

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

      3. distribute-frac-neg2 21.0%

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

      4. distribute-frac-neg 21.0%

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

      5. neg-mul-1 21.0%

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

      6. distribute-lft-neg-out 21.0%

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

      7. *-commutative 21.0%

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

      8. distribute-lft-neg-in 21.0%

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

      9. times-frac 21.0%

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

      10. metadata-eval 21.0%

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

      11. metadata-eval 21.0%

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

      12. associate--l+ 21.0%

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

      13. fma-define 21.0%

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

   3. Simplified 21.0%

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

      1. fma-undefine 21.0%

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

      2. associate--l+ 21.0%

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

      3. add-sqr-sqrt 21.0%

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

      4. difference-of-squares 24.7%

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

      5. hypot-define 27.5%

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

      6. hypot-define 27.5%

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

   6. Applied egg-rr 27.5%

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

   7. Taylor expanded in x around 0 24.7%

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

      1. associate-/l* 81.8%

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

   9. Simplified 81.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{-130}:\\ \;\;\;\;\frac{0.5}{\frac{y}{\left(x + z\right) \cdot \left(x - z\right)}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+131}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, \left(y + z\right) \cdot \left(y - z\right)\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\_m\right) \cdot \frac{y - z\_m}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\_m\right) \cdot \frac{x - z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* x x) 5e-122)
   (* 0.5 (* (+ y z_m) (/ (- y z_m) y)))
   (if (<= (* x x) 2e+265)
     (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))
     (* 0.5 (* (+ x z_m) (/ (- x z_m) y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if ((x * x) <= 5e-122) {
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	} else if ((x * x) <= 2e+265) {
		tmp = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	} else {
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((x * x) <= 5d-122) then
        tmp = 0.5d0 * ((y + z_m) * ((y - z_m) / y))
    else if ((x * x) <= 2d+265) then
        tmp = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)
    else
        tmp = 0.5d0 * ((x + z_m) * ((x - z_m) / y))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if ((x * x) <= 5e-122) {
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	} else if ((x * x) <= 2e+265) {
		tmp = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	} else {
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if (x * x) <= 5e-122:
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y))
	elif (x * x) <= 2e+265:
		tmp = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	else:
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(x * x) <= 5e-122)
		tmp = Float64(0.5 * Float64(Float64(y + z_m) * Float64(Float64(y - z_m) / y)));
	elseif (Float64(x * x) <= 2e+265)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0));
	else
		tmp = Float64(0.5 * Float64(Float64(x + z_m) * Float64(Float64(x - z_m) / y)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if ((x * x) <= 5e-122)
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	elseif ((x * x) <= 2e+265)
		tmp = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	else
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-122], N[(0.5 * N[(N[(y + z$95$m), $MachinePrecision] * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+265], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x + z$95$m), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.9999999999999999e-122

   1. Initial program 76.8%

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

      1. remove-double-neg 76.8%

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

      2. distribute-lft-neg-out 76.8%

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

      3. distribute-frac-neg2 76.8%

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

      4. distribute-frac-neg 76.8%

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

      5. neg-mul-1 76.8%

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

      6. distribute-lft-neg-out 76.8%

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

      7. *-commutative 76.8%

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

      8. distribute-lft-neg-in 76.8%

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

      9. times-frac 76.8%

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

      10. metadata-eval 76.8%

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

      11. metadata-eval 76.8%

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

      12. associate--l+ 76.8%

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

      13. fma-define 76.8%

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

   3. Simplified 76.8%

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

      1. fma-undefine 76.8%

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

      2. associate--l+ 76.8%

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

      3. add-sqr-sqrt 76.8%

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

      4. difference-of-squares 77.0%

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

      5. hypot-define 77.0%

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

      6. hypot-define 77.0%

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

   6. Applied egg-rr 77.0%

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

   7. Taylor expanded in x around 0 74.2%

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

      1. associate-/l* 97.1%

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

   9. Simplified 97.1%

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

   if 4.9999999999999999e-122 < (*.f64 x x) < 2.00000000000000013e265

   1. Initial program 82.8%

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

   if 2.00000000000000013e265 < (*.f64 x x)

   1. Initial program 52.7%

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

      1. remove-double-neg 52.7%

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

      2. distribute-lft-neg-out 52.7%

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

      3. distribute-frac-neg2 52.7%

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

      4. distribute-frac-neg 52.7%

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

      5. neg-mul-1 52.7%

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

      6. distribute-lft-neg-out 52.7%

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

      7. *-commutative 52.7%

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

      8. distribute-lft-neg-in 52.7%

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

      9. times-frac 52.7%

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

      10. metadata-eval 52.7%

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

      11. metadata-eval 52.7%

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

      12. associate--l+ 52.7%

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

      13. fma-define 63.9%

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

   3. Simplified 63.9%

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

      1. fma-undefine 52.7%

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

      2. associate--l+ 52.7%

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

      3. add-sqr-sqrt 52.7%

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

      4. difference-of-squares 65.7%

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

      5. hypot-define 72.7%

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

      6. hypot-define 76.8%

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

   6. Applied egg-rr 76.8%

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

   7. Taylor expanded in y around 0 76.8%

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

      1. associate-/l* 90.0%

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

      2. +-commutative 90.0%

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

   9. Simplified 90.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-122}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+265}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 2.9 \cdot 10^{-168}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{y}{x}}{x + z\_m}}\\ \mathbf{elif}\;z\_m \leq 2.3 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(z\_m \cdot \frac{y - z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= z_m 2.9e-168)
   (/ 0.5 (/ (/ y x) (+ x z_m)))
   (if (<= z_m 2.3e+71) (* 0.5 y) (* 0.5 (* z_m (/ (- y z_m) y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2.9e-168) {
		tmp = 0.5 / ((y / x) / (x + z_m));
	} else if (z_m <= 2.3e+71) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (z_m * ((y - z_m) / y));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 2.9d-168) then
        tmp = 0.5d0 / ((y / x) / (x + z_m))
    else if (z_m <= 2.3d+71) then
        tmp = 0.5d0 * y
    else
        tmp = 0.5d0 * (z_m * ((y - z_m) / y))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2.9e-168) {
		tmp = 0.5 / ((y / x) / (x + z_m));
	} else if (z_m <= 2.3e+71) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (z_m * ((y - z_m) / y));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if z_m <= 2.9e-168:
		tmp = 0.5 / ((y / x) / (x + z_m))
	elif z_m <= 2.3e+71:
		tmp = 0.5 * y
	else:
		tmp = 0.5 * (z_m * ((y - z_m) / y))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (z_m <= 2.9e-168)
		tmp = Float64(0.5 / Float64(Float64(y / x) / Float64(x + z_m)));
	elseif (z_m <= 2.3e+71)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(0.5 * Float64(z_m * Float64(Float64(y - z_m) / y)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (z_m <= 2.9e-168)
		tmp = 0.5 / ((y / x) / (x + z_m));
	elseif (z_m <= 2.3e+71)
		tmp = 0.5 * y;
	else
		tmp = 0.5 * (z_m * ((y - z_m) / y));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[z$95$m, 2.9e-168], N[(0.5 / N[(N[(y / x), $MachinePrecision] / N[(x + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.3e+71], N[(0.5 * y), $MachinePrecision], N[(0.5 * N[(z$95$m * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.8999999999999998e-168

   1. Initial program 70.0%

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

      1. remove-double-neg 70.0%

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

      2. distribute-lft-neg-out 70.0%

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

      3. distribute-frac-neg2 70.0%

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

      4. distribute-frac-neg 70.0%

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

      5. neg-mul-1 70.0%

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

      6. distribute-lft-neg-out 70.0%

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

      7. *-commutative 70.0%

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

      8. distribute-lft-neg-in 70.0%

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

      9. times-frac 69.6%

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

      10. metadata-eval 69.6%

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

      11. metadata-eval 69.6%

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

      12. associate--l+ 69.6%

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

      13. fma-define 72.5%

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

   3. Simplified 72.5%

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

      1. clear-num 72.4%

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

      2. un-div-inv 72.8%

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

      3. fma-undefine 69.9%

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

      4. associate--l+ 69.9%

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

      5. add-sqr-sqrt 69.9%

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

      6. pow2 69.9%

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

      7. hypot-define 69.9%

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

      8. pow2 69.9%

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

   6. Applied egg-rr 69.9%

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

      1. *-un-lft-identity 69.9%

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

      2. unpow2 69.9%

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

      3. unpow2 69.9%

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

      4. difference-of-squares 77.8%

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

      5. times-frac 99.7%

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

   8. Applied egg-rr 99.7%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{\mathsf{hypot}\left(x, y\right) + z} \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. hypot-undefine 79.4%

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

      4. unpow2 79.4%

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

      5. unpow2 79.4%

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

      6. +-commutative 79.4%

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

      7. unpow2 79.4%

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

      8. unpow2 79.4%

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

      9. hypot-define 99.8%

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

      10. +-commutative 99.8%

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

      11. hypot-undefine 76.0%

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

      12. unpow2 76.0%

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

      13. unpow2 76.0%

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

      14. +-commutative 76.0%

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

      15. unpow2 76.0%

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

      16. unpow2 76.0%

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

      17. hypot-define 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 25.4%

       \[\leadsto \frac{0.5}{\frac{\color{blue}{\frac{y}{x}}}{z + \mathsf{hypot}\left(y, x\right)}} \]

   12. Taylor expanded in y around 0 38.3%

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

       1. associate-/r* 42.3%

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

       2. +-commutative 42.3%

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

   14. Simplified 42.3%

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

   if 2.8999999999999998e-168 < z < 2.3000000000000002e71

   1. Initial program 78.8%

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

      1. remove-double-neg 78.8%

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

      2. distribute-lft-neg-out 78.8%

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

      3. distribute-frac-neg2 78.8%

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

      4. distribute-frac-neg 78.8%

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

      5. neg-mul-1 78.8%

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

      6. distribute-lft-neg-out 78.8%

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

      7. *-commutative 78.8%

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

      8. distribute-lft-neg-in 78.8%

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

      9. times-frac 78.8%

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

      10. metadata-eval 78.8%

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

      11. metadata-eval 78.8%

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

      12. associate--l+ 78.8%

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

      13. fma-define 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in y around inf 44.2%

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

   if 2.3000000000000002e71 < z

   1. Initial program 72.5%

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

      1. remove-double-neg 72.5%

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

      2. distribute-lft-neg-out 72.5%

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

      3. distribute-frac-neg2 72.5%

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

      4. distribute-frac-neg 72.5%

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

      5. neg-mul-1 72.5%

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

      6. distribute-lft-neg-out 72.5%

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

      7. *-commutative 72.5%

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

      8. distribute-lft-neg-in 72.5%

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

      9. times-frac 72.5%

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

      10. metadata-eval 72.5%

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

      11. metadata-eval 72.5%

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

      12. associate--l+ 72.5%

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

      13. fma-define 79.1%

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

   3. Simplified 79.1%

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

      1. fma-undefine 72.5%

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

      2. associate--l+ 72.5%

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

      3. add-sqr-sqrt 72.5%

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

      4. difference-of-squares 77.2%

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

      5. hypot-define 77.2%

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

      6. hypot-define 85.9%

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

   6. Applied egg-rr 85.9%

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

   7. Taylor expanded in x around 0 77.3%

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

      1. associate-/l* 89.2%

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

   9. Simplified 89.2%

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

   10. Taylor expanded in y around 0 76.9%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-168}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{y}{x}}{x + z}}\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \frac{y - z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{elif}\;z\_m \leq 2.7 \cdot 10^{+71}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(z\_m \cdot \frac{y - z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= z_m 5.6e-163)
   (* (* x x) (/ 0.5 y))
   (if (<= z_m 2.7e+71) (* 0.5 y) (* 0.5 (* z_m (/ (- y z_m) y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.6e-163) {
		tmp = (x * x) * (0.5 / y);
	} else if (z_m <= 2.7e+71) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (z_m * ((y - z_m) / y));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 5.6d-163) then
        tmp = (x * x) * (0.5d0 / y)
    else if (z_m <= 2.7d+71) then
        tmp = 0.5d0 * y
    else
        tmp = 0.5d0 * (z_m * ((y - z_m) / y))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.6e-163) {
		tmp = (x * x) * (0.5 / y);
	} else if (z_m <= 2.7e+71) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (z_m * ((y - z_m) / y));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if z_m <= 5.6e-163:
		tmp = (x * x) * (0.5 / y)
	elif z_m <= 2.7e+71:
		tmp = 0.5 * y
	else:
		tmp = 0.5 * (z_m * ((y - z_m) / y))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (z_m <= 5.6e-163)
		tmp = Float64(Float64(x * x) * Float64(0.5 / y));
	elseif (z_m <= 2.7e+71)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(0.5 * Float64(z_m * Float64(Float64(y - z_m) / y)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (z_m <= 5.6e-163)
		tmp = (x * x) * (0.5 / y);
	elseif (z_m <= 2.7e+71)
		tmp = 0.5 * y;
	else
		tmp = 0.5 * (z_m * ((y - z_m) / y));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[z$95$m, 5.6e-163], N[(N[(x * x), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.7e+71], N[(0.5 * y), $MachinePrecision], N[(0.5 * N[(z$95$m * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.5999999999999999e-163

   1. Initial program 70.0%

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

      1. remove-double-neg 70.0%

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

      2. distribute-lft-neg-out 70.0%

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

      3. distribute-frac-neg2 70.0%

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

      4. distribute-frac-neg 70.0%

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

      5. neg-mul-1 70.0%

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

      6. distribute-lft-neg-out 70.0%

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

      7. *-commutative 70.0%

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

      8. distribute-lft-neg-in 70.0%

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

      9. times-frac 69.6%

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

      10. metadata-eval 69.6%

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

      11. metadata-eval 69.6%

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

      12. associate--l+ 69.6%

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

      13. fma-define 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in x around inf 30.2%

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

      1. *-commutative 30.2%

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

      2. associate-*l/ 30.7%

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

      3. associate-*r/ 30.7%

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

   7. Simplified 30.7%

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

      1. unpow2 30.7%

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

   9. Applied egg-rr 30.7%

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

   if 5.5999999999999999e-163 < z < 2.69999999999999997e71

   1. Initial program 78.8%

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

      1. remove-double-neg 78.8%

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

      2. distribute-lft-neg-out 78.8%

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

      3. distribute-frac-neg2 78.8%

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

      4. distribute-frac-neg 78.8%

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

      5. neg-mul-1 78.8%

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

      6. distribute-lft-neg-out 78.8%

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

      7. *-commutative 78.8%

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

      8. distribute-lft-neg-in 78.8%

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

      9. times-frac 78.8%

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

      10. metadata-eval 78.8%

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

      11. metadata-eval 78.8%

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

      12. associate--l+ 78.8%

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

      13. fma-define 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in y around inf 44.2%

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

   if 2.69999999999999997e71 < z

   1. Initial program 72.5%

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

      1. remove-double-neg 72.5%

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

      2. distribute-lft-neg-out 72.5%

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

      3. distribute-frac-neg2 72.5%

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

      4. distribute-frac-neg 72.5%

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

      5. neg-mul-1 72.5%

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

      6. distribute-lft-neg-out 72.5%

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

      7. *-commutative 72.5%

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

      8. distribute-lft-neg-in 72.5%

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

      9. times-frac 72.5%

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

      10. metadata-eval 72.5%

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

      11. metadata-eval 72.5%

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

      12. associate--l+ 72.5%

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

      13. fma-define 79.1%

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

   3. Simplified 79.1%

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

      1. fma-undefine 72.5%

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

      2. associate--l+ 72.5%

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

      3. add-sqr-sqrt 72.5%

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

      4. difference-of-squares 77.2%

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

      5. hypot-define 77.2%

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

      6. hypot-define 85.9%

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

   6. Applied egg-rr 85.9%

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

   7. Taylor expanded in x around 0 77.3%

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

      1. associate-/l* 89.2%

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

   9. Simplified 89.2%

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

   10. Taylor expanded in y around 0 76.9%

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

Alternative 8: 51.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{elif}\;z\_m \leq 1.8 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z\_m \cdot \frac{z\_m}{y}\right) \cdot \left(-0.5\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= z_m 5.6e-163)
   (* (* x x) (/ 0.5 y))
   (if (<= z_m 1.8e+70) (* 0.5 y) (* (* z_m (/ z_m y)) (- 0.5)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.6e-163) {
		tmp = (x * x) * (0.5 / y);
	} else if (z_m <= 1.8e+70) {
		tmp = 0.5 * y;
	} else {
		tmp = (z_m * (z_m / y)) * -0.5;
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 5.6d-163) then
        tmp = (x * x) * (0.5d0 / y)
    else if (z_m <= 1.8d+70) then
        tmp = 0.5d0 * y
    else
        tmp = (z_m * (z_m / y)) * -0.5d0
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.6e-163) {
		tmp = (x * x) * (0.5 / y);
	} else if (z_m <= 1.8e+70) {
		tmp = 0.5 * y;
	} else {
		tmp = (z_m * (z_m / y)) * -0.5;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if z_m <= 5.6e-163:
		tmp = (x * x) * (0.5 / y)
	elif z_m <= 1.8e+70:
		tmp = 0.5 * y
	else:
		tmp = (z_m * (z_m / y)) * -0.5
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (z_m <= 5.6e-163)
		tmp = Float64(Float64(x * x) * Float64(0.5 / y));
	elseif (z_m <= 1.8e+70)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(z_m * Float64(z_m / y)) * Float64(-0.5));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (z_m <= 5.6e-163)
		tmp = (x * x) * (0.5 / y);
	elseif (z_m <= 1.8e+70)
		tmp = 0.5 * y;
	else
		tmp = (z_m * (z_m / y)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[z$95$m, 5.6e-163], N[(N[(x * x), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.8e+70], N[(0.5 * y), $MachinePrecision], N[(N[(z$95$m * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] * (-0.5)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 5.5999999999999999e-163

   1. Initial program 70.0%

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

      1. remove-double-neg 70.0%

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

      2. distribute-lft-neg-out 70.0%

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

      3. distribute-frac-neg2 70.0%

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

      4. distribute-frac-neg 70.0%

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

      5. neg-mul-1 70.0%

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

      6. distribute-lft-neg-out 70.0%

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

      7. *-commutative 70.0%

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

      8. distribute-lft-neg-in 70.0%

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

      9. times-frac 69.6%

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

      10. metadata-eval 69.6%

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

      11. metadata-eval 69.6%

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

      12. associate--l+ 69.6%

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

      13. fma-define 72.5%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in x around inf 30.2%

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

      1. *-commutative 30.2%

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

      2. associate-*l/ 30.7%

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

      3. associate-*r/ 30.7%

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

   7. Simplified 30.7%

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

      1. unpow2 30.7%

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

   9. Applied egg-rr 30.7%

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

   if 5.5999999999999999e-163 < z < 1.8e70

   1. Initial program 78.8%

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

      1. remove-double-neg 78.8%

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

      2. distribute-lft-neg-out 78.8%

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

      3. distribute-frac-neg2 78.8%

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

      4. distribute-frac-neg 78.8%

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

      5. neg-mul-1 78.8%

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

      6. distribute-lft-neg-out 78.8%

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

      7. *-commutative 78.8%

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

      8. distribute-lft-neg-in 78.8%

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

      9. times-frac 78.8%

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

      10. metadata-eval 78.8%

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

      11. metadata-eval 78.8%

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

      12. associate--l+ 78.8%

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

      13. fma-define 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in y around inf 44.2%

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

   if 1.8e70 < z

   1. Initial program 72.5%

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

      1. remove-double-neg 72.5%

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

      2. distribute-lft-neg-out 72.5%

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

      3. distribute-frac-neg2 72.5%

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

      4. distribute-frac-neg 72.5%

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

      5. neg-mul-1 72.5%

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

      6. distribute-lft-neg-out 72.5%

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

      7. *-commutative 72.5%

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

      8. distribute-lft-neg-in 72.5%

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

      9. times-frac 72.5%

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

      10. metadata-eval 72.5%

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

      11. metadata-eval 72.5%

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

      12. associate--l+ 72.5%

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

      13. fma-define 79.1%

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

   3. Simplified 79.1%

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

      1. fma-undefine 72.5%

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

      2. associate--l+ 72.5%

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

      3. add-sqr-sqrt 72.5%

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

      4. difference-of-squares 77.2%

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

      5. hypot-define 77.2%

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

      6. hypot-define 85.9%

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

   6. Applied egg-rr 85.9%

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

   7. Taylor expanded in x around 0 77.3%

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

      1. associate-/l* 89.2%

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

   9. Simplified 89.2%

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

   10. Taylor expanded in y around 0 76.9%

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

   11. Taylor expanded in y around 0 76.1%

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

       1. neg-mul-1 76.1%

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

       2. distribute-neg-frac2 76.1%

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

   13. Simplified 76.1%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.6 \cdot 10^{-163}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+70}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot \left(-0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\_m\right) \cdot \frac{y - z\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\_m\right) \cdot \frac{x - z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (* x x) 5e+151)
   (* 0.5 (* (+ y z_m) (/ (- y z_m) y)))
   (* 0.5 (* (+ x z_m) (/ (- x z_m) y)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if ((x * x) <= 5e+151) {
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	} else {
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if ((x * x) <= 5d+151) then
        tmp = 0.5d0 * ((y + z_m) * ((y - z_m) / y))
    else
        tmp = 0.5d0 * ((x + z_m) * ((x - z_m) / y))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if ((x * x) <= 5e+151) {
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	} else {
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if (x * x) <= 5e+151:
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y))
	else:
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(x * x) <= 5e+151)
		tmp = Float64(0.5 * Float64(Float64(y + z_m) * Float64(Float64(y - z_m) / y)));
	else
		tmp = Float64(0.5 * Float64(Float64(x + z_m) * Float64(Float64(x - z_m) / y)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if ((x * x) <= 5e+151)
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	else
		tmp = 0.5 * ((x + z_m) * ((x - z_m) / y));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e+151], N[(0.5 * N[(N[(y + z$95$m), $MachinePrecision] * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x + z$95$m), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.0000000000000002e151

   1. Initial program 79.4%

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

      1. remove-double-neg 79.4%

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

      2. distribute-lft-neg-out 79.4%

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

      3. distribute-frac-neg2 79.4%

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

      4. distribute-frac-neg 79.4%

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

      5. neg-mul-1 79.4%

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

      6. distribute-lft-neg-out 79.4%

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

      7. *-commutative 79.4%

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

      8. distribute-lft-neg-in 79.4%

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

      9. times-frac 79.4%

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

      10. metadata-eval 79.4%

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

      11. metadata-eval 79.4%

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

      12. associate--l+ 79.4%

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

      13. fma-define 79.4%

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

   3. Simplified 79.4%

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

      1. fma-undefine 79.4%

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

      2. associate--l+ 79.4%

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

      3. add-sqr-sqrt 79.4%

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

      4. difference-of-squares 79.6%

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

      5. hypot-define 80.2%

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

      6. hypot-define 80.8%

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in x around 0 72.1%

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

      1. associate-/l* 91.1%

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

   9. Simplified 91.1%

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

   if 5.0000000000000002e151 < (*.f64 x x)

   1. Initial program 59.8%

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

      1. remove-double-neg 59.8%

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

      2. distribute-lft-neg-out 59.8%

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

      3. distribute-frac-neg2 59.8%

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

      4. distribute-frac-neg 59.8%

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

      5. neg-mul-1 59.8%

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

      6. distribute-lft-neg-out 59.8%

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

      7. *-commutative 59.8%

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

      8. distribute-lft-neg-in 59.8%

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

      9. times-frac 59.0%

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

      10. metadata-eval 59.0%

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

      11. metadata-eval 59.0%

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

      12. associate--l+ 59.0%

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

      13. fma-define 67.1%

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

   3. Simplified 67.1%

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

      1. fma-undefine 59.0%

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

      2. associate--l+ 59.0%

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

      3. add-sqr-sqrt 59.0%

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

      4. difference-of-squares 68.5%

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

      5. hypot-define 73.6%

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

      6. hypot-define 76.6%

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

   6. Applied egg-rr 76.6%

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

   7. Taylor expanded in y around 0 75.4%

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

      1. associate-/l* 85.0%

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

      2. +-commutative 85.0%

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

   9. Simplified 85.0%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+176}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\_m\right) \cdot \frac{y - z\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{y}{x}}{x + z\_m}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= x 7e+176)
   (* 0.5 (* (+ y z_m) (/ (- y z_m) y)))
   (/ 0.5 (/ (/ y x) (+ x z_m)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (x <= 7e+176) {
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	} else {
		tmp = 0.5 / ((y / x) / (x + z_m));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 7d+176) then
        tmp = 0.5d0 * ((y + z_m) * ((y - z_m) / y))
    else
        tmp = 0.5d0 / ((y / x) / (x + z_m))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (x <= 7e+176) {
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	} else {
		tmp = 0.5 / ((y / x) / (x + z_m));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if x <= 7e+176:
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y))
	else:
		tmp = 0.5 / ((y / x) / (x + z_m))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (x <= 7e+176)
		tmp = Float64(0.5 * Float64(Float64(y + z_m) * Float64(Float64(y - z_m) / y)));
	else
		tmp = Float64(0.5 / Float64(Float64(y / x) / Float64(x + z_m)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (x <= 7e+176)
		tmp = 0.5 * ((y + z_m) * ((y - z_m) / y));
	else
		tmp = 0.5 / ((y / x) / (x + z_m));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[x, 7e+176], N[(0.5 * N[(N[(y + z$95$m), $MachinePrecision] * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(N[(y / x), $MachinePrecision] / N[(x + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.00000000000000005e176

   1. Initial program 74.7%

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

      1. remove-double-neg 74.7%

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

      2. distribute-lft-neg-out 74.7%

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

      3. distribute-frac-neg2 74.7%

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

      4. distribute-frac-neg 74.7%

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

      5. neg-mul-1 74.7%

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

      6. distribute-lft-neg-out 74.7%

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

      7. *-commutative 74.7%

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

      8. distribute-lft-neg-in 74.7%

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

      9. times-frac 74.3%

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

      10. metadata-eval 74.3%

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

      11. metadata-eval 74.3%

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

      12. associate--l+ 74.3%

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

      13. fma-define 76.1%

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

   3. Simplified 76.1%

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

      1. fma-undefine 74.3%

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

      2. associate--l+ 74.3%

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

      3. add-sqr-sqrt 74.3%

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

      4. difference-of-squares 76.7%

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

      5. hypot-define 78.0%

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

      6. hypot-define 79.3%

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

   6. Applied egg-rr 79.3%

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

   7. Taylor expanded in x around 0 58.0%

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

      1. associate-/l* 74.6%

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

   9. Simplified 74.6%

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

   if 7.00000000000000005e176 < x

   1. Initial program 45.6%

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

      1. remove-double-neg 45.6%

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

      2. distribute-lft-neg-out 45.6%

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

      3. distribute-frac-neg2 45.6%

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

      4. distribute-frac-neg 45.6%

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

      5. neg-mul-1 45.6%

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

      6. distribute-lft-neg-out 45.6%

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

      7. *-commutative 45.6%

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

      8. distribute-lft-neg-in 45.6%

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

      9. times-frac 45.6%

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

      10. metadata-eval 45.6%

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

      11. metadata-eval 45.6%

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

      12. associate--l+ 45.6%

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

      13. fma-define 61.6%

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

   3. Simplified 61.6%

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

      1. clear-num 61.6%

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

      2. un-div-inv 61.6%

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

      3. fma-undefine 45.6%

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

      4. associate--l+ 45.6%

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

      5. add-sqr-sqrt 45.6%

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

      6. pow2 45.6%

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

      7. hypot-define 45.6%

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

      8. pow2 45.6%

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

   6. Applied egg-rr 45.6%

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

      1. *-un-lft-identity 45.6%

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

      2. unpow2 45.6%

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

      3. unpow2 45.6%

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

      4. difference-of-squares 78.1%

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

      5. times-frac 99.9%

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

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{\mathsf{hypot}\left(x, y\right) + z} \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

      3. hypot-undefine 74.1%

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

      4. unpow2 74.1%

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

      5. unpow2 74.1%

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

      6. +-commutative 74.1%

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

      7. unpow2 74.1%

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

      8. unpow2 74.1%

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

      9. hypot-define 100.0%

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

      10. +-commutative 100.0%

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

      11. hypot-undefine 66.1%

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

      12. unpow2 66.1%

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

      13. unpow2 66.1%

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

      14. +-commutative 66.1%

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

      15. unpow2 66.1%

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

      16. unpow2 66.1%

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

      17. hypot-define 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in x around inf 92.3%

       \[\leadsto \frac{0.5}{\frac{\color{blue}{\frac{y}{x}}}{z + \mathsf{hypot}\left(y, x\right)}} \]

   12. Taylor expanded in y around 0 74.1%

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

       1. associate-/r* 92.2%

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

       2. +-commutative 92.2%

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

   14. Simplified 92.2%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{+176}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{\frac{y}{x}}{x + z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 6.6 \cdot 10^{+75}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= x 6.6e+75) (* 0.5 y) (* (* x x) (/ 0.5 y))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (x <= 6.6e+75) {
		tmp = 0.5 * y;
	} else {
		tmp = (x * x) * (0.5 / y);
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 6.6d+75) then
        tmp = 0.5d0 * y
    else
        tmp = (x * x) * (0.5d0 / y)
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (x <= 6.6e+75) {
		tmp = 0.5 * y;
	} else {
		tmp = (x * x) * (0.5 / y);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if x <= 6.6e+75:
		tmp = 0.5 * y
	else:
		tmp = (x * x) * (0.5 / y)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (x <= 6.6e+75)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x * x) * Float64(0.5 / y));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (x <= 6.6e+75)
		tmp = 0.5 * y;
	else
		tmp = (x * x) * (0.5 / y);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[x, 6.6e+75], N[(0.5 * y), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(0.5 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.59999999999999996e75

   1. Initial program 74.1%

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

      1. remove-double-neg 74.1%

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

      2. distribute-lft-neg-out 74.1%

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

      3. distribute-frac-neg2 74.1%

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

      4. distribute-frac-neg 74.1%

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

      5. neg-mul-1 74.1%

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

      6. distribute-lft-neg-out 74.1%

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

      7. *-commutative 74.1%

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

      8. distribute-lft-neg-in 74.1%

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

      9. times-frac 73.7%

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

      10. metadata-eval 73.7%

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

      11. metadata-eval 73.7%

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

      12. associate--l+ 73.7%

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

      13. fma-define 75.2%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in y around inf 36.4%

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

   if 6.59999999999999996e75 < x

   1. Initial program 63.4%

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

      1. remove-double-neg 63.4%

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

      2. distribute-lft-neg-out 63.4%

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

      3. distribute-frac-neg2 63.4%

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

      4. distribute-frac-neg 63.4%

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

      5. neg-mul-1 63.4%

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

      6. distribute-lft-neg-out 63.4%

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

      7. *-commutative 63.4%

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

      8. distribute-lft-neg-in 63.4%

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

      9. times-frac 63.4%

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

      10. metadata-eval 63.4%

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

      11. metadata-eval 63.4%

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

      12. associate--l+ 63.4%

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

      13. fma-define 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in x around inf 50.6%

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

      1. *-commutative 50.6%

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

      2. associate-*l/ 50.6%

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

      3. associate-*r/ 50.7%

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

   7. Simplified 50.7%

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

      1. unpow2 50.7%

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

   9. Applied egg-rr 50.7%

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

Alternative 12: 36.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 2.35 \cdot 10^{+207}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{z\_m}{y}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= x 2.35e+207) (* 0.5 y) (* 0.5 (* x (/ z_m y)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (x <= 2.35e+207) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (x * (z_m / y));
	}
	return tmp;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 2.35d+207) then
        tmp = 0.5d0 * y
    else
        tmp = 0.5d0 * (x * (z_m / y))
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (x <= 2.35e+207) {
		tmp = 0.5 * y;
	} else {
		tmp = 0.5 * (x * (z_m / y));
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if x <= 2.35e+207:
		tmp = 0.5 * y
	else:
		tmp = 0.5 * (x * (z_m / y))
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (x <= 2.35e+207)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(0.5 * Float64(x * Float64(z_m / y)));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (x <= 2.35e+207)
		tmp = 0.5 * y;
	else
		tmp = 0.5 * (x * (z_m / y));
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[x, 2.35e+207], N[(0.5 * y), $MachinePrecision], N[(0.5 * N[(x * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.34999999999999988e207

   1. Initial program 73.6%

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

      1. remove-double-neg 73.6%

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

      2. distribute-lft-neg-out 73.6%

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

      3. distribute-frac-neg2 73.6%

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

      4. distribute-frac-neg 73.6%

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

      5. neg-mul-1 73.6%

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

      6. distribute-lft-neg-out 73.6%

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

      7. *-commutative 73.6%

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

      8. distribute-lft-neg-in 73.6%

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

      9. times-frac 73.2%

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

      10. metadata-eval 73.2%

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

      11. metadata-eval 73.2%

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

      12. associate--l+ 73.2%

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

      13. fma-define 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in y around inf 34.0%

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

   if 2.34999999999999988e207 < x

   1. Initial program 51.4%

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

      1. remove-double-neg 51.4%

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

      2. distribute-lft-neg-out 51.4%

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

      3. distribute-frac-neg2 51.4%

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

      4. distribute-frac-neg 51.4%

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

      5. neg-mul-1 51.4%

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

      6. distribute-lft-neg-out 51.4%

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

      7. *-commutative 51.4%

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

      8. distribute-lft-neg-in 51.4%

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

      9. times-frac 51.4%

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

      10. metadata-eval 51.4%

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

      11. metadata-eval 51.4%

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

      12. associate--l+ 51.4%

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

      13. fma-define 61.4%

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

   3. Simplified 61.4%

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

      1. clear-num 61.4%

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

      2. un-div-inv 61.4%

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

      3. fma-undefine 51.4%

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

      4. associate--l+ 51.4%

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

      5. add-sqr-sqrt 51.4%

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

      6. pow2 51.4%

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

      7. hypot-define 51.4%

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

      8. pow2 51.4%

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

   6. Applied egg-rr 51.4%

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

      1. *-un-lft-identity 51.4%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}} \]

      2. unpow2 51.4%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)} - {z}^{2}}} \]

      3. unpow2 51.4%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right) - \color{blue}{z \cdot z}}} \]

      4. difference-of-squares 82.0%

         \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{\left(\mathsf{hypot}\left(x, y\right) + z\right) \cdot \left(\mathsf{hypot}\left(x, y\right) - z\right)}}} \]

      5. times-frac 99.9%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{\mathsf{hypot}\left(x, y\right) + z} \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{\mathsf{hypot}\left(x, y\right) + z} \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}} \]
   9. Step-by-step derivation

      1. associate-*l/ 100.0%

         \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}{\mathsf{hypot}\left(x, y\right) + z}}} \]

      2. *-lft-identity 100.0%

         \[\leadsto \frac{0.5}{\frac{\color{blue}{\frac{y}{\mathsf{hypot}\left(x, y\right) - z}}}{\mathsf{hypot}\left(x, y\right) + z}} \]

      3. hypot-undefine 82.0%

         \[\leadsto \frac{0.5}{\frac{\frac{y}{\color{blue}{\sqrt{x \cdot x + y \cdot y}} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

      4. unpow2 82.0%

         \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{\color{blue}{{x}^{2}} + y \cdot y} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

      5. unpow2 82.0%

         \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{{x}^{2} + \color{blue}{{y}^{2}}} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

      6. +-commutative 82.0%

         \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{\color{blue}{{y}^{2} + {x}^{2}}} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

      7. unpow2 82.0%

         \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{\color{blue}{y \cdot y} + {x}^{2}} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

      8. unpow2 82.0%

         \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{y \cdot y + \color{blue}{x \cdot x}} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

      9. hypot-define 100.0%

         \[\leadsto \frac{0.5}{\frac{\frac{y}{\color{blue}{\mathsf{hypot}\left(y, x\right)} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

      10. +-commutative 100.0%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{\color{blue}{z + \mathsf{hypot}\left(x, y\right)}}} \]

      11. hypot-undefine 72.0%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \color{blue}{\sqrt{x \cdot x + y \cdot y}}}} \]

      12. unpow2 72.0%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \sqrt{\color{blue}{{x}^{2}} + y \cdot y}}} \]

      13. unpow2 72.0%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \sqrt{{x}^{2} + \color{blue}{{y}^{2}}}}} \]

      14. +-commutative 72.0%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \sqrt{\color{blue}{{y}^{2} + {x}^{2}}}}} \]

      15. unpow2 72.0%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \sqrt{\color{blue}{y \cdot y} + {x}^{2}}}} \]

      16. unpow2 72.0%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \sqrt{y \cdot y + \color{blue}{x \cdot x}}}} \]

      17. hypot-define 100.0%

          \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \color{blue}{\mathsf{hypot}\left(y, x\right)}}} \]

   10. Simplified 100.0%

       \[\leadsto \frac{0.5}{\color{blue}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \mathsf{hypot}\left(y, x\right)}}} \]

   11. Taylor expanded in x around inf 95.3%

       \[\leadsto \frac{0.5}{\frac{\color{blue}{\frac{y}{x}}}{z + \mathsf{hypot}\left(y, x\right)}} \]

   12. Taylor expanded in z around inf 25.5%

       \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot z}{y}} \]
   13. Step-by-step derivation

       1. associate-/l* 25.5%

          \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \frac{z}{y}\right)} \]

   14. Simplified 25.5%

       \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \frac{z}{y}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 33.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ 0.5 \cdot y \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m) :precision binary64 (* 0.5 y))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	return 0.5 * y;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = 0.5d0 * y
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	return 0.5 * y;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	return 0.5 * y

Julia

z_m = abs(z)
function code(x, y, z_m)
	return Float64(0.5 * y)
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m)
	tmp = 0.5 * y;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 71.9%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
2. Step-by-step derivation

   1. remove-double-neg 71.9%

      \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{-\left(-y \cdot 2\right)}} \]

   2. distribute-lft-neg-out 71.9%

      \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{-\color{blue}{\left(-y\right) \cdot 2}} \]

   3. distribute-frac-neg2 71.9%

      \[\leadsto \color{blue}{-\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\left(-y\right) \cdot 2}} \]

   4. distribute-frac-neg 71.9%

      \[\leadsto \color{blue}{\frac{-\left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}{\left(-y\right) \cdot 2}} \]

   5. neg-mul-1 71.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}}{\left(-y\right) \cdot 2} \]

   6. distribute-lft-neg-out 71.9%

      \[\leadsto \frac{-1 \cdot \left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}{\color{blue}{-y \cdot 2}} \]

   7. *-commutative 71.9%

      \[\leadsto \frac{-1 \cdot \left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}{-\color{blue}{2 \cdot y}} \]

   8. distribute-lft-neg-in 71.9%

      \[\leadsto \frac{-1 \cdot \left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}{\color{blue}{\left(-2\right) \cdot y}} \]

   9. times-frac 71.5%

      \[\leadsto \color{blue}{\frac{-1}{-2} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}} \]

   10. metadata-eval 71.5%

       \[\leadsto \frac{-1}{\color{blue}{-2}} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

   11. metadata-eval 71.5%

       \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

   12. associate--l+ 71.5%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

   13. fma-define 74.7%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

3. Simplified 74.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
4. Add Preprocessing

5. Taylor expanded in y around inf 31.9%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
6. Add Preprocessing

Alternative 14: 2.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ 0.5 \cdot x \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m) :precision binary64 (* 0.5 x))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	return 0.5 * x;
}

Fortran

z_m = abs(z)
real(8) function code(x, y, z_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = 0.5d0 * x
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	return 0.5 * x;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	return 0.5 * x

Julia

z_m = abs(z)
function code(x, y, z_m)
	return Float64(0.5 * x)
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m)
	tmp = 0.5 * x;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(0.5 * x), $MachinePrecision]

Derivation

1. Initial program 71.9%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
2. Step-by-step derivation

   1. remove-double-neg 71.9%

      \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{-\left(-y \cdot 2\right)}} \]

   2. distribute-lft-neg-out 71.9%

      \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{-\color{blue}{\left(-y\right) \cdot 2}} \]

   3. distribute-frac-neg2 71.9%

      \[\leadsto \color{blue}{-\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\left(-y\right) \cdot 2}} \]

   4. distribute-frac-neg 71.9%

      \[\leadsto \color{blue}{\frac{-\left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}{\left(-y\right) \cdot 2}} \]

   5. neg-mul-1 71.9%

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}}{\left(-y\right) \cdot 2} \]

   6. distribute-lft-neg-out 71.9%

      \[\leadsto \frac{-1 \cdot \left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}{\color{blue}{-y \cdot 2}} \]

   7. *-commutative 71.9%

      \[\leadsto \frac{-1 \cdot \left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}{-\color{blue}{2 \cdot y}} \]

   8. distribute-lft-neg-in 71.9%

      \[\leadsto \frac{-1 \cdot \left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}{\color{blue}{\left(-2\right) \cdot y}} \]

   9. times-frac 71.5%

      \[\leadsto \color{blue}{\frac{-1}{-2} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y}} \]

   10. metadata-eval 71.5%

       \[\leadsto \frac{-1}{\color{blue}{-2}} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

   11. metadata-eval 71.5%

       \[\leadsto \color{blue}{0.5} \cdot \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y} \]

   12. associate--l+ 71.5%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}{y} \]

   13. fma-define 74.7%

       \[\leadsto 0.5 \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}{y} \]

3. Simplified 74.7%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num 74.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{y}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}} \]

   2. un-div-inv 74.9%

      \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}}} \]

   3. fma-undefine 71.7%

      \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{x \cdot x + \left(y \cdot y - z \cdot z\right)}}} \]

   4. associate--l+ 71.7%

      \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{\left(x \cdot x + y \cdot y\right) - z \cdot z}}} \]

   5. add-sqr-sqrt 71.7%

      \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{\sqrt{x \cdot x + y \cdot y} \cdot \sqrt{x \cdot x + y \cdot y}} - z \cdot z}} \]

   6. pow2 71.7%

      \[\leadsto \frac{0.5}{\frac{y}{\color{blue}{{\left(\sqrt{x \cdot x + y \cdot y}\right)}^{2}} - z \cdot z}} \]

   7. hypot-define 71.7%

      \[\leadsto \frac{0.5}{\frac{y}{{\color{blue}{\left(\mathsf{hypot}\left(x, y\right)\right)}}^{2} - z \cdot z}} \]

   8. pow2 71.7%

      \[\leadsto \frac{0.5}{\frac{y}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - \color{blue}{{z}^{2}}}} \]

6. Applied egg-rr 71.7%

   \[\leadsto \color{blue}{\frac{0.5}{\frac{y}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 71.7%

      \[\leadsto \frac{0.5}{\frac{\color{blue}{1 \cdot y}}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2} - {z}^{2}}} \]

   2. unpow2 71.7%

      \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right)} - {z}^{2}}} \]

   3. unpow2 71.7%

      \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\mathsf{hypot}\left(x, y\right) \cdot \mathsf{hypot}\left(x, y\right) - \color{blue}{z \cdot z}}} \]

   4. difference-of-squares 79.4%

      \[\leadsto \frac{0.5}{\frac{1 \cdot y}{\color{blue}{\left(\mathsf{hypot}\left(x, y\right) + z\right) \cdot \left(\mathsf{hypot}\left(x, y\right) - z\right)}}} \]

   5. times-frac 99.7%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{\mathsf{hypot}\left(x, y\right) + z} \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}} \]

8. Applied egg-rr 99.7%

   \[\leadsto \frac{0.5}{\color{blue}{\frac{1}{\mathsf{hypot}\left(x, y\right) + z} \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}} \]
9. Step-by-step derivation

   1. associate-*l/ 99.8%

      \[\leadsto \frac{0.5}{\color{blue}{\frac{1 \cdot \frac{y}{\mathsf{hypot}\left(x, y\right) - z}}{\mathsf{hypot}\left(x, y\right) + z}}} \]

   2. *-lft-identity 99.8%

      \[\leadsto \frac{0.5}{\frac{\color{blue}{\frac{y}{\mathsf{hypot}\left(x, y\right) - z}}}{\mathsf{hypot}\left(x, y\right) + z}} \]

   3. hypot-undefine 79.2%

      \[\leadsto \frac{0.5}{\frac{\frac{y}{\color{blue}{\sqrt{x \cdot x + y \cdot y}} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

   4. unpow2 79.2%

      \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{\color{blue}{{x}^{2}} + y \cdot y} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

   5. unpow2 79.2%

      \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{{x}^{2} + \color{blue}{{y}^{2}}} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

   6. +-commutative 79.2%

      \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{\color{blue}{{y}^{2} + {x}^{2}}} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

   7. unpow2 79.2%

      \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{\color{blue}{y \cdot y} + {x}^{2}} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

   8. unpow2 79.2%

      \[\leadsto \frac{0.5}{\frac{\frac{y}{\sqrt{y \cdot y + \color{blue}{x \cdot x}} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

   9. hypot-define 99.8%

      \[\leadsto \frac{0.5}{\frac{\frac{y}{\color{blue}{\mathsf{hypot}\left(y, x\right)} - z}}{\mathsf{hypot}\left(x, y\right) + z}} \]

   10. +-commutative 99.8%

       \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{\color{blue}{z + \mathsf{hypot}\left(x, y\right)}}} \]

   11. hypot-undefine 78.5%

       \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \color{blue}{\sqrt{x \cdot x + y \cdot y}}}} \]

   12. unpow2 78.5%

       \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \sqrt{\color{blue}{{x}^{2}} + y \cdot y}}} \]

   13. unpow2 78.5%

       \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \sqrt{{x}^{2} + \color{blue}{{y}^{2}}}}} \]

   14. +-commutative 78.5%

       \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \sqrt{\color{blue}{{y}^{2} + {x}^{2}}}}} \]

   15. unpow2 78.5%

       \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \sqrt{\color{blue}{y \cdot y} + {x}^{2}}}} \]

   16. unpow2 78.5%

       \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \sqrt{y \cdot y + \color{blue}{x \cdot x}}}} \]

   17. hypot-define 99.8%

       \[\leadsto \frac{0.5}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \color{blue}{\mathsf{hypot}\left(y, x\right)}}} \]

10. Simplified 99.8%

    \[\leadsto \frac{0.5}{\color{blue}{\frac{\frac{y}{\mathsf{hypot}\left(y, x\right) - z}}{z + \mathsf{hypot}\left(y, x\right)}}} \]

11. Taylor expanded in x around inf 23.0%

    \[\leadsto \frac{0.5}{\frac{\color{blue}{\frac{y}{x}}}{z + \mathsf{hypot}\left(y, x\right)}} \]

12. Taylor expanded in y around inf 3.0%

    \[\leadsto \color{blue}{0.5 \cdot x} \]
13. 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 2024133 
(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)))