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

Percentage Accurate: 69.2% → 93.4%

Time: 8.4s

Alternatives: 9

Speedup: 0.7×

• 42.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% 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: 93.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, y\_m + z\right)\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -5 \cdot 10^{-135}:\\ \;\;\;\;\frac{z}{y\_m} \cdot \frac{z}{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 \cdot \frac{t\_0}{y\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (hypot x (+ y_m z))))
   (*
    y_s
    (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -5e-135)
      (* (/ z y_m) (/ z -2.0))
      (* 0.5 (* t_0 (/ t_0 y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = hypot(x, (y_m + z));
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -5e-135) {
		tmp = (z / y_m) * (z / -2.0);
	} else {
		tmp = 0.5 * (t_0 * (t_0 / y_m));
	}
	return y_s * tmp;
}

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = Math.hypot(x, (y_m + z));
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -5e-135) {
		tmp = (z / y_m) * (z / -2.0);
	} else {
		tmp = 0.5 * (t_0 * (t_0 / y_m));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = math.hypot(x, (y_m + z))
	tmp = 0
	if ((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -5e-135:
		tmp = (z / y_m) * (z / -2.0)
	else:
		tmp = 0.5 * (t_0 * (t_0 / y_m))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = hypot(x, Float64(y_m + z))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -5e-135)
		tmp = Float64(Float64(z / y_m) * Float64(z / -2.0));
	else
		tmp = Float64(0.5 * Float64(t_0 * Float64(t_0 / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = hypot(x, (y_m + z));
	tmp = 0.0;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -5e-135)
		tmp = (z / y_m) * (z / -2.0);
	else
		tmp = 0.5 * (t_0 * (t_0 / y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(y$95$m + z), $MachinePrecision] ^ 2], $MachinePrecision]}, N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -5e-135], N[(N[(z / y$95$m), $MachinePrecision] * N[(z / -2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 * N[(t$95$0 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -5.0000000000000002e-135

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf 28.5%

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

      1. associate-*r/ 28.5%

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

      2. metadata-eval 28.5%

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

      3. distribute-lft-neg-in 28.5%

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

      4. *-commutative 28.5%

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

      5. distribute-neg-frac 28.5%

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

      6. associate-*r/ 28.5%

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

      7. distribute-rgt-neg-in 28.5%

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

      8. distribute-neg-frac 28.5%

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

      9. metadata-eval 28.5%

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

   5. Simplified 28.5%

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

      1. associate-*r/ 28.5%

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

      2. clear-num 28.5%

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

   7. Applied egg-rr 28.5%

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

      1. clear-num 28.5%

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

      2. associate-/l* 28.5%

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

      3. metadata-eval 28.5%

         \[\leadsto {z}^{2} \cdot \frac{\color{blue}{\frac{1}{-2}}}{y} \]

      4. associate-/r* 28.5%

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{1}{-2 \cdot y}} \]

      5. *-commutative 28.5%

         \[\leadsto {z}^{2} \cdot \frac{1}{\color{blue}{y \cdot -2}} \]

      6. div-inv 28.5%

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

      7. unpow2 28.5%

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

      8. times-frac 30.1%

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

   9. Applied egg-rr 30.1%

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

   if -5.0000000000000002e-135 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.6%

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

      1. remove-double-neg 61.6%

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

      2. distribute-lft-neg-out 61.6%

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

      3. distribute-frac-neg2 61.6%

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

      4. distribute-frac-neg 61.6%

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

      5. neg-mul-1 61.6%

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

      6. distribute-lft-neg-out 61.6%

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

      7. *-commutative 61.6%

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

      8. distribute-lft-neg-in 61.6%

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

      9. times-frac 61.6%

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

      10. metadata-eval 61.6%

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

      11. metadata-eval 61.6%

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

      12. associate--l+ 61.6%

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

      13. fma-define 64.4%

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

   3. Simplified 64.4%

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

      1. prod-diff 52.7%

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

      2. fma-neg 52.7%

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

      3. difference-of-squares 53.9%

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

      4. fma-define 57.4%

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

      5. pow2 57.4%

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

   6. Applied egg-rr 57.4%

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

      1. add-sqr-sqrt 42.4%

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

      2. associate-/l* 42.5%

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

   8. Applied egg-rr 62.9%

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

Alternative 2: 88.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -5 \cdot 10^{-135}:\\ \;\;\;\;\frac{z}{y\_m} \cdot \frac{z}{-2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m + \frac{{x}^{2}}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -5e-135)
    (* (/ z y_m) (/ z -2.0))
    (* 0.5 (+ y_m (/ (pow x 2.0) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -5e-135) {
		tmp = (z / y_m) * (z / -2.0);
	} else {
		tmp = 0.5 * (y_m + (pow(x, 2.0) / y_m));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)) <= (-5d-135)) then
        tmp = (z / y_m) * (z / (-2.0d0))
    else
        tmp = 0.5d0 * (y_m + ((x ** 2.0d0) / y_m))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -5e-135) {
		tmp = (z / y_m) * (z / -2.0);
	} else {
		tmp = 0.5 * (y_m + (Math.pow(x, 2.0) / y_m));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if ((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -5e-135:
		tmp = (z / y_m) * (z / -2.0)
	else:
		tmp = 0.5 * (y_m + (math.pow(x, 2.0) / y_m))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -5e-135)
		tmp = Float64(Float64(z / y_m) * Float64(z / -2.0));
	else
		tmp = Float64(0.5 * Float64(y_m + Float64((x ^ 2.0) / y_m)));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -5e-135)
		tmp = (z / y_m) * (z / -2.0);
	else
		tmp = 0.5 * (y_m + ((x ^ 2.0) / y_m));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -5e-135], N[(N[(z / y$95$m), $MachinePrecision] * N[(z / -2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m + N[(N[Power[x, 2.0], $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -5.0000000000000002e-135

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf 28.5%

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

      1. associate-*r/ 28.5%

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

      2. metadata-eval 28.5%

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

      3. distribute-lft-neg-in 28.5%

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

      4. *-commutative 28.5%

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

      5. distribute-neg-frac 28.5%

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

      6. associate-*r/ 28.5%

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

      7. distribute-rgt-neg-in 28.5%

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

      8. distribute-neg-frac 28.5%

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

      9. metadata-eval 28.5%

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

   5. Simplified 28.5%

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

      1. associate-*r/ 28.5%

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

      2. clear-num 28.5%

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

   7. Applied egg-rr 28.5%

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

      1. clear-num 28.5%

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

      2. associate-/l* 28.5%

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

      3. metadata-eval 28.5%

         \[\leadsto {z}^{2} \cdot \frac{\color{blue}{\frac{1}{-2}}}{y} \]

      4. associate-/r* 28.5%

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{1}{-2 \cdot y}} \]

      5. *-commutative 28.5%

         \[\leadsto {z}^{2} \cdot \frac{1}{\color{blue}{y \cdot -2}} \]

      6. div-inv 28.5%

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

      7. unpow2 28.5%

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

      8. times-frac 30.1%

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

   9. Applied egg-rr 30.1%

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

   if -5.0000000000000002e-135 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.6%

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

   3. Taylor expanded in z around 0 42.3%

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

      1. associate-*r/ 42.3%

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

      2. rem-square-sqrt 42.3%

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

      3. unpow2 42.3%

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

      4. unpow2 42.3%

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

      5. hypot-undefine 42.3%

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

      6. unpow2 42.3%

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

      7. unpow2 42.3%

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

      8. hypot-undefine 42.3%

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

      9. unpow2 42.3%

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

      10. associate-*l/ 42.3%

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

      11. *-commutative 42.3%

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

      12. hypot-undefine 42.3%

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

      13. unpow2 42.3%

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

      14. unpow2 42.3%

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

      15. +-commutative 42.3%

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

      16. unpow2 42.3%

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

      17. unpow2 42.3%

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

      18. hypot-define 42.3%

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

   5. Simplified 42.3%

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

   6. Taylor expanded in x around 0 59.2%

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

      1. distribute-lft-out 59.2%

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

   8. Simplified 59.2%

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

Alternative 3: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.95 \cdot 10^{+145}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.95e+145)
    (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
    (* y_m 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.95e+145) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.95d+145) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.95e+145) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 1.95e+145:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.95e+145)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.95e+145)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.95e+145], N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.9499999999999999e145

   1. Initial program 72.7%

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

   if 1.9499999999999999e145 < y

   1. Initial program 4.4%

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

   3. Taylor expanded in y around inf 66.2%

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

      1. *-commutative 66.2%

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

   5. Simplified 66.2%

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

Alternative 4: 51.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{z}{y\_m} \cdot \frac{z}{-2}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 5e-29) (* (/ z y_m) (/ z -2.0)) (* y_m 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-29) {
		tmp = (z / y_m) * (z / -2.0);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 5d-29) then
        tmp = (z / y_m) * (z / (-2.0d0))
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 5e-29) {
		tmp = (z / y_m) * (z / -2.0);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 5e-29:
		tmp = (z / y_m) * (z / -2.0)
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 5e-29)
		tmp = Float64(Float64(z / y_m) * Float64(z / -2.0));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 5e-29)
		tmp = (z / y_m) * (z / -2.0);
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 5e-29], N[(N[(z / y$95$m), $MachinePrecision] * N[(z / -2.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999986e-29

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf 35.6%

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

      1. associate-*r/ 35.6%

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

      2. metadata-eval 35.6%

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

      3. distribute-lft-neg-in 35.6%

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

      4. *-commutative 35.6%

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

      5. distribute-neg-frac 35.6%

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

      6. associate-*r/ 35.6%

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

      7. distribute-rgt-neg-in 35.6%

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

      8. distribute-neg-frac 35.6%

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

      9. metadata-eval 35.6%

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

   5. Simplified 35.6%

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

      1. associate-*r/ 35.6%

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

      2. clear-num 35.6%

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

   7. Applied egg-rr 35.6%

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

      1. clear-num 35.6%

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

      2. associate-/l* 35.6%

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

      3. metadata-eval 35.6%

         \[\leadsto {z}^{2} \cdot \frac{\color{blue}{\frac{1}{-2}}}{y} \]

      4. associate-/r* 35.6%

         \[\leadsto {z}^{2} \cdot \color{blue}{\frac{1}{-2 \cdot y}} \]

      5. *-commutative 35.6%

         \[\leadsto {z}^{2} \cdot \frac{1}{\color{blue}{y \cdot -2}} \]

      6. div-inv 35.6%

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

      7. unpow2 35.6%

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

      8. times-frac 37.1%

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

   9. Applied egg-rr 37.1%

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

   if 4.99999999999999986e-29 < y

   1. Initial program 52.9%

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

   3. Taylor expanded in y around inf 54.5%

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

      1. *-commutative 54.5%

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

   5. Simplified 54.5%

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

Alternative 5: 51.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.2 \cdot 10^{-28}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 1.2e-28) (* z (* z (/ -0.5 y_m))) (* y_m 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.2e-28) {
		tmp = z * (z * (-0.5 / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.2d-28) then
        tmp = z * (z * ((-0.5d0) / y_m))
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.2e-28) {
		tmp = z * (z * (-0.5 / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 1.2e-28:
		tmp = z * (z * (-0.5 / y_m))
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.2e-28)
		tmp = Float64(z * Float64(z * Float64(-0.5 / y_m)));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.2e-28)
		tmp = z * (z * (-0.5 / y_m));
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.2e-28], N[(z * N[(z * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.2000000000000001e-28

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf 35.6%

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

      1. associate-*r/ 35.6%

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

      2. metadata-eval 35.6%

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

      3. distribute-lft-neg-in 35.6%

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

      4. *-commutative 35.6%

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

      5. distribute-neg-frac 35.6%

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

      6. associate-*r/ 35.6%

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

      7. distribute-rgt-neg-in 35.6%

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

      8. distribute-neg-frac 35.6%

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

      9. metadata-eval 35.6%

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

   5. Simplified 35.6%

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

      1. associate-*r/ 35.6%

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

      2. clear-num 35.6%

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

   7. Applied egg-rr 35.6%

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

      1. clear-num 35.6%

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

      2. associate-/l* 35.6%

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

      3. unpow2 35.6%

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

      4. metadata-eval 35.6%

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

      5. associate-/r* 35.6%

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

      6. *-commutative 35.6%

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

      7. associate-*l* 37.1%

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

      8. *-commutative 37.1%

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

      9. associate-/r* 37.1%

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

      10. metadata-eval 37.1%

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

   9. Applied egg-rr 37.1%

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

   if 1.2000000000000001e-28 < y

   1. Initial program 52.9%

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

   3. Taylor expanded in y around inf 54.5%

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

      1. *-commutative 54.5%

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

   5. Simplified 54.5%

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

Alternative 6: 35.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{+173}:\\ \;\;\;\;y\_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot z}{y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= x 7.6e+173) (* y_m 0.5) (* 0.5 (/ (* x z) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 7.6e+173) {
		tmp = y_m * 0.5;
	} else {
		tmp = 0.5 * ((x * z) / y_m);
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 7.6d+173) then
        tmp = y_m * 0.5d0
    else
        tmp = 0.5d0 * ((x * z) / y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 7.6e+173) {
		tmp = y_m * 0.5;
	} else {
		tmp = 0.5 * ((x * z) / y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 7.6e+173:
		tmp = y_m * 0.5
	else:
		tmp = 0.5 * ((x * z) / y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 7.6e+173)
		tmp = Float64(y_m * 0.5);
	else
		tmp = Float64(0.5 * Float64(Float64(x * z) / y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 7.6e+173)
		tmp = y_m * 0.5;
	else
		tmp = 0.5 * ((x * z) / y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 7.6e+173], N[(y$95$m * 0.5), $MachinePrecision], N[(0.5 * N[(N[(x * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.60000000000000022e173

   1. Initial program 64.7%

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

   3. Taylor expanded in y around inf 39.3%

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

      1. *-commutative 39.3%

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

   5. Simplified 39.3%

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

   if 7.60000000000000022e173 < x

   1. Initial program 67.1%

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

      1. remove-double-neg 67.1%

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

      2. distribute-lft-neg-out 67.1%

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

      3. distribute-frac-neg2 67.1%

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

      4. distribute-frac-neg 67.1%

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

      5. neg-mul-1 67.1%

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

      6. distribute-lft-neg-out 67.1%

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

      7. *-commutative 67.1%

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

      8. distribute-lft-neg-in 67.1%

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

      9. times-frac 67.1%

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

      10. metadata-eval 67.1%

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

      11. metadata-eval 67.1%

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

      12. associate--l+ 67.1%

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

      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. Step-by-step derivation

      1. prod-diff 67.1%

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

      2. fma-neg 67.1%

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

      3. difference-of-squares 67.5%

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

      4. fma-define 67.5%

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

      5. pow2 67.5%

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

   6. Applied egg-rr 67.5%

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

      1. add-sqr-sqrt 67.5%

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

      2. associate-/l* 67.5%

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

   8. Applied egg-rr 91.3%

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

   9. Taylor expanded in x around inf 79.3%

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

   10. Taylor expanded in z around inf 27.6%

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

Alternative 7: 40.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.7 \cdot 10^{-20}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 4.7e-20) (* 0.5 (* x (/ z y_m))) (* y_m 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4.7e-20) {
		tmp = 0.5 * (x * (z / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 4.7d-20) then
        tmp = 0.5d0 * (x * (z / y_m))
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4.7e-20) {
		tmp = 0.5 * (x * (z / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 4.7e-20:
		tmp = 0.5 * (x * (z / y_m))
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 4.7e-20)
		tmp = Float64(0.5 * Float64(x * Float64(z / y_m)));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 4.7e-20)
		tmp = 0.5 * (x * (z / y_m));
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 4.7e-20], N[(0.5 * N[(x * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.70000000000000015e-20

   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 70.0%

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

      10. metadata-eval 70.0%

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

      11. metadata-eval 70.0%

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

      12. associate--l+ 70.0%

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

      13. fma-define 72.2%

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

   3. Simplified 72.2%

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

      1. prod-diff 55.9%

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

      2. fma-neg 55.9%

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

      3. difference-of-squares 56.7%

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

      4. fma-define 58.8%

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

      5. pow2 58.8%

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

   6. Applied egg-rr 58.8%

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

      1. add-sqr-sqrt 42.2%

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

      2. associate-/l* 42.2%

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

   8. Applied egg-rr 63.1%

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

   9. Taylor expanded in x around inf 19.7%

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

   10. Taylor expanded in z around inf 15.9%

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

       1. associate-/l* 16.9%

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

   12. Simplified 16.9%

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

   if 4.70000000000000015e-20 < y

   1. Initial program 50.0%

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

   3. Taylor expanded in y around inf 57.6%

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

      1. *-commutative 57.6%

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

   5. Simplified 57.6%

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

Alternative 8: 34.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(y\_m \cdot 0.5\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* y_m 0.5)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m * 0.5);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (y_m * 0.5d0)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (y_m * 0.5);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (y_m * 0.5)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(y_m * 0.5))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (y_m * 0.5);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.9%

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

3. Taylor expanded in y around inf 37.2%

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

   1. *-commutative 37.2%

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

5. Simplified 37.2%

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

Alternative 9: 3.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x \cdot 0.5\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* x 0.5)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (x * 0.5);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x * 0.5d0)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (x * 0.5);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (x * 0.5)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(x * 0.5))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (x * 0.5);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.9%

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

   1. remove-double-neg 64.9%

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

   2. distribute-lft-neg-out 64.9%

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

   3. distribute-frac-neg2 64.9%

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

   4. distribute-frac-neg 64.9%

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

   5. neg-mul-1 64.9%

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

   6. distribute-lft-neg-out 64.9%

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

   7. *-commutative 64.9%

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

   8. distribute-lft-neg-in 64.9%

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

   9. times-frac 64.9%

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

   10. metadata-eval 64.9%

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

   11. metadata-eval 64.9%

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

   12. associate--l+ 64.9%

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

   13. fma-define 66.5%

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

3. Simplified 66.5%

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

   1. prod-diff 52.4%

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

   2. fma-neg 52.4%

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

   3. difference-of-squares 53.1%

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

   4. fma-define 55.0%

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

   5. pow2 55.0%

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

6. Applied egg-rr 55.0%

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

   1. add-sqr-sqrt 42.2%

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

   2. associate-/l* 42.3%

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

8. Applied egg-rr 66.4%

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

9. Taylor expanded in x around inf 18.0%

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

10. Taylor expanded in y around inf 2.9%

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

11. Final simplification 2.9%

    \[\leadsto x \cdot 0.5 \]
12. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024103 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

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

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