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

Percentage Accurate: 69.6% → 95.7%

Time: 11.8s

Alternatives: 9

Speedup: 0.7×

• 41.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.6% 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: 95.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 7.6 \cdot 10^{+86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m - z, y\_m + z, x \cdot x\right)}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m - \frac{z}{\frac{y\_m}{z}}\right) + x \cdot \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 7.6e+86)
    (/ (fma (- y_m z) (+ y_m z) (* x x)) (* y_m 2.0))
    (* 0.5 (+ (- y_m (/ z (/ y_m z))) (* x (/ x 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 (y_m <= 7.6e+86) {
		tmp = fma((y_m - z), (y_m + z), (x * x)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * ((y_m - (z / (y_m / z))) + (x * (x / 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 (y_m <= 7.6e+86)
		tmp = Float64(fma(Float64(y_m - z), Float64(y_m + z), Float64(x * x)) / Float64(y_m * 2.0));
	else
		tmp = Float64(0.5 * Float64(Float64(y_m - Float64(z / Float64(y_m / z))) + Float64(x * Float64(x / y_m))));
	end
	return Float64(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, 7.6e+86], N[(N[(N[(y$95$m - z), $MachinePrecision] * N[(y$95$m + z), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y$95$m - N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.59999999999999956e86

   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. associate--l+ 74.7%

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

      2. +-commutative 74.7%

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

      3. sqr-neg 74.7%

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

      4. difference-of-squares 76.3%

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

      5. fma-def 78.2%

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

      6. sub-neg 78.2%

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

      7. sub-neg 78.2%

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

      8. remove-double-neg 78.2%

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

   3. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   if 7.59999999999999956e86 < y

   1. Initial program 31.2%

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

   3. Taylor expanded in x around 0 31.2%

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

      1. fma-def 31.2%

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

      2. div-sub 31.2%

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

      3. unpow2 31.2%

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

      4. associate-/l* 61.3%

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

      5. *-inverses 61.3%

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

      6. /-rgt-identity 61.3%

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

      7. fma-def 61.3%

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

      8. distribute-lft-out 61.3%

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

   5. Simplified 61.3%

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

      1. unpow2 61.3%

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

      2. *-un-lft-identity 61.3%

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

      3. times-frac 84.3%

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

   7. Applied egg-rr 84.3%

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

      1. unpow2 84.3%

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

      2. *-un-lft-identity 84.3%

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

      3. times-frac 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. /-rgt-identity 99.9%

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

       2. clear-num 99.9%

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

       3. un-div-inv 99.9%

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

   11. Applied egg-rr 99.9%

       \[\leadsto 0.5 \cdot \left(\left(y - \color{blue}{\frac{z}{\frac{y}{z}}}\right) + \frac{x}{1} \cdot \frac{x}{y}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{+86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y - \frac{z}{\frac{y}{z}}\right) + x \cdot \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\frac{x}{y\_m}}{\frac{2}{x}}\\ t_1 := \frac{y\_m - z}{y\_m} \cdot \frac{y\_m + z}{2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 10^{+158}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y\_m}{x}}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+207}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (/ x y_m) (/ 2.0 x)))
        (t_1 (* (/ (- y_m z) y_m) (/ (+ y_m z) 2.0))))
   (*
    y_s
    (if (<= x 1.02e+42)
      t_1
      (if (<= x 1.5e+87)
        t_0
        (if (<= x 3.15e+131)
          t_1
          (if (<= x 1e+158)
            (/ (* x 0.5) (/ y_m x))
            (if (<= x 2.2e+207) t_1 t_0))))))))

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 = (x / y_m) / (2.0 / x);
	double t_1 = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	double tmp;
	if (x <= 1.02e+42) {
		tmp = t_1;
	} else if (x <= 1.5e+87) {
		tmp = t_0;
	} else if (x <= 3.15e+131) {
		tmp = t_1;
	} else if (x <= 1e+158) {
		tmp = (x * 0.5) / (y_m / x);
	} else if (x <= 2.2e+207) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / y_m) / (2.0d0 / x)
    t_1 = ((y_m - z) / y_m) * ((y_m + z) / 2.0d0)
    if (x <= 1.02d+42) then
        tmp = t_1
    else if (x <= 1.5d+87) then
        tmp = t_0
    else if (x <= 3.15d+131) then
        tmp = t_1
    else if (x <= 1d+158) then
        tmp = (x * 0.5d0) / (y_m / x)
    else if (x <= 2.2d+207) then
        tmp = t_1
    else
        tmp = t_0
    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 t_0 = (x / y_m) / (2.0 / x);
	double t_1 = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	double tmp;
	if (x <= 1.02e+42) {
		tmp = t_1;
	} else if (x <= 1.5e+87) {
		tmp = t_0;
	} else if (x <= 3.15e+131) {
		tmp = t_1;
	} else if (x <= 1e+158) {
		tmp = (x * 0.5) / (y_m / x);
	} else if (x <= 2.2e+207) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 = (x / y_m) / (2.0 / x)
	t_1 = ((y_m - z) / y_m) * ((y_m + z) / 2.0)
	tmp = 0
	if x <= 1.02e+42:
		tmp = t_1
	elif x <= 1.5e+87:
		tmp = t_0
	elif x <= 3.15e+131:
		tmp = t_1
	elif x <= 1e+158:
		tmp = (x * 0.5) / (y_m / x)
	elif x <= 2.2e+207:
		tmp = t_1
	else:
		tmp = t_0
	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 = Float64(Float64(x / y_m) / Float64(2.0 / x))
	t_1 = Float64(Float64(Float64(y_m - z) / y_m) * Float64(Float64(y_m + z) / 2.0))
	tmp = 0.0
	if (x <= 1.02e+42)
		tmp = t_1;
	elseif (x <= 1.5e+87)
		tmp = t_0;
	elseif (x <= 3.15e+131)
		tmp = t_1;
	elseif (x <= 1e+158)
		tmp = Float64(Float64(x * 0.5) / Float64(y_m / x));
	elseif (x <= 2.2e+207)
		tmp = t_1;
	else
		tmp = t_0;
	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 = (x / y_m) / (2.0 / x);
	t_1 = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	tmp = 0.0;
	if (x <= 1.02e+42)
		tmp = t_1;
	elseif (x <= 1.5e+87)
		tmp = t_0;
	elseif (x <= 3.15e+131)
		tmp = t_1;
	elseif (x <= 1e+158)
		tmp = (x * 0.5) / (y_m / x);
	elseif (x <= 2.2e+207)
		tmp = t_1;
	else
		tmp = t_0;
	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[(N[(x / y$95$m), $MachinePrecision] / N[(2.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(y$95$m + z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[x, 1.02e+42], t$95$1, If[LessEqual[x, 1.5e+87], t$95$0, If[LessEqual[x, 3.15e+131], t$95$1, If[LessEqual[x, 1e+158], N[(N[(x * 0.5), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2e+207], t$95$1, t$95$0]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.01999999999999996e42 or 1.4999999999999999e87 < x < 3.14999999999999998e131 or 9.99999999999999953e157 < x < 2.20000000000000009e207

   1. Initial program 65.2%

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

      1. associate--l+ 65.2%

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

      2. +-commutative 65.2%

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

      3. sqr-neg 65.2%

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

      4. difference-of-squares 68.3%

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

      5. fma-def 70.1%

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

      6. sub-neg 70.1%

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

      7. sub-neg 70.1%

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

      8. remove-double-neg 70.1%

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

   3. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 51.7%

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

      1. *-commutative 51.7%

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

      2. times-frac 74.4%

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

   7. Applied egg-rr 74.4%

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

   if 1.01999999999999996e42 < x < 1.4999999999999999e87 or 2.20000000000000009e207 < x

   1. Initial program 81.6%

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

   3. Taylor expanded in x around inf 78.9%

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

      1. unpow2 78.9%

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

      2. times-frac 87.6%

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

   5. Applied egg-rr 87.6%

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

      1. clear-num 87.5%

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

      2. un-div-inv 87.6%

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

   7. Applied egg-rr 87.6%

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

   if 3.14999999999999998e131 < x < 9.99999999999999953e157

   1. Initial program 68.5%

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

   3. Taylor expanded in x around inf 35.1%

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

      1. unpow2 35.1%

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

      2. times-frac 66.1%

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

   5. Applied egg-rr 66.1%

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

      1. div-inv 66.1%

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

      2. metadata-eval 66.1%

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

      3. *-commutative 66.1%

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

      4. remove-double-div 65.8%

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

      5. div-inv 65.8%

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

      6. times-frac 66.7%

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

      7. un-div-inv 66.7%

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

   7. Applied egg-rr 66.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.02 \cdot 10^{+42}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{2}{x}}\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{+131}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \mathbf{elif}\;x \leq 10^{+158}:\\ \;\;\;\;\frac{x \cdot 0.5}{\frac{y}{x}}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{+207}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{2}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 9.5 \cdot 10^{+147}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m - z}{y\_m} \cdot \frac{y\_m + z}{2}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 9.5e+147)
    (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
    (* (/ (- y_m z) y_m) (/ (+ y_m z) 2.0)))))

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 <= 9.5e+147) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	}
	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 <= 9.5d+147) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0d0)
    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 <= 9.5e+147) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	}
	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 <= 9.5e+147:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0)
	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 <= 9.5e+147)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(Float64(Float64(y_m - z) / y_m) * Float64(Float64(y_m + z) / 2.0));
	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 <= 9.5e+147)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	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, 9.5e+147], 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[(N[(N[(y$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(y$95$m + z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.4999999999999996e147

   1. Initial program 74.5%

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

   if 9.4999999999999996e147 < y

   1. Initial program 16.8%

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

      1. associate--l+ 16.8%

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

      2. +-commutative 16.8%

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

      3. sqr-neg 16.8%

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

      4. difference-of-squares 27.5%

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

      5. fma-def 30.6%

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

      6. sub-neg 30.6%

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

      7. sub-neg 30.6%

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

      8. remove-double-neg 30.6%

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

   3. Simplified 30.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 30.6%

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

      1. *-commutative 30.6%

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

      2. times-frac 81.7%

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

   7. Applied egg-rr 81.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{+147}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 4.1 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m - \frac{z}{\frac{y\_m}{z}}\right) + x \cdot \frac{x}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m - z}{y\_m} \cdot \frac{y\_m + z}{2}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 4.1e+149)
    (* 0.5 (+ (- y_m (/ z (/ y_m z))) (* x (/ x y_m))))
    (* (/ (- y_m z) y_m) (/ (+ y_m z) 2.0)))))

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 (z <= 4.1e+149) {
		tmp = 0.5 * ((y_m - (z / (y_m / z))) + (x * (x / y_m)));
	} else {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	}
	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 (z <= 4.1d+149) then
        tmp = 0.5d0 * ((y_m - (z / (y_m / z))) + (x * (x / y_m)))
    else
        tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0d0)
    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 (z <= 4.1e+149) {
		tmp = 0.5 * ((y_m - (z / (y_m / z))) + (x * (x / y_m)));
	} else {
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	}
	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 z <= 4.1e+149:
		tmp = 0.5 * ((y_m - (z / (y_m / z))) + (x * (x / y_m)))
	else:
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0)
	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 (z <= 4.1e+149)
		tmp = Float64(0.5 * Float64(Float64(y_m - Float64(z / Float64(y_m / z))) + Float64(x * Float64(x / y_m))));
	else
		tmp = Float64(Float64(Float64(y_m - z) / y_m) * Float64(Float64(y_m + z) / 2.0));
	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 (z <= 4.1e+149)
		tmp = 0.5 * ((y_m - (z / (y_m / z))) + (x * (x / y_m)));
	else
		tmp = ((y_m - z) / y_m) * ((y_m + z) / 2.0);
	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[z, 4.1e+149], N[(0.5 * N[(N[(y$95$m - N[(z / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(y$95$m + z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.0999999999999996e149

   1. Initial program 69.0%

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

   3. Taylor expanded in x around 0 64.1%

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

      1. fma-def 64.1%

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

      2. div-sub 64.1%

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

      3. unpow2 64.1%

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

      4. associate-/l* 79.6%

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

      5. *-inverses 79.6%

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

      6. /-rgt-identity 79.6%

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

      7. fma-def 79.6%

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

      8. distribute-lft-out 79.6%

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

   5. Simplified 79.6%

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

      1. unpow2 79.6%

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

      2. *-un-lft-identity 79.6%

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

      3. times-frac 87.6%

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

   7. Applied egg-rr 87.6%

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

      1. unpow2 87.5%

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

      2. *-un-lft-identity 87.5%

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

      3. times-frac 91.4%

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

   9. Applied egg-rr 91.4%

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

       1. /-rgt-identity 91.4%

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

       2. clear-num 91.4%

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

       3. un-div-inv 91.4%

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

   11. Applied egg-rr 91.4%

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

   if 4.0999999999999996e149 < z

   1. Initial program 57.0%

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

      1. associate--l+ 57.0%

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

      2. +-commutative 57.0%

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

      3. sqr-neg 57.0%

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

      4. difference-of-squares 63.5%

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

      5. fma-def 71.6%

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

      6. sub-neg 71.6%

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

      7. sub-neg 71.6%

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

      8. remove-double-neg 71.6%

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

   3. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 74.5%

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

      1. *-commutative 74.5%

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

      2. times-frac 91.9%

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

   7. Applied egg-rr 91.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.1 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot \left(\left(y - \frac{z}{\frac{y}{z}}\right) + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot \frac{y + z}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.05 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{x}{y\_m}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{1}{y\_m - z}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.05e+53)
    (/ (/ x y_m) (/ 2.0 x))
    (/ 1.0 (* 2.0 (/ 1.0 (- y_m z)))))))

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 <= 1.05e+53) {
		tmp = (x / y_m) / (2.0 / x);
	} else {
		tmp = 1.0 / (2.0 * (1.0 / (y_m - z)));
	}
	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 <= 1.05d+53) then
        tmp = (x / y_m) / (2.0d0 / x)
    else
        tmp = 1.0d0 / (2.0d0 * (1.0d0 / (y_m - z)))
    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 <= 1.05e+53) {
		tmp = (x / y_m) / (2.0 / x);
	} else {
		tmp = 1.0 / (2.0 * (1.0 / (y_m - z)));
	}
	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 <= 1.05e+53:
		tmp = (x / y_m) / (2.0 / x)
	else:
		tmp = 1.0 / (2.0 * (1.0 / (y_m - z)))
	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 <= 1.05e+53)
		tmp = Float64(Float64(x / y_m) / Float64(2.0 / x));
	else
		tmp = Float64(1.0 / Float64(2.0 * Float64(1.0 / Float64(y_m - z))));
	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 <= 1.05e+53)
		tmp = (x / y_m) / (2.0 / x);
	else
		tmp = 1.0 / (2.0 * (1.0 / (y_m - z)));
	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, 1.05e+53], N[(N[(x / y$95$m), $MachinePrecision] / N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(2.0 * N[(1.0 / N[(y$95$m - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.0500000000000001e53

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 35.9%

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

      1. unpow2 35.9%

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

      2. times-frac 37.2%

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

   5. Applied egg-rr 37.2%

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

      1. clear-num 37.2%

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

      2. un-div-inv 37.2%

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

   7. Applied egg-rr 37.2%

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

   if 1.0500000000000001e53 < y

   1. Initial program 39.4%

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

      1. associate--l+ 39.4%

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

      2. +-commutative 39.4%

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

      3. sqr-neg 39.4%

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

      4. difference-of-squares 46.2%

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

      5. fma-def 50.2%

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

      6. sub-neg 50.2%

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

      7. sub-neg 50.2%

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

      8. remove-double-neg 50.2%

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

   3. Simplified 50.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 43.9%

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

      1. clear-num 43.8%

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

      2. inv-pow 43.8%

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

      3. *-commutative 43.8%

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

      4. *-un-lft-identity 43.8%

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

      5. times-frac 43.8%

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

      6. metadata-eval 43.8%

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

   7. Applied egg-rr 43.8%

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

      1. unpow-1 43.8%

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

      2. associate-/r* 76.5%

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

      3. +-commutative 76.5%

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

   9. Simplified 76.5%

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

   10. Taylor expanded in y around inf 55.6%

       \[\leadsto \frac{1}{2 \cdot \frac{\color{blue}{1}}{y - z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{+53}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2 \cdot \frac{1}{y - z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.5 \cdot 10^{+53}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 1.5e+53) (* x (* x (/ 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 <= 1.5e+53) {
		tmp = x * (x * (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 <= 1.5d+53) then
        tmp = x * (x * (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 <= 1.5e+53) {
		tmp = x * (x * (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 <= 1.5e+53:
		tmp = x * (x * (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 <= 1.5e+53)
		tmp = Float64(x * Float64(x * 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 <= 1.5e+53)
		tmp = x * (x * (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, 1.5e+53], N[(x * N[(x * 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 < 1.49999999999999999e53

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 35.9%

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

      1. div-inv 35.9%

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

      2. unpow2 35.9%

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

      3. metadata-eval 35.9%

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

      4. div-inv 35.9%

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

      5. clear-num 35.9%

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

      6. associate-*l* 37.2%

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

   5. Applied egg-rr 37.2%

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

   if 1.49999999999999999e53 < y

   1. Initial program 39.4%

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

   3. Taylor expanded in y around inf 55.0%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{+53}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 5.5e+52) (/ (* x (* x 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 <= 5.5e+52) {
		tmp = (x * (x * 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 <= 5.5d+52) then
        tmp = (x * (x * 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 <= 5.5e+52) {
		tmp = (x * (x * 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 <= 5.5e+52:
		tmp = (x * (x * 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 <= 5.5e+52)
		tmp = Float64(Float64(x * Float64(x * 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 <= 5.5e+52)
		tmp = (x * (x * 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, 5.5e+52], N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.49999999999999996e52

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 35.9%

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

      1. div-inv 35.9%

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

      2. unpow2 35.9%

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

      3. metadata-eval 35.9%

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

      4. div-inv 35.9%

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

      5. clear-num 35.9%

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

      6. associate-*l* 37.2%

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

   5. Applied egg-rr 37.2%

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

      1. /-rgt-identity 37.2%

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

      2. associate-*r/ 37.2%

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

      3. frac-times 35.9%

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

      4. *-un-lft-identity 35.9%

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

   7. Applied egg-rr 35.9%

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

   if 5.49999999999999996e52 < y

   1. Initial program 39.4%

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

   3. Taylor expanded in y around inf 55.0%

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

4. Final simplification 39.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{x \cdot \left(x \cdot 0.5\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{x}{y\_m}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 8e+52) (/ (/ x y_m) (/ 2.0 x)) (* 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 <= 8e+52) {
		tmp = (x / y_m) / (2.0 / x);
	} 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 <= 8d+52) then
        tmp = (x / y_m) / (2.0d0 / x)
    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 <= 8e+52) {
		tmp = (x / y_m) / (2.0 / x);
	} 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 <= 8e+52:
		tmp = (x / y_m) / (2.0 / x)
	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 <= 8e+52)
		tmp = Float64(Float64(x / y_m) / Float64(2.0 / x));
	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 <= 8e+52)
		tmp = (x / y_m) / (2.0 / x);
	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, 8e+52], N[(N[(x / y$95$m), $MachinePrecision] / N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 7.9999999999999999e52

   1. Initial program 74.0%

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

   3. Taylor expanded in x around inf 35.9%

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

      1. unpow2 35.9%

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

      2. times-frac 37.2%

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

   5. Applied egg-rr 37.2%

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

      1. clear-num 37.2%

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

      2. un-div-inv 37.2%

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

   7. Applied egg-rr 37.2%

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

   if 7.9999999999999999e52 < y

   1. Initial program 39.4%

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

   3. Taylor expanded in y around inf 55.0%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{2}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.9% 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 1 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 67.2%

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

3. Taylor expanded in y around inf 31.7%

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

4. Final simplification 31.7%

   \[\leadsto y \cdot 0.5 \]
5. Add Preprocessing

Developer target: 99.8% 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 2024040 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

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

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