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

Percentage Accurate: 69.9% → 90.7%

Time: 11.7s

Alternatives: 10

Speedup: 0.7×

• 41.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.9% 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: 90.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.3 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y_m - z, y_m + z, x \cdot x\right)}{y_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m + z}{\frac{y_m}{\left(y_m - z\right) \cdot 0.5}}\\ \end{array} \end{array} \]

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 1.3e+126)
    (/ (fma (- y_m z) (+ y_m z) (* x x)) (* y_m 2.0))
    (/ (+ y_m z) (/ y_m (* (- y_m z) 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 <= 1.3e+126) {
		tmp = fma((y_m - z), (y_m + z), (x * x)) / (y_m * 2.0);
	} else {
		tmp = (y_m + z) / (y_m / ((y_m - z) * 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 <= 1.3e+126)
		tmp = Float64(fma(Float64(y_m - z), Float64(y_m + z), Float64(x * x)) / Float64(y_m * 2.0));
	else
		tmp = Float64(Float64(y_m + z) / Float64(y_m / Float64(Float64(y_m - z) * 0.5)));
	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, 1.3e+126], 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[(N[(y$95$m + z), $MachinePrecision] / N[(y$95$m / N[(N[(y$95$m - z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.3e126

   1. Initial program 77.6%

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

      1. associate--l+ 77.6%

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

      2. +-commutative 77.6%

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

      3. sqr-neg 77.6%

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

      4. difference-of-squares 78.3%

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

      5. fma-def 80.1%

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

      6. sub-neg 80.1%

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

      7. sub-neg 80.1%

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

      8. remove-double-neg 80.1%

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

   3. Simplified 80.1%

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

   if 1.3e126 < y

   1. Initial program 29.0%

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

      1. associate--l+ 29.0%

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

      2. +-commutative 29.0%

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

      3. sqr-neg 29.0%

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

      4. difference-of-squares 29.7%

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

      5. fma-def 29.7%

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

      6. sub-neg 29.7%

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

      7. sub-neg 29.7%

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

      8. remove-double-neg 29.7%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in x around 0 29.7%

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

      1. associate-/l* 87.2%

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

      2. div-inv 87.1%

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

      3. associate-/l* 87.1%

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

      4. div-inv 87.1%

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

      5. metadata-eval 87.1%

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

   7. Applied egg-rr 87.1%

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

      1. associate-*r/ 87.2%

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

      2. *-rgt-identity 87.2%

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

      3. *-commutative 87.2%

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

   9. Simplified 87.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{+126}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y - z, y + z, x \cdot x\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + z}{\frac{y}{\left(y - z\right) \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot \frac{-0.5}{y_m}\right)\\ t_1 := \frac{x}{y_m} \cdot \frac{x}{2}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.35 \cdot 10^{-276}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y_m \leq 5.2 \cdot 10^{-219}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 2.2 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y_m \leq 1.18 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 6.2 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y_m \leq 49000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 7.8 \cdot 10^{+20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y_m \leq 3.15 \cdot 10^{+32} \lor \neg \left(y_m \leq 6.8 \cdot 10^{+68}\right):\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* z (* z (/ -0.5 y_m)))) (t_1 (* (/ x y_m) (/ x 2.0))))
   (*
    y_s
    (if (<= y_m 1.35e-276)
      t_1
      (if (<= y_m 5.2e-219)
        t_0
        (if (<= y_m 2.2e-181)
          t_1
          (if (<= y_m 1.18e-153)
            t_0
            (if (<= y_m 6.2e-60)
              t_1
              (if (<= y_m 49000000.0)
                t_0
                (if (<= y_m 7.8e+20)
                  t_1
                  (if (or (<= y_m 3.15e+32) (not (<= y_m 6.8e+68)))
                    (* y_m 0.5)
                    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 = z * (z * (-0.5 / y_m));
	double t_1 = (x / y_m) * (x / 2.0);
	double tmp;
	if (y_m <= 1.35e-276) {
		tmp = t_1;
	} else if (y_m <= 5.2e-219) {
		tmp = t_0;
	} else if (y_m <= 2.2e-181) {
		tmp = t_1;
	} else if (y_m <= 1.18e-153) {
		tmp = t_0;
	} else if (y_m <= 6.2e-60) {
		tmp = t_1;
	} else if (y_m <= 49000000.0) {
		tmp = t_0;
	} else if (y_m <= 7.8e+20) {
		tmp = t_1;
	} else if ((y_m <= 3.15e+32) || !(y_m <= 6.8e+68)) {
		tmp = y_m * 0.5;
	} 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 = z * (z * ((-0.5d0) / y_m))
    t_1 = (x / y_m) * (x / 2.0d0)
    if (y_m <= 1.35d-276) then
        tmp = t_1
    else if (y_m <= 5.2d-219) then
        tmp = t_0
    else if (y_m <= 2.2d-181) then
        tmp = t_1
    else if (y_m <= 1.18d-153) then
        tmp = t_0
    else if (y_m <= 6.2d-60) then
        tmp = t_1
    else if (y_m <= 49000000.0d0) then
        tmp = t_0
    else if (y_m <= 7.8d+20) then
        tmp = t_1
    else if ((y_m <= 3.15d+32) .or. (.not. (y_m <= 6.8d+68))) then
        tmp = y_m * 0.5d0
    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 = z * (z * (-0.5 / y_m));
	double t_1 = (x / y_m) * (x / 2.0);
	double tmp;
	if (y_m <= 1.35e-276) {
		tmp = t_1;
	} else if (y_m <= 5.2e-219) {
		tmp = t_0;
	} else if (y_m <= 2.2e-181) {
		tmp = t_1;
	} else if (y_m <= 1.18e-153) {
		tmp = t_0;
	} else if (y_m <= 6.2e-60) {
		tmp = t_1;
	} else if (y_m <= 49000000.0) {
		tmp = t_0;
	} else if (y_m <= 7.8e+20) {
		tmp = t_1;
	} else if ((y_m <= 3.15e+32) || !(y_m <= 6.8e+68)) {
		tmp = y_m * 0.5;
	} 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 = z * (z * (-0.5 / y_m))
	t_1 = (x / y_m) * (x / 2.0)
	tmp = 0
	if y_m <= 1.35e-276:
		tmp = t_1
	elif y_m <= 5.2e-219:
		tmp = t_0
	elif y_m <= 2.2e-181:
		tmp = t_1
	elif y_m <= 1.18e-153:
		tmp = t_0
	elif y_m <= 6.2e-60:
		tmp = t_1
	elif y_m <= 49000000.0:
		tmp = t_0
	elif y_m <= 7.8e+20:
		tmp = t_1
	elif (y_m <= 3.15e+32) or not (y_m <= 6.8e+68):
		tmp = y_m * 0.5
	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(z * Float64(z * Float64(-0.5 / y_m)))
	t_1 = Float64(Float64(x / y_m) * Float64(x / 2.0))
	tmp = 0.0
	if (y_m <= 1.35e-276)
		tmp = t_1;
	elseif (y_m <= 5.2e-219)
		tmp = t_0;
	elseif (y_m <= 2.2e-181)
		tmp = t_1;
	elseif (y_m <= 1.18e-153)
		tmp = t_0;
	elseif (y_m <= 6.2e-60)
		tmp = t_1;
	elseif (y_m <= 49000000.0)
		tmp = t_0;
	elseif (y_m <= 7.8e+20)
		tmp = t_1;
	elseif ((y_m <= 3.15e+32) || !(y_m <= 6.8e+68))
		tmp = Float64(y_m * 0.5);
	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 = z * (z * (-0.5 / y_m));
	t_1 = (x / y_m) * (x / 2.0);
	tmp = 0.0;
	if (y_m <= 1.35e-276)
		tmp = t_1;
	elseif (y_m <= 5.2e-219)
		tmp = t_0;
	elseif (y_m <= 2.2e-181)
		tmp = t_1;
	elseif (y_m <= 1.18e-153)
		tmp = t_0;
	elseif (y_m <= 6.2e-60)
		tmp = t_1;
	elseif (y_m <= 49000000.0)
		tmp = t_0;
	elseif (y_m <= 7.8e+20)
		tmp = t_1;
	elseif ((y_m <= 3.15e+32) || ~((y_m <= 6.8e+68)))
		tmp = y_m * 0.5;
	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[(z * N[(z * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 1.35e-276], t$95$1, If[LessEqual[y$95$m, 5.2e-219], t$95$0, If[LessEqual[y$95$m, 2.2e-181], t$95$1, If[LessEqual[y$95$m, 1.18e-153], t$95$0, If[LessEqual[y$95$m, 6.2e-60], t$95$1, If[LessEqual[y$95$m, 49000000.0], t$95$0, If[LessEqual[y$95$m, 7.8e+20], t$95$1, If[Or[LessEqual[y$95$m, 3.15e+32], N[Not[LessEqual[y$95$m, 6.8e+68]], $MachinePrecision]], N[(y$95$m * 0.5), $MachinePrecision], t$95$0]]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.34999999999999993e-276 or 5.20000000000000004e-219 < y < 2.19999999999999997e-181 or 1.1800000000000001e-153 < y < 6.19999999999999976e-60 or 4.9e7 < y < 7.8e20

   1. Initial program 73.1%

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

   3. Taylor expanded in x around inf 37.7%

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

      1. unpow2 37.7%

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

      2. times-frac 38.7%

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

   5. Applied egg-rr 38.7%

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

   if 1.34999999999999993e-276 < y < 5.20000000000000004e-219 or 2.19999999999999997e-181 < y < 1.1800000000000001e-153 or 6.19999999999999976e-60 < y < 4.9e7 or 3.1500000000000001e32 < y < 6.8000000000000003e68

   1. Initial program 88.4%

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

   3. Taylor expanded in z around inf 61.2%

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

      1. associate-*r/ 61.2%

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

      2. associate-/l* 61.3%

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

   5. Simplified 61.3%

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

      1. associate-/r/ 61.1%

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

      2. unpow2 61.1%

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

      3. associate-*r* 61.1%

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

   7. Applied egg-rr 61.1%

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

   if 7.8e20 < y < 3.1500000000000001e32 or 6.8000000000000003e68 < y

   1. Initial program 48.6%

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

   3. Taylor expanded in y around inf 67.6%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-276}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-219}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-181}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-153}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 49000000:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 3.15 \cdot 10^{+32} \lor \neg \left(y \leq 6.8 \cdot 10^{+68}\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{-0.5}{\frac{\frac{y_m}{z}}{z}}\\ t_1 := z \cdot \left(z \cdot \frac{-0.5}{y_m}\right)\\ t_2 := \frac{x}{y_m} \cdot \frac{x}{2}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.25 \cdot 10^{-276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y_m \leq 1.4 \cdot 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 2.3 \cdot 10^{-180}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y_m \leq 6.8 \cdot 10^{-152}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y_m \leq 36000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y_m \leq 8.5 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y_m \leq 6.4 \cdot 10^{+33} \lor \neg \left(y_m \leq 1.25 \cdot 10^{+67}\right):\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ -0.5 (/ (/ y_m z) z)))
        (t_1 (* z (* z (/ -0.5 y_m))))
        (t_2 (* (/ x y_m) (/ x 2.0))))
   (*
    y_s
    (if (<= y_m 1.25e-276)
      t_2
      (if (<= y_m 1.4e-217)
        t_0
        (if (<= y_m 2.3e-180)
          t_2
          (if (<= y_m 6.8e-152)
            t_0
            (if (<= y_m 8.5e-61)
              t_2
              (if (<= y_m 36000000.0)
                t_1
                (if (<= y_m 8.5e+20)
                  t_2
                  (if (or (<= y_m 6.4e+33) (not (<= y_m 1.25e+67)))
                    (* y_m 0.5)
                    t_1)))))))))))

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 = -0.5 / ((y_m / z) / z);
	double t_1 = z * (z * (-0.5 / y_m));
	double t_2 = (x / y_m) * (x / 2.0);
	double tmp;
	if (y_m <= 1.25e-276) {
		tmp = t_2;
	} else if (y_m <= 1.4e-217) {
		tmp = t_0;
	} else if (y_m <= 2.3e-180) {
		tmp = t_2;
	} else if (y_m <= 6.8e-152) {
		tmp = t_0;
	} else if (y_m <= 8.5e-61) {
		tmp = t_2;
	} else if (y_m <= 36000000.0) {
		tmp = t_1;
	} else if (y_m <= 8.5e+20) {
		tmp = t_2;
	} else if ((y_m <= 6.4e+33) || !(y_m <= 1.25e+67)) {
		tmp = y_m * 0.5;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (-0.5d0) / ((y_m / z) / z)
    t_1 = z * (z * ((-0.5d0) / y_m))
    t_2 = (x / y_m) * (x / 2.0d0)
    if (y_m <= 1.25d-276) then
        tmp = t_2
    else if (y_m <= 1.4d-217) then
        tmp = t_0
    else if (y_m <= 2.3d-180) then
        tmp = t_2
    else if (y_m <= 6.8d-152) then
        tmp = t_0
    else if (y_m <= 8.5d-61) then
        tmp = t_2
    else if (y_m <= 36000000.0d0) then
        tmp = t_1
    else if (y_m <= 8.5d+20) then
        tmp = t_2
    else if ((y_m <= 6.4d+33) .or. (.not. (y_m <= 1.25d+67))) then
        tmp = y_m * 0.5d0
    else
        tmp = t_1
    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 = -0.5 / ((y_m / z) / z);
	double t_1 = z * (z * (-0.5 / y_m));
	double t_2 = (x / y_m) * (x / 2.0);
	double tmp;
	if (y_m <= 1.25e-276) {
		tmp = t_2;
	} else if (y_m <= 1.4e-217) {
		tmp = t_0;
	} else if (y_m <= 2.3e-180) {
		tmp = t_2;
	} else if (y_m <= 6.8e-152) {
		tmp = t_0;
	} else if (y_m <= 8.5e-61) {
		tmp = t_2;
	} else if (y_m <= 36000000.0) {
		tmp = t_1;
	} else if (y_m <= 8.5e+20) {
		tmp = t_2;
	} else if ((y_m <= 6.4e+33) || !(y_m <= 1.25e+67)) {
		tmp = y_m * 0.5;
	} else {
		tmp = t_1;
	}
	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 = -0.5 / ((y_m / z) / z)
	t_1 = z * (z * (-0.5 / y_m))
	t_2 = (x / y_m) * (x / 2.0)
	tmp = 0
	if y_m <= 1.25e-276:
		tmp = t_2
	elif y_m <= 1.4e-217:
		tmp = t_0
	elif y_m <= 2.3e-180:
		tmp = t_2
	elif y_m <= 6.8e-152:
		tmp = t_0
	elif y_m <= 8.5e-61:
		tmp = t_2
	elif y_m <= 36000000.0:
		tmp = t_1
	elif y_m <= 8.5e+20:
		tmp = t_2
	elif (y_m <= 6.4e+33) or not (y_m <= 1.25e+67):
		tmp = y_m * 0.5
	else:
		tmp = t_1
	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(-0.5 / Float64(Float64(y_m / z) / z))
	t_1 = Float64(z * Float64(z * Float64(-0.5 / y_m)))
	t_2 = Float64(Float64(x / y_m) * Float64(x / 2.0))
	tmp = 0.0
	if (y_m <= 1.25e-276)
		tmp = t_2;
	elseif (y_m <= 1.4e-217)
		tmp = t_0;
	elseif (y_m <= 2.3e-180)
		tmp = t_2;
	elseif (y_m <= 6.8e-152)
		tmp = t_0;
	elseif (y_m <= 8.5e-61)
		tmp = t_2;
	elseif (y_m <= 36000000.0)
		tmp = t_1;
	elseif (y_m <= 8.5e+20)
		tmp = t_2;
	elseif ((y_m <= 6.4e+33) || !(y_m <= 1.25e+67))
		tmp = Float64(y_m * 0.5);
	else
		tmp = t_1;
	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 = -0.5 / ((y_m / z) / z);
	t_1 = z * (z * (-0.5 / y_m));
	t_2 = (x / y_m) * (x / 2.0);
	tmp = 0.0;
	if (y_m <= 1.25e-276)
		tmp = t_2;
	elseif (y_m <= 1.4e-217)
		tmp = t_0;
	elseif (y_m <= 2.3e-180)
		tmp = t_2;
	elseif (y_m <= 6.8e-152)
		tmp = t_0;
	elseif (y_m <= 8.5e-61)
		tmp = t_2;
	elseif (y_m <= 36000000.0)
		tmp = t_1;
	elseif (y_m <= 8.5e+20)
		tmp = t_2;
	elseif ((y_m <= 6.4e+33) || ~((y_m <= 1.25e+67)))
		tmp = y_m * 0.5;
	else
		tmp = t_1;
	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[(-0.5 / N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(z * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 1.25e-276], t$95$2, If[LessEqual[y$95$m, 1.4e-217], t$95$0, If[LessEqual[y$95$m, 2.3e-180], t$95$2, If[LessEqual[y$95$m, 6.8e-152], t$95$0, If[LessEqual[y$95$m, 8.5e-61], t$95$2, If[LessEqual[y$95$m, 36000000.0], t$95$1, If[LessEqual[y$95$m, 8.5e+20], t$95$2, If[Or[LessEqual[y$95$m, 6.4e+33], N[Not[LessEqual[y$95$m, 1.25e+67]], $MachinePrecision]], N[(y$95$m * 0.5), $MachinePrecision], t$95$1]]]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.24999999999999992e-276 or 1.4e-217 < y < 2.29999999999999996e-180 or 6.79999999999999968e-152 < y < 8.50000000000000016e-61 or 3.6e7 < y < 8.5e20

   1. Initial program 73.1%

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

   3. Taylor expanded in x around inf 37.7%

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

      1. unpow2 37.7%

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

      2. times-frac 38.7%

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

   5. Applied egg-rr 38.7%

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

   if 1.24999999999999992e-276 < y < 1.4e-217 or 2.29999999999999996e-180 < y < 6.79999999999999968e-152

   1. Initial program 88.1%

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

   3. Taylor expanded in z around inf 64.4%

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

      1. associate-*r/ 64.4%

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

      2. associate-/l* 64.6%

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

   5. Simplified 64.6%

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

      1. *-un-lft-identity 64.6%

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

      2. unpow2 64.6%

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

      3. times-frac 64.5%

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

   7. Applied egg-rr 64.5%

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

      1. associate-*l/ 64.6%

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

      2. *-lft-identity 64.6%

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

   9. Simplified 64.6%

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

   if 8.50000000000000016e-61 < y < 3.6e7 or 6.40000000000000034e33 < y < 1.24999999999999994e67

   1. Initial program 88.8%

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

   3. Taylor expanded in z around inf 58.2%

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

      1. associate-*r/ 58.2%

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

      2. associate-/l* 58.1%

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

   5. Simplified 58.1%

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

      1. associate-/r/ 57.9%

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

      2. unpow2 57.9%

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

      3. associate-*r* 58.0%

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

   7. Applied egg-rr 58.0%

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

   if 8.5e20 < y < 6.40000000000000034e33 or 1.24999999999999994e67 < y

   1. Initial program 48.6%

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

   3. Taylor expanded in y around inf 67.6%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-276}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-217}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-152}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 36000000:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+33} \lor \neg \left(y \leq 1.25 \cdot 10^{+67}\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{-0.5}{\frac{\frac{y_m}{z}}{z}}\\ t_1 := z \cdot \left(z \cdot \frac{-0.5}{y_m}\right)\\ t_2 := \frac{x}{y_m} \cdot \frac{x}{2}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.15 \cdot 10^{-276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y_m \leq 3.9 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 1.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{y_m \cdot \frac{2}{x}}\\ \mathbf{elif}\;y_m \leq 1.42 \cdot 10^{-151}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 1.05 \cdot 10^{-60}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y_m \leq 215000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y_m \leq 1.5 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y_m \leq 6.2 \cdot 10^{+33} \lor \neg \left(y_m \leq 6.6 \cdot 10^{+69}\right):\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ -0.5 (/ (/ y_m z) z)))
        (t_1 (* z (* z (/ -0.5 y_m))))
        (t_2 (* (/ x y_m) (/ x 2.0))))
   (*
    y_s
    (if (<= y_m 1.15e-276)
      t_2
      (if (<= y_m 3.9e-218)
        t_0
        (if (<= y_m 1.4e-180)
          (/ x (* y_m (/ 2.0 x)))
          (if (<= y_m 1.42e-151)
            t_0
            (if (<= y_m 1.05e-60)
              t_2
              (if (<= y_m 215000000.0)
                t_1
                (if (<= y_m 1.5e+21)
                  t_2
                  (if (or (<= y_m 6.2e+33) (not (<= y_m 6.6e+69)))
                    (* y_m 0.5)
                    t_1)))))))))))

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 = -0.5 / ((y_m / z) / z);
	double t_1 = z * (z * (-0.5 / y_m));
	double t_2 = (x / y_m) * (x / 2.0);
	double tmp;
	if (y_m <= 1.15e-276) {
		tmp = t_2;
	} else if (y_m <= 3.9e-218) {
		tmp = t_0;
	} else if (y_m <= 1.4e-180) {
		tmp = x / (y_m * (2.0 / x));
	} else if (y_m <= 1.42e-151) {
		tmp = t_0;
	} else if (y_m <= 1.05e-60) {
		tmp = t_2;
	} else if (y_m <= 215000000.0) {
		tmp = t_1;
	} else if (y_m <= 1.5e+21) {
		tmp = t_2;
	} else if ((y_m <= 6.2e+33) || !(y_m <= 6.6e+69)) {
		tmp = y_m * 0.5;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (-0.5d0) / ((y_m / z) / z)
    t_1 = z * (z * ((-0.5d0) / y_m))
    t_2 = (x / y_m) * (x / 2.0d0)
    if (y_m <= 1.15d-276) then
        tmp = t_2
    else if (y_m <= 3.9d-218) then
        tmp = t_0
    else if (y_m <= 1.4d-180) then
        tmp = x / (y_m * (2.0d0 / x))
    else if (y_m <= 1.42d-151) then
        tmp = t_0
    else if (y_m <= 1.05d-60) then
        tmp = t_2
    else if (y_m <= 215000000.0d0) then
        tmp = t_1
    else if (y_m <= 1.5d+21) then
        tmp = t_2
    else if ((y_m <= 6.2d+33) .or. (.not. (y_m <= 6.6d+69))) then
        tmp = y_m * 0.5d0
    else
        tmp = t_1
    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 = -0.5 / ((y_m / z) / z);
	double t_1 = z * (z * (-0.5 / y_m));
	double t_2 = (x / y_m) * (x / 2.0);
	double tmp;
	if (y_m <= 1.15e-276) {
		tmp = t_2;
	} else if (y_m <= 3.9e-218) {
		tmp = t_0;
	} else if (y_m <= 1.4e-180) {
		tmp = x / (y_m * (2.0 / x));
	} else if (y_m <= 1.42e-151) {
		tmp = t_0;
	} else if (y_m <= 1.05e-60) {
		tmp = t_2;
	} else if (y_m <= 215000000.0) {
		tmp = t_1;
	} else if (y_m <= 1.5e+21) {
		tmp = t_2;
	} else if ((y_m <= 6.2e+33) || !(y_m <= 6.6e+69)) {
		tmp = y_m * 0.5;
	} else {
		tmp = t_1;
	}
	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 = -0.5 / ((y_m / z) / z)
	t_1 = z * (z * (-0.5 / y_m))
	t_2 = (x / y_m) * (x / 2.0)
	tmp = 0
	if y_m <= 1.15e-276:
		tmp = t_2
	elif y_m <= 3.9e-218:
		tmp = t_0
	elif y_m <= 1.4e-180:
		tmp = x / (y_m * (2.0 / x))
	elif y_m <= 1.42e-151:
		tmp = t_0
	elif y_m <= 1.05e-60:
		tmp = t_2
	elif y_m <= 215000000.0:
		tmp = t_1
	elif y_m <= 1.5e+21:
		tmp = t_2
	elif (y_m <= 6.2e+33) or not (y_m <= 6.6e+69):
		tmp = y_m * 0.5
	else:
		tmp = t_1
	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(-0.5 / Float64(Float64(y_m / z) / z))
	t_1 = Float64(z * Float64(z * Float64(-0.5 / y_m)))
	t_2 = Float64(Float64(x / y_m) * Float64(x / 2.0))
	tmp = 0.0
	if (y_m <= 1.15e-276)
		tmp = t_2;
	elseif (y_m <= 3.9e-218)
		tmp = t_0;
	elseif (y_m <= 1.4e-180)
		tmp = Float64(x / Float64(y_m * Float64(2.0 / x)));
	elseif (y_m <= 1.42e-151)
		tmp = t_0;
	elseif (y_m <= 1.05e-60)
		tmp = t_2;
	elseif (y_m <= 215000000.0)
		tmp = t_1;
	elseif (y_m <= 1.5e+21)
		tmp = t_2;
	elseif ((y_m <= 6.2e+33) || !(y_m <= 6.6e+69))
		tmp = Float64(y_m * 0.5);
	else
		tmp = t_1;
	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 = -0.5 / ((y_m / z) / z);
	t_1 = z * (z * (-0.5 / y_m));
	t_2 = (x / y_m) * (x / 2.0);
	tmp = 0.0;
	if (y_m <= 1.15e-276)
		tmp = t_2;
	elseif (y_m <= 3.9e-218)
		tmp = t_0;
	elseif (y_m <= 1.4e-180)
		tmp = x / (y_m * (2.0 / x));
	elseif (y_m <= 1.42e-151)
		tmp = t_0;
	elseif (y_m <= 1.05e-60)
		tmp = t_2;
	elseif (y_m <= 215000000.0)
		tmp = t_1;
	elseif (y_m <= 1.5e+21)
		tmp = t_2;
	elseif ((y_m <= 6.2e+33) || ~((y_m <= 6.6e+69)))
		tmp = y_m * 0.5;
	else
		tmp = t_1;
	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[(-0.5 / N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(z * N[(z * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 1.15e-276], t$95$2, If[LessEqual[y$95$m, 3.9e-218], t$95$0, If[LessEqual[y$95$m, 1.4e-180], N[(x / N[(y$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.42e-151], t$95$0, If[LessEqual[y$95$m, 1.05e-60], t$95$2, If[LessEqual[y$95$m, 215000000.0], t$95$1, If[LessEqual[y$95$m, 1.5e+21], t$95$2, If[Or[LessEqual[y$95$m, 6.2e+33], N[Not[LessEqual[y$95$m, 6.6e+69]], $MachinePrecision]], N[(y$95$m * 0.5), $MachinePrecision], t$95$1]]]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if y < 1.14999999999999991e-276 or 1.42000000000000002e-151 < y < 1.04999999999999996e-60 or 2.15e8 < y < 1.5e21

   1. Initial program 72.9%

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

   3. Taylor expanded in x around inf 35.7%

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

      1. unpow2 35.7%

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

      2. times-frac 36.8%

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

   5. Applied egg-rr 36.8%

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

   if 1.14999999999999991e-276 < y < 3.9e-218 or 1.39999999999999999e-180 < y < 1.42000000000000002e-151

   1. Initial program 88.1%

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

   3. Taylor expanded in z around inf 64.4%

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

      1. associate-*r/ 64.4%

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

      2. associate-/l* 64.6%

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

   5. Simplified 64.6%

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

      1. *-un-lft-identity 64.6%

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

      2. unpow2 64.6%

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

      3. times-frac 64.5%

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

   7. Applied egg-rr 64.5%

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

      1. associate-*l/ 64.6%

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

      2. *-lft-identity 64.6%

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

   9. Simplified 64.6%

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

   if 3.9e-218 < y < 1.39999999999999999e-180

   1. Initial program 80.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. unpow2 100.0%

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

      2. times-frac 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-times 100.0%

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

      4. *-un-lft-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 1.04999999999999996e-60 < y < 2.15e8 or 6.2e33 < y < 6.5999999999999997e69

   1. Initial program 88.8%

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

   3. Taylor expanded in z around inf 58.2%

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

      1. associate-*r/ 58.2%

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

      2. associate-/l* 58.1%

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

   5. Simplified 58.1%

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

      1. associate-/r/ 57.9%

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

      2. unpow2 57.9%

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

      3. associate-*r* 58.0%

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

   7. Applied egg-rr 58.0%

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

   if 1.5e21 < y < 6.2e33 or 6.5999999999999997e69 < y

   1. Initial program 48.6%

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

   3. Taylor expanded in y around inf 67.6%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-276}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-218}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{-151}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 215000000:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+33} \lor \neg \left(y \leq 6.6 \cdot 10^{+69}\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{-0.5}{\frac{\frac{y_m}{z}}{z}}\\ t_1 := \frac{z \cdot -0.5}{\frac{y_m}{z}}\\ t_2 := \frac{x}{y_m} \cdot \frac{x}{2}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 1.1 \cdot 10^{-276}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y_m \leq 3.6 \cdot 10^{-217}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 6.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{x}{y_m \cdot \frac{2}{x}}\\ \mathbf{elif}\;y_m \leq 2.8 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 1.25 \cdot 10^{-59}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y_m \leq 1250000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y_m \leq 2.16 \cdot 10^{+21}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y_m \leq 9.2 \cdot 10^{+32} \lor \neg \left(y_m \leq 9.4 \cdot 10^{+67}\right):\\ \;\;\;\;y_m \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ -0.5 (/ (/ y_m z) z)))
        (t_1 (/ (* z -0.5) (/ y_m z)))
        (t_2 (* (/ x y_m) (/ x 2.0))))
   (*
    y_s
    (if (<= y_m 1.1e-276)
      t_2
      (if (<= y_m 3.6e-217)
        t_0
        (if (<= y_m 6.5e-181)
          (/ x (* y_m (/ 2.0 x)))
          (if (<= y_m 2.8e-153)
            t_0
            (if (<= y_m 1.25e-59)
              t_2
              (if (<= y_m 1250000000.0)
                t_1
                (if (<= y_m 2.16e+21)
                  t_2
                  (if (or (<= y_m 9.2e+32) (not (<= y_m 9.4e+67)))
                    (* y_m 0.5)
                    t_1)))))))))))

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 = -0.5 / ((y_m / z) / z);
	double t_1 = (z * -0.5) / (y_m / z);
	double t_2 = (x / y_m) * (x / 2.0);
	double tmp;
	if (y_m <= 1.1e-276) {
		tmp = t_2;
	} else if (y_m <= 3.6e-217) {
		tmp = t_0;
	} else if (y_m <= 6.5e-181) {
		tmp = x / (y_m * (2.0 / x));
	} else if (y_m <= 2.8e-153) {
		tmp = t_0;
	} else if (y_m <= 1.25e-59) {
		tmp = t_2;
	} else if (y_m <= 1250000000.0) {
		tmp = t_1;
	} else if (y_m <= 2.16e+21) {
		tmp = t_2;
	} else if ((y_m <= 9.2e+32) || !(y_m <= 9.4e+67)) {
		tmp = y_m * 0.5;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (-0.5d0) / ((y_m / z) / z)
    t_1 = (z * (-0.5d0)) / (y_m / z)
    t_2 = (x / y_m) * (x / 2.0d0)
    if (y_m <= 1.1d-276) then
        tmp = t_2
    else if (y_m <= 3.6d-217) then
        tmp = t_0
    else if (y_m <= 6.5d-181) then
        tmp = x / (y_m * (2.0d0 / x))
    else if (y_m <= 2.8d-153) then
        tmp = t_0
    else if (y_m <= 1.25d-59) then
        tmp = t_2
    else if (y_m <= 1250000000.0d0) then
        tmp = t_1
    else if (y_m <= 2.16d+21) then
        tmp = t_2
    else if ((y_m <= 9.2d+32) .or. (.not. (y_m <= 9.4d+67))) then
        tmp = y_m * 0.5d0
    else
        tmp = t_1
    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 = -0.5 / ((y_m / z) / z);
	double t_1 = (z * -0.5) / (y_m / z);
	double t_2 = (x / y_m) * (x / 2.0);
	double tmp;
	if (y_m <= 1.1e-276) {
		tmp = t_2;
	} else if (y_m <= 3.6e-217) {
		tmp = t_0;
	} else if (y_m <= 6.5e-181) {
		tmp = x / (y_m * (2.0 / x));
	} else if (y_m <= 2.8e-153) {
		tmp = t_0;
	} else if (y_m <= 1.25e-59) {
		tmp = t_2;
	} else if (y_m <= 1250000000.0) {
		tmp = t_1;
	} else if (y_m <= 2.16e+21) {
		tmp = t_2;
	} else if ((y_m <= 9.2e+32) || !(y_m <= 9.4e+67)) {
		tmp = y_m * 0.5;
	} else {
		tmp = t_1;
	}
	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 = -0.5 / ((y_m / z) / z)
	t_1 = (z * -0.5) / (y_m / z)
	t_2 = (x / y_m) * (x / 2.0)
	tmp = 0
	if y_m <= 1.1e-276:
		tmp = t_2
	elif y_m <= 3.6e-217:
		tmp = t_0
	elif y_m <= 6.5e-181:
		tmp = x / (y_m * (2.0 / x))
	elif y_m <= 2.8e-153:
		tmp = t_0
	elif y_m <= 1.25e-59:
		tmp = t_2
	elif y_m <= 1250000000.0:
		tmp = t_1
	elif y_m <= 2.16e+21:
		tmp = t_2
	elif (y_m <= 9.2e+32) or not (y_m <= 9.4e+67):
		tmp = y_m * 0.5
	else:
		tmp = t_1
	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(-0.5 / Float64(Float64(y_m / z) / z))
	t_1 = Float64(Float64(z * -0.5) / Float64(y_m / z))
	t_2 = Float64(Float64(x / y_m) * Float64(x / 2.0))
	tmp = 0.0
	if (y_m <= 1.1e-276)
		tmp = t_2;
	elseif (y_m <= 3.6e-217)
		tmp = t_0;
	elseif (y_m <= 6.5e-181)
		tmp = Float64(x / Float64(y_m * Float64(2.0 / x)));
	elseif (y_m <= 2.8e-153)
		tmp = t_0;
	elseif (y_m <= 1.25e-59)
		tmp = t_2;
	elseif (y_m <= 1250000000.0)
		tmp = t_1;
	elseif (y_m <= 2.16e+21)
		tmp = t_2;
	elseif ((y_m <= 9.2e+32) || !(y_m <= 9.4e+67))
		tmp = Float64(y_m * 0.5);
	else
		tmp = t_1;
	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 = -0.5 / ((y_m / z) / z);
	t_1 = (z * -0.5) / (y_m / z);
	t_2 = (x / y_m) * (x / 2.0);
	tmp = 0.0;
	if (y_m <= 1.1e-276)
		tmp = t_2;
	elseif (y_m <= 3.6e-217)
		tmp = t_0;
	elseif (y_m <= 6.5e-181)
		tmp = x / (y_m * (2.0 / x));
	elseif (y_m <= 2.8e-153)
		tmp = t_0;
	elseif (y_m <= 1.25e-59)
		tmp = t_2;
	elseif (y_m <= 1250000000.0)
		tmp = t_1;
	elseif (y_m <= 2.16e+21)
		tmp = t_2;
	elseif ((y_m <= 9.2e+32) || ~((y_m <= 9.4e+67)))
		tmp = y_m * 0.5;
	else
		tmp = t_1;
	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[(-0.5 / N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * -0.5), $MachinePrecision] / N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 1.1e-276], t$95$2, If[LessEqual[y$95$m, 3.6e-217], t$95$0, If[LessEqual[y$95$m, 6.5e-181], N[(x / N[(y$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 2.8e-153], t$95$0, If[LessEqual[y$95$m, 1.25e-59], t$95$2, If[LessEqual[y$95$m, 1250000000.0], t$95$1, If[LessEqual[y$95$m, 2.16e+21], t$95$2, If[Or[LessEqual[y$95$m, 9.2e+32], N[Not[LessEqual[y$95$m, 9.4e+67]], $MachinePrecision]], N[(y$95$m * 0.5), $MachinePrecision], t$95$1]]]]]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 5 regimes

2. if y < 1.0999999999999999e-276 or 2.8000000000000001e-153 < y < 1.25e-59 or 1.25e9 < y < 2.16e21

   1. Initial program 72.9%

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

   3. Taylor expanded in x around inf 35.7%

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

      1. unpow2 35.7%

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

      2. times-frac 36.8%

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

   5. Applied egg-rr 36.8%

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

   if 1.0999999999999999e-276 < y < 3.59999999999999981e-217 or 6.4999999999999997e-181 < y < 2.8000000000000001e-153

   1. Initial program 88.1%

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

   3. Taylor expanded in z around inf 64.4%

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

      1. associate-*r/ 64.4%

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

      2. associate-/l* 64.6%

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

   5. Simplified 64.6%

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

      1. *-un-lft-identity 64.6%

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

      2. unpow2 64.6%

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

      3. times-frac 64.5%

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

   7. Applied egg-rr 64.5%

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

      1. associate-*l/ 64.6%

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

      2. *-lft-identity 64.6%

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

   9. Simplified 64.6%

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

   if 3.59999999999999981e-217 < y < 6.4999999999999997e-181

   1. Initial program 80.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. unpow2 100.0%

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

      2. times-frac 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-times 100.0%

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

      4. *-un-lft-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 1.25e-59 < y < 1.25e9 or 9.1999999999999998e32 < y < 9.40000000000000035e67

   1. Initial program 88.8%

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

   3. Taylor expanded in z around inf 58.2%

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

      1. associate-*r/ 58.2%

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

      2. associate-/l* 58.1%

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

   5. Simplified 58.1%

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

      1. associate-/r/ 57.9%

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

      2. unpow2 57.9%

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

      3. associate-*r* 58.0%

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

   7. Applied egg-rr 58.0%

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

      1. associate-*l/ 58.2%

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

      2. associate-/l* 58.0%

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

      3. associate-*l/ 58.2%

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

      4. *-commutative 58.2%

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

   9. Applied egg-rr 58.2%

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

   if 2.16e21 < y < 9.1999999999999998e32 or 9.40000000000000035e67 < y

   1. Initial program 48.6%

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

   3. Taylor expanded in y around inf 67.6%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-276}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-217}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-181}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-153}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-59}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 1250000000:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 2.16 \cdot 10^{+21}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+32} \lor \neg \left(y \leq 9.4 \cdot 10^{+67}\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{-0.5}{\frac{\frac{y_m}{z}}{z}}\\ t_1 := \frac{x}{y_m} \cdot \frac{x}{2}\\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 9 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y_m \leq 1.32 \cdot 10^{-216}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 2.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{x}{y_m \cdot \frac{2}{x}}\\ \mathbf{elif}\;y_m \leq 3.2 \cdot 10^{-153}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y_m \leq 8 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y_m \leq 2.6 \cdot 10^{+162}:\\ \;\;\;\;\frac{\left(y_m - z\right) \cdot \left(y_m + z\right)}{y_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ -0.5 (/ (/ y_m z) z))) (t_1 (* (/ x y_m) (/ x 2.0))))
   (*
    y_s
    (if (<= y_m 9e-277)
      t_1
      (if (<= y_m 1.32e-216)
        t_0
        (if (<= y_m 2.4e-182)
          (/ x (* y_m (/ 2.0 x)))
          (if (<= y_m 3.2e-153)
            t_0
            (if (<= y_m 8e-61)
              t_1
              (if (<= y_m 2.6e+162)
                (/ (* (- y_m z) (+ y_m 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 t_0 = -0.5 / ((y_m / z) / z);
	double t_1 = (x / y_m) * (x / 2.0);
	double tmp;
	if (y_m <= 9e-277) {
		tmp = t_1;
	} else if (y_m <= 1.32e-216) {
		tmp = t_0;
	} else if (y_m <= 2.4e-182) {
		tmp = x / (y_m * (2.0 / x));
	} else if (y_m <= 3.2e-153) {
		tmp = t_0;
	} else if (y_m <= 8e-61) {
		tmp = t_1;
	} else if (y_m <= 2.6e+162) {
		tmp = ((y_m - z) * (y_m + 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-0.5d0) / ((y_m / z) / z)
    t_1 = (x / y_m) * (x / 2.0d0)
    if (y_m <= 9d-277) then
        tmp = t_1
    else if (y_m <= 1.32d-216) then
        tmp = t_0
    else if (y_m <= 2.4d-182) then
        tmp = x / (y_m * (2.0d0 / x))
    else if (y_m <= 3.2d-153) then
        tmp = t_0
    else if (y_m <= 8d-61) then
        tmp = t_1
    else if (y_m <= 2.6d+162) then
        tmp = ((y_m - z) * (y_m + 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 t_0 = -0.5 / ((y_m / z) / z);
	double t_1 = (x / y_m) * (x / 2.0);
	double tmp;
	if (y_m <= 9e-277) {
		tmp = t_1;
	} else if (y_m <= 1.32e-216) {
		tmp = t_0;
	} else if (y_m <= 2.4e-182) {
		tmp = x / (y_m * (2.0 / x));
	} else if (y_m <= 3.2e-153) {
		tmp = t_0;
	} else if (y_m <= 8e-61) {
		tmp = t_1;
	} else if (y_m <= 2.6e+162) {
		tmp = ((y_m - z) * (y_m + 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):
	t_0 = -0.5 / ((y_m / z) / z)
	t_1 = (x / y_m) * (x / 2.0)
	tmp = 0
	if y_m <= 9e-277:
		tmp = t_1
	elif y_m <= 1.32e-216:
		tmp = t_0
	elif y_m <= 2.4e-182:
		tmp = x / (y_m * (2.0 / x))
	elif y_m <= 3.2e-153:
		tmp = t_0
	elif y_m <= 8e-61:
		tmp = t_1
	elif y_m <= 2.6e+162:
		tmp = ((y_m - z) * (y_m + 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)
	t_0 = Float64(-0.5 / Float64(Float64(y_m / z) / z))
	t_1 = Float64(Float64(x / y_m) * Float64(x / 2.0))
	tmp = 0.0
	if (y_m <= 9e-277)
		tmp = t_1;
	elseif (y_m <= 1.32e-216)
		tmp = t_0;
	elseif (y_m <= 2.4e-182)
		tmp = Float64(x / Float64(y_m * Float64(2.0 / x)));
	elseif (y_m <= 3.2e-153)
		tmp = t_0;
	elseif (y_m <= 8e-61)
		tmp = t_1;
	elseif (y_m <= 2.6e+162)
		tmp = Float64(Float64(Float64(y_m - z) * Float64(y_m + 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)
	t_0 = -0.5 / ((y_m / z) / z);
	t_1 = (x / y_m) * (x / 2.0);
	tmp = 0.0;
	if (y_m <= 9e-277)
		tmp = t_1;
	elseif (y_m <= 1.32e-216)
		tmp = t_0;
	elseif (y_m <= 2.4e-182)
		tmp = x / (y_m * (2.0 / x));
	elseif (y_m <= 3.2e-153)
		tmp = t_0;
	elseif (y_m <= 8e-61)
		tmp = t_1;
	elseif (y_m <= 2.6e+162)
		tmp = ((y_m - z) * (y_m + 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_] := Block[{t$95$0 = N[(-0.5 / N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 9e-277], t$95$1, If[LessEqual[y$95$m, 1.32e-216], t$95$0, If[LessEqual[y$95$m, 2.4e-182], N[(x / N[(y$95$m * N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 3.2e-153], t$95$0, If[LessEqual[y$95$m, 8e-61], t$95$1, If[LessEqual[y$95$m, 2.6e+162], N[(N[(N[(y$95$m - z), $MachinePrecision] * N[(y$95$m + z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 5 regimes

2. if y < 8.99999999999999985e-277 or 3.1999999999999999e-153 < y < 8.0000000000000003e-61

   1. Initial program 72.6%

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

   3. Taylor expanded in x around inf 34.9%

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

      1. unpow2 34.9%

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

      2. times-frac 36.0%

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

   5. Applied egg-rr 36.0%

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

   if 8.99999999999999985e-277 < y < 1.31999999999999997e-216 or 2.3999999999999998e-182 < y < 3.1999999999999999e-153

   1. Initial program 88.1%

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

   3. Taylor expanded in z around inf 64.4%

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

      1. associate-*r/ 64.4%

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

      2. associate-/l* 64.6%

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

   5. Simplified 64.6%

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

      1. *-un-lft-identity 64.6%

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

      2. unpow2 64.6%

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

      3. times-frac 64.5%

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

   7. Applied egg-rr 64.5%

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

      1. associate-*l/ 64.6%

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

      2. *-lft-identity 64.6%

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

   9. Simplified 64.6%

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

   if 1.31999999999999997e-216 < y < 2.3999999999999998e-182

   1. Initial program 80.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. unpow2 100.0%

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

      2. times-frac 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. frac-times 100.0%

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

      4. *-un-lft-identity 100.0%

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

   7. Applied egg-rr 100.0%

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

   if 8.0000000000000003e-61 < y < 2.6e162

   1. Initial program 87.9%

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

      1. associate--l+ 87.9%

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

      2. +-commutative 87.9%

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

      3. sqr-neg 87.9%

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

      4. difference-of-squares 87.9%

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

      5. fma-def 87.9%

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

      6. sub-neg 87.9%

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

      7. sub-neg 87.9%

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

      8. remove-double-neg 87.9%

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

   3. Simplified 87.9%

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

   5. Taylor expanded in x around 0 72.1%

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

   if 2.6e162 < y

   1. Initial program 15.0%

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

   3. Taylor expanded in y around inf 82.9%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-277}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{-216}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-182}:\\ \;\;\;\;\frac{x}{y \cdot \frac{2}{x}}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-153}:\\ \;\;\;\;\frac{-0.5}{\frac{\frac{y}{z}}{z}}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+162}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot \left(y + z\right)}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+104} \lor \neg \left(x \leq 5.9 \cdot 10^{+166}\right) \land x \leq 5.3 \cdot 10^{+172}:\\ \;\;\;\;\frac{y_m + z}{\frac{y_m}{\left(y_m - z\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y_m} \cdot \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (or (<= x 4.8e+104) (and (not (<= x 5.9e+166)) (<= x 5.3e+172)))
    (/ (+ y_m z) (/ y_m (* (- y_m z) 0.5)))
    (* (/ x y_m) (/ x 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 ((x <= 4.8e+104) || (!(x <= 5.9e+166) && (x <= 5.3e+172))) {
		tmp = (y_m + z) / (y_m / ((y_m - z) * 0.5));
	} else {
		tmp = (x / y_m) * (x / 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 ((x <= 4.8d+104) .or. (.not. (x <= 5.9d+166)) .and. (x <= 5.3d+172)) then
        tmp = (y_m + z) / (y_m / ((y_m - z) * 0.5d0))
    else
        tmp = (x / y_m) * (x / 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 ((x <= 4.8e+104) || (!(x <= 5.9e+166) && (x <= 5.3e+172))) {
		tmp = (y_m + z) / (y_m / ((y_m - z) * 0.5));
	} else {
		tmp = (x / y_m) * (x / 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 (x <= 4.8e+104) or (not (x <= 5.9e+166) and (x <= 5.3e+172)):
		tmp = (y_m + z) / (y_m / ((y_m - z) * 0.5))
	else:
		tmp = (x / y_m) * (x / 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 ((x <= 4.8e+104) || (!(x <= 5.9e+166) && (x <= 5.3e+172)))
		tmp = Float64(Float64(y_m + z) / Float64(y_m / Float64(Float64(y_m - z) * 0.5)));
	else
		tmp = Float64(Float64(x / y_m) * Float64(x / 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 ((x <= 4.8e+104) || (~((x <= 5.9e+166)) && (x <= 5.3e+172)))
		tmp = (y_m + z) / (y_m / ((y_m - z) * 0.5));
	else
		tmp = (x / y_m) * (x / 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[Or[LessEqual[x, 4.8e+104], And[N[Not[LessEqual[x, 5.9e+166]], $MachinePrecision], LessEqual[x, 5.3e+172]]], N[(N[(y$95$m + z), $MachinePrecision] / N[(y$95$m / N[(N[(y$95$m - z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y$95$m), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.8e104 or 5.90000000000000012e166 < x < 5.3e172

   1. Initial program 72.9%

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

      1. associate--l+ 72.9%

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

      2. +-commutative 72.9%

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

      3. sqr-neg 72.9%

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

      4. difference-of-squares 73.6%

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

      5. fma-def 74.5%

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

      6. sub-neg 74.5%

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

      7. sub-neg 74.5%

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

      8. remove-double-neg 74.5%

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

   3. Simplified 74.5%

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

   5. Taylor expanded in x around 0 53.8%

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

      1. associate-/l* 75.3%

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

      2. div-inv 75.2%

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

      3. associate-/l* 75.2%

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

      4. div-inv 75.2%

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

      5. metadata-eval 75.2%

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

   7. Applied egg-rr 75.2%

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

      1. associate-*r/ 75.3%

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

      2. *-rgt-identity 75.3%

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

      3. *-commutative 75.3%

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

   9. Simplified 75.3%

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

   if 4.8e104 < x < 5.90000000000000012e166 or 5.3e172 < x

   1. Initial program 67.5%

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

   3. Taylor expanded in x around inf 73.2%

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

      1. unpow2 73.2%

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

      2. times-frac 80.7%

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

   5. Applied egg-rr 80.7%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+104} \lor \neg \left(x \leq 5.9 \cdot 10^{+166}\right) \land x \leq 5.3 \cdot 10^{+172}:\\ \;\;\;\;\frac{y + z}{\frac{y}{\left(y - z\right) \cdot 0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{y_m + z}{2}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+82} \lor \neg \left(z \leq 5.6 \cdot 10^{+113}\right):\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 4.2e+70)
    (/ (+ y_m z) 2.0)
    (if (or (<= z 2.9e+82) (not (<= z 5.6e+113)))
      (* 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 (z <= 4.2e+70) {
		tmp = (y_m + z) / 2.0;
	} else if ((z <= 2.9e+82) || !(z <= 5.6e+113)) {
		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 (z <= 4.2d+70) then
        tmp = (y_m + z) / 2.0d0
    else if ((z <= 2.9d+82) .or. (.not. (z <= 5.6d+113))) 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 (z <= 4.2e+70) {
		tmp = (y_m + z) / 2.0;
	} else if ((z <= 2.9e+82) || !(z <= 5.6e+113)) {
		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 z <= 4.2e+70:
		tmp = (y_m + z) / 2.0
	elif (z <= 2.9e+82) or not (z <= 5.6e+113):
		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 (z <= 4.2e+70)
		tmp = Float64(Float64(y_m + z) / 2.0);
	elseif ((z <= 2.9e+82) || !(z <= 5.6e+113))
		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 (z <= 4.2e+70)
		tmp = (y_m + z) / 2.0;
	elseif ((z <= 2.9e+82) || ~((z <= 5.6e+113)))
		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[z, 4.2e+70], N[(N[(y$95$m + z), $MachinePrecision] / 2.0), $MachinePrecision], If[Or[LessEqual[z, 2.9e+82], N[Not[LessEqual[z, 5.6e+113]], $MachinePrecision]], 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 3 regimes

2. if z < 4.20000000000000015e70

   1. Initial program 74.2%

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

      1. associate--l+ 74.2%

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

      2. +-commutative 74.2%

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

      3. sqr-neg 74.2%

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

      4. difference-of-squares 74.3%

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

      5. fma-def 75.3%

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

      6. sub-neg 75.3%

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

      7. sub-neg 75.3%

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

      8. remove-double-neg 75.3%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in x around 0 43.8%

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

      1. associate-/l* 63.4%

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

      2. div-inv 63.4%

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

      3. associate-/l* 63.4%

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

      4. div-inv 63.4%

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

      5. metadata-eval 63.4%

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

   7. Applied egg-rr 63.4%

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

      1. associate-*r/ 63.4%

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

      2. *-rgt-identity 63.4%

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

      3. *-commutative 63.4%

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

   9. Simplified 63.4%

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

   10. Taylor expanded in y around inf 42.3%

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

   if 4.20000000000000015e70 < z < 2.9000000000000001e82 or 5.59999999999999995e113 < z

   1. Initial program 71.6%

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

   3. Taylor expanded in z around inf 79.2%

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

      1. associate-*r/ 79.2%

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

      2. associate-/l* 79.2%

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

   5. Simplified 79.2%

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

      1. associate-/r/ 79.2%

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

      2. unpow2 79.2%

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

      3. associate-*r* 85.6%

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

   7. Applied egg-rr 85.6%

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

   if 2.9000000000000001e82 < z < 5.59999999999999995e113

   1. Initial program 27.8%

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

   3. Taylor expanded in y around inf 67.9%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+70}:\\ \;\;\;\;\frac{y + z}{2}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+82} \lor \neg \left(z \leq 5.6 \cdot 10^{+113}\right):\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 3.8 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(x \cdot x + y_m \cdot y_m\right) - z \cdot z}{y_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m + z}{\frac{y_m}{\left(y_m - z\right) \cdot 0.5}}\\ \end{array} \end{array} \]

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 3.8e+125)
    (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
    (/ (+ y_m z) (/ y_m (* (- y_m z) 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 <= 3.8e+125) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = (y_m + z) / (y_m / ((y_m - z) * 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 <= 3.8d+125) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = (y_m + z) / (y_m / ((y_m - z) * 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 <= 3.8e+125) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = (y_m + z) / (y_m / ((y_m - z) * 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 <= 3.8e+125:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = (y_m + z) / (y_m / ((y_m - z) * 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 <= 3.8e+125)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(Float64(y_m + z) / Float64(y_m / Float64(Float64(y_m - z) * 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 <= 3.8e+125)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = (y_m + z) / (y_m / ((y_m - z) * 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, 3.8e+125], 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[(y$95$m + z), $MachinePrecision] / N[(y$95$m / N[(N[(y$95$m - z), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.80000000000000002e125

   1. Initial program 77.6%

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

   if 3.80000000000000002e125 < y

   1. Initial program 29.0%

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

      1. associate--l+ 29.0%

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

      2. +-commutative 29.0%

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

      3. sqr-neg 29.0%

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

      4. difference-of-squares 29.7%

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

      5. fma-def 29.7%

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

      6. sub-neg 29.7%

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

      7. sub-neg 29.7%

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

      8. remove-double-neg 29.7%

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

   3. Simplified 29.7%

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

   5. Taylor expanded in x around 0 29.7%

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

      1. associate-/l* 87.2%

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

      2. div-inv 87.1%

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

      3. associate-/l* 87.1%

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

      4. div-inv 87.1%

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

      5. metadata-eval 87.1%

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

   7. Applied egg-rr 87.1%

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

      1. associate-*r/ 87.2%

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

      2. *-rgt-identity 87.2%

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

      3. *-commutative 87.2%

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

   9. Simplified 87.2%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + z}{\frac{y}{\left(y - z\right) \cdot 0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.6% 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 72.1%

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

3. Taylor expanded in y around inf 37.0%

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

4. Final simplification 37.0%

   \[\leadsto y \cdot 0.5 \]
5. 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 2024018 
(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)))