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

Percentage Accurate: 69.5% → 96.0%

Time: 11.0s

Alternatives: 8

Speedup: 1.1×

• 41.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.4 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y\_m \cdot y\_m - z \cdot z\right)}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m + \frac{x}{\frac{y\_m}{x}}\right) - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 2.4e+44)
    (* 0.5 (/ (fma x x (- (* y_m y_m) (* z z))) y_m))
    (* 0.5 (- (+ y_m (/ x (/ y_m x))) (* z (/ z y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.4e+44) {
		tmp = 0.5 * (fma(x, x, ((y_m * y_m) - (z * z))) / y_m);
	} else {
		tmp = 0.5 * ((y_m + (x / (y_m / x))) - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 2.4e+44)
		tmp = Float64(0.5 * Float64(fma(x, x, Float64(Float64(y_m * y_m) - Float64(z * z))) / y_m));
	else
		tmp = Float64(0.5 * Float64(Float64(y_m + Float64(x / Float64(y_m / x))) - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 2.4e+44], N[(0.5 * N[(N[(x * x + N[(N[(y$95$m * y$95$m), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y$95$m + N[(x / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.40000000000000013e44

   1. Initial program 80.1%

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

      1. remove-double-neg 80.1%

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

      2. distribute-lft-neg-out 80.1%

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

      3. distribute-frac-neg2 80.1%

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

      4. distribute-frac-neg 80.1%

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

      5. neg-mul-1 80.1%

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

      6. distribute-lft-neg-out 80.1%

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

      7. *-commutative 80.1%

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

      8. distribute-lft-neg-in 80.1%

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

      9. times-frac 80.1%

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

      10. metadata-eval 80.1%

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

      11. metadata-eval 80.1%

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

      12. associate--l+ 80.1%

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

      13. fma-define 82.1%

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

   3. Simplified 82.1%

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

   if 2.40000000000000013e44 < y

   1. Initial program 27.8%

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

      1. remove-double-neg 27.8%

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

      2. distribute-lft-neg-out 27.8%

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

      3. distribute-frac-neg2 27.8%

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

      4. distribute-frac-neg 27.8%

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

      5. neg-mul-1 27.8%

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

      6. distribute-lft-neg-out 27.8%

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

      7. *-commutative 27.8%

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

      8. distribute-lft-neg-in 27.8%

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

      9. times-frac 27.8%

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

      10. metadata-eval 27.8%

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

      11. metadata-eval 27.8%

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

      12. associate--l+ 27.8%

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

      13. fma-define 27.8%

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

   3. Simplified 27.8%

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

   5. Taylor expanded in x around 0 74.6%

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

      1. unpow2 74.6%

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

      2. *-un-lft-identity 74.6%

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

      3. times-frac 83.8%

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

   7. Applied egg-rr 83.8%

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

      1. unpow2 83.8%

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

      2. *-un-lft-identity 83.8%

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

      3. times-frac 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. /-rgt-identity 99.9%

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

       2. clear-num 99.9%

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

       3. un-div-inv 99.9%

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

   11. Applied egg-rr 99.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{+44}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0.5 \cdot \left(y\_m + \frac{1}{\frac{-1}{z} \cdot \frac{y\_m}{z}}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;0.5 \cdot \left(y\_m + x \cdot \frac{x}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* 0.5 (+ y_m (/ 1.0 (* (/ -1.0 z) (/ y_m z)))))
      (if (<= t_0 INFINITY)
        (* 0.5 (+ y_m (* x (/ x y_m))))
        (* 0.5 (- y_m (* z (/ z y_m)))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * (y_m + (1.0 / ((-1.0 / z) * (y_m / z))));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 0.5 * (y_m + (x * (x / y_m)));
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 0.5 * (y_m + (1.0 / ((-1.0 / z) * (y_m / z))));
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 0.5 * (y_m + (x * (x / y_m)));
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 0.5 * (y_m + (1.0 / ((-1.0 / z) * (y_m / z))))
	elif t_0 <= math.inf:
		tmp = 0.5 * (y_m + (x * (x / y_m)))
	else:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	return y_s * tmp

Julia

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

MATLAB

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(0.5 * N[(y$95$m + N[(1.0 / N[(N[(-1.0 / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(0.5 * N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 78.3%

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

      1. remove-double-neg 78.3%

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

      2. distribute-lft-neg-out 78.3%

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

      3. distribute-frac-neg2 78.3%

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

      4. distribute-frac-neg 78.3%

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

      5. neg-mul-1 78.3%

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

      6. distribute-lft-neg-out 78.3%

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

      7. *-commutative 78.3%

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

      8. distribute-lft-neg-in 78.3%

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

      9. times-frac 78.3%

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

      10. metadata-eval 78.3%

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

      11. metadata-eval 78.3%

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

      12. associate--l+ 78.3%

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

      13. fma-define 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in x around 0 86.4%

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

      1. associate--l+ 86.4%

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

      2. div-sub 91.4%

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

   7. Simplified 91.4%

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

      1. clear-num 91.4%

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

      2. inv-pow 91.4%

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

   9. Applied egg-rr 91.4%

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

       1. unpow-1 91.4%

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

   11. Simplified 91.4%

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

   12. Taylor expanded in x around 0 62.1%

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

       1. neg-mul-1 62.1%

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

       2. distribute-neg-frac 62.1%

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

   14. Simplified 62.1%

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

       1. neg-mul-1 62.1%

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

       2. unpow2 62.1%

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

       3. times-frac 66.0%

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

   16. Applied egg-rr 66.0%

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

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

   1. Initial program 72.3%

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

      1. remove-double-neg 72.3%

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

      2. distribute-lft-neg-out 72.3%

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

      3. distribute-frac-neg2 72.3%

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

      4. distribute-frac-neg 72.3%

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

      5. neg-mul-1 72.3%

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

      6. distribute-lft-neg-out 72.3%

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

      7. *-commutative 72.3%

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

      8. distribute-lft-neg-in 72.3%

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

      9. times-frac 72.3%

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

      10. metadata-eval 72.3%

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

      11. metadata-eval 72.3%

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

      12. associate--l+ 72.3%

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

      13. fma-define 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in x around 0 87.3%

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

      1. associate--l+ 87.3%

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

      2. div-sub 92.5%

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

   7. Simplified 92.5%

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

   8. Taylor expanded in z around 0 59.0%

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

      1. +-commutative 59.0%

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

      2. unpow2 59.0%

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

      3. associate-*r/ 64.6%

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

      4. fma-define 64.6%

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

   10. Simplified 64.6%

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

       1. fma-undefine 64.6%

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

   12. Applied egg-rr 64.6%

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

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

   1. Initial program 0.0%

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

      1. remove-double-neg 0.0%

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

      2. distribute-lft-neg-out 0.0%

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

      3. distribute-frac-neg2 0.0%

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

      4. distribute-frac-neg 0.0%

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

      5. neg-mul-1 0.0%

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

      6. distribute-lft-neg-out 0.0%

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

      7. *-commutative 0.0%

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

      8. distribute-lft-neg-in 0.0%

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

      9. times-frac 0.0%

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

      10. metadata-eval 0.0%

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

      11. metadata-eval 0.0%

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

      12. associate--l+ 0.0%

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

      13. fma-define 20.0%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in x around 0 10.9%

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

      1. associate--l+ 10.9%

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

      2. div-sub 10.9%

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

   7. Simplified 10.9%

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

      1. clear-num 10.9%

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

      2. inv-pow 10.9%

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

   9. Applied egg-rr 10.9%

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

       1. unpow-1 10.9%

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

   11. Simplified 10.9%

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

   12. Taylor expanded in x around 0 46.3%

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

       1. neg-mul-1 46.3%

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

       2. distribute-neg-frac 46.3%

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

   14. Simplified 46.3%

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

       1. clear-num 46.3%

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

       2. distribute-frac-neg2 46.3%

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

       3. unpow2 46.3%

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

       4. *-un-lft-identity 46.3%

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

       5. frac-times 85.0%

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

       6. /-rgt-identity 85.0%

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

       7. *-commutative 85.0%

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

       8. distribute-rgt-neg-in 85.0%

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

   16. Applied egg-rr 85.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2} \leq 0:\\ \;\;\;\;0.5 \cdot \left(y + \frac{1}{\frac{-1}{z} \cdot \frac{y}{z}}\right)\\ \mathbf{elif}\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.45 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(y\_m \cdot y\_m + x \cdot x\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y\_m + \frac{x}{\frac{y\_m}{x}}\right) - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 2.45e+29)
    (/ (- (+ (* y_m y_m) (* x x)) (* z z)) (* y_m 2.0))
    (* 0.5 (- (+ y_m (/ x (/ y_m x))) (* z (/ z y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.45e+29) {
		tmp = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * ((y_m + (x / (y_m / x))) - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 2.45d+29) then
        tmp = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = 0.5d0 * ((y_m + (x / (y_m / x))) - (z * (z / y_m)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.45e+29) {
		tmp = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = 0.5 * ((y_m + (x / (y_m / x))) - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 2.45e+29:
		tmp = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = 0.5 * ((y_m + (x / (y_m / x))) - (z * (z / y_m)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 2.45e+29)
		tmp = Float64(Float64(Float64(Float64(y_m * y_m) + Float64(x * x)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(0.5 * Float64(Float64(y_m + Float64(x / Float64(y_m / x))) - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 2.45e+29)
		tmp = (((y_m * y_m) + (x * x)) - (z * z)) / (y_m * 2.0);
	else
		tmp = 0.5 * ((y_m + (x / (y_m / x))) - (z * (z / y_m)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 2.45e+29], N[(N[(N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y$95$m + N[(x / N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.4500000000000001e29

   1. Initial program 80.1%

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

   if 2.4500000000000001e29 < y

   1. Initial program 32.4%

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

      1. remove-double-neg 32.4%

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

      2. distribute-lft-neg-out 32.4%

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

      3. distribute-frac-neg2 32.4%

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

      4. distribute-frac-neg 32.4%

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

      5. neg-mul-1 32.4%

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

      6. distribute-lft-neg-out 32.4%

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

      7. *-commutative 32.4%

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

      8. distribute-lft-neg-in 32.4%

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

      9. times-frac 32.4%

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

      10. metadata-eval 32.4%

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

      11. metadata-eval 32.4%

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

      12. associate--l+ 32.4%

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

      13. fma-define 34.1%

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

   3. Simplified 34.1%

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

   5. Taylor expanded in x around 0 75.0%

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

      1. unpow2 75.0%

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

      2. *-un-lft-identity 75.0%

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

      3. times-frac 83.5%

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

   7. Applied egg-rr 83.5%

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

      1. unpow2 83.5%

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

      2. *-un-lft-identity 83.5%

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

      3. times-frac 98.1%

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

   9. Applied egg-rr 98.1%

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

       1. /-rgt-identity 98.1%

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

       2. clear-num 98.2%

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

       3. un-div-inv 98.2%

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

   11. Applied egg-rr 98.2%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.45 \cdot 10^{+29}:\\ \;\;\;\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.45 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(y\_m + x \cdot \frac{x}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= z 2.45e+39)
    (* 0.5 (+ y_m (* x (/ x y_m))))
    (* 0.5 (- y_m (* z (/ z y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 2.45e+39) {
		tmp = 0.5 * (y_m + (x * (x / y_m)));
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.45d+39) then
        tmp = 0.5d0 * (y_m + (x * (x / y_m)))
    else
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (z <= 2.45e+39) {
		tmp = 0.5 * (y_m + (x * (x / y_m)));
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if z <= 2.45e+39:
		tmp = 0.5 * (y_m + (x * (x / y_m)))
	else:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (z <= 2.45e+39)
		tmp = Float64(0.5 * Float64(y_m + Float64(x * Float64(x / y_m))));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (z <= 2.45e+39)
		tmp = 0.5 * (y_m + (x * (x / y_m)));
	else
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[z, 2.45e+39], N[(0.5 * N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.44999999999999994e39

   1. Initial program 70.7%

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

      1. remove-double-neg 70.7%

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

      2. distribute-lft-neg-out 70.7%

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

      3. distribute-frac-neg2 70.7%

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

      4. distribute-frac-neg 70.7%

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

      5. neg-mul-1 70.7%

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

      6. distribute-lft-neg-out 70.7%

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

      7. *-commutative 70.7%

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

      8. distribute-lft-neg-in 70.7%

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

      9. times-frac 70.7%

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

      10. metadata-eval 70.7%

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

      11. metadata-eval 70.7%

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

      12. associate--l+ 70.7%

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

      13. fma-define 71.8%

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

   3. Simplified 71.8%

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

   5. Taylor expanded in x around 0 84.5%

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

      1. associate--l+ 84.5%

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

      2. div-sub 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in z around 0 66.3%

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

      1. +-commutative 66.3%

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

      2. unpow2 66.3%

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

      3. associate-*r/ 71.1%

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

      4. fma-define 71.1%

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

   10. Simplified 71.1%

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

       1. fma-undefine 71.1%

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

   12. Applied egg-rr 71.1%

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

   if 2.44999999999999994e39 < z

   1. Initial program 65.6%

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

      1. remove-double-neg 65.6%

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

      2. distribute-lft-neg-out 65.6%

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

      3. distribute-frac-neg2 65.6%

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

      4. distribute-frac-neg 65.6%

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

      5. neg-mul-1 65.6%

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

      6. distribute-lft-neg-out 65.6%

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

      7. *-commutative 65.6%

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

      8. distribute-lft-neg-in 65.6%

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

      9. times-frac 65.6%

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

      10. metadata-eval 65.6%

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

      11. metadata-eval 65.6%

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

      12. associate--l+ 65.6%

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

      13. fma-define 68.8%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in x around 0 69.9%

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

      1. associate--l+ 69.9%

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

      2. div-sub 77.9%

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

   7. Simplified 77.9%

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

      1. clear-num 78.0%

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

      2. inv-pow 78.0%

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

   9. Applied egg-rr 78.0%

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

       1. unpow-1 78.0%

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

   11. Simplified 78.0%

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

   12. Taylor expanded in x around 0 73.6%

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

       1. neg-mul-1 73.6%

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

       2. distribute-neg-frac 73.6%

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

   14. Simplified 73.6%

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

       1. clear-num 73.6%

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

       2. distribute-frac-neg2 73.6%

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

       3. unpow2 73.6%

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

       4. *-un-lft-identity 73.6%

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

       5. frac-times 87.5%

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

       6. /-rgt-identity 87.5%

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

       7. *-commutative 87.5%

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

       8. distribute-rgt-neg-in 87.5%

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

   16. Applied egg-rr 87.5%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.45 \cdot 10^{+39}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y - z \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 9.6 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 9.6e+129) (* x (* x (/ 0.5 y_m))) (* y_m 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 9.6e+129) {
		tmp = x * (x * (0.5 / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 9.6d+129) then
        tmp = x * (x * (0.5d0 / y_m))
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 9.6e+129) {
		tmp = x * (x * (0.5 / y_m));
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 9.6e+129:
		tmp = x * (x * (0.5 / y_m))
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 9.6e+129)
		tmp = Float64(x * Float64(x * Float64(0.5 / y_m)));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 9.6e+129)
		tmp = x * (x * (0.5 / y_m));
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 9.6e+129], N[(x * N[(x * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.5999999999999995e129

   1. Initial program 79.3%

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

      1. remove-double-neg 79.3%

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

      2. distribute-lft-neg-out 79.3%

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

      3. distribute-frac-neg2 79.3%

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

      4. distribute-frac-neg 79.3%

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

      5. neg-mul-1 79.3%

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

      6. distribute-lft-neg-out 79.3%

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

      7. *-commutative 79.3%

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

      8. distribute-lft-neg-in 79.3%

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

      9. times-frac 79.3%

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

      10. metadata-eval 79.3%

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

      11. metadata-eval 79.3%

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

      12. associate--l+ 79.3%

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

      13. fma-define 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around inf 34.9%

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

      1. *-commutative 34.9%

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

      2. associate-*l/ 34.9%

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

      3. associate-*r/ 34.9%

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

   7. Simplified 34.9%

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

      1. add-sqr-sqrt 15.5%

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

      2. pow2 15.5%

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

      3. sqrt-prod 15.2%

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

      4. sqrt-pow1 16.6%

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

      5. metadata-eval 16.6%

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

      6. pow1 16.6%

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

   9. Applied egg-rr 16.6%

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

       1. unpow2 16.6%

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

       2. swap-sqr 15.2%

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

       3. add-sqr-sqrt 34.9%

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

       4. *-commutative 34.9%

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

       5. associate-*r* 36.7%

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

   11. Applied egg-rr 36.7%

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

   if 9.5999999999999995e129 < y

   1. Initial program 13.3%

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

      1. remove-double-neg 13.3%

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

      2. distribute-lft-neg-out 13.3%

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

      3. distribute-frac-neg2 13.3%

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

      4. distribute-frac-neg 13.3%

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

      5. neg-mul-1 13.3%

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

      6. distribute-lft-neg-out 13.3%

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

      7. *-commutative 13.3%

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

      8. distribute-lft-neg-in 13.3%

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

      9. times-frac 13.3%

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

      10. metadata-eval 13.3%

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

      11. metadata-eval 13.3%

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

      12. associate--l+ 13.3%

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

      13. fma-define 13.3%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in y around inf 84.9%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.6 \cdot 10^{+129}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 9.6 \cdot 10^{+129}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 9.6e+129) (* (* x x) (/ 0.5 y_m)) (* y_m 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 9.6e+129) {
		tmp = (x * x) * (0.5 / y_m);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
real(8) function code(y_s, x, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 9.6d+129) then
        tmp = (x * x) * (0.5d0 / y_m)
    else
        tmp = y_m * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 9.6e+129) {
		tmp = (x * x) * (0.5 / y_m);
	} else {
		tmp = y_m * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 9.6e+129:
		tmp = (x * x) * (0.5 / y_m)
	else:
		tmp = y_m * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 9.6e+129)
		tmp = Float64(Float64(x * x) * Float64(0.5 / y_m));
	else
		tmp = Float64(y_m * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 9.6e+129)
		tmp = (x * x) * (0.5 / y_m);
	else
		tmp = y_m * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 9.6e+129], N[(N[(x * x), $MachinePrecision] * N[(0.5 / y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 9.5999999999999995e129

   1. Initial program 79.3%

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

      1. remove-double-neg 79.3%

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

      2. distribute-lft-neg-out 79.3%

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

      3. distribute-frac-neg2 79.3%

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

      4. distribute-frac-neg 79.3%

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

      5. neg-mul-1 79.3%

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

      6. distribute-lft-neg-out 79.3%

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

      7. *-commutative 79.3%

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

      8. distribute-lft-neg-in 79.3%

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

      9. times-frac 79.3%

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

      10. metadata-eval 79.3%

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

      11. metadata-eval 79.3%

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

      12. associate--l+ 79.3%

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

      13. fma-define 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around inf 34.9%

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

      1. *-commutative 34.9%

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

      2. associate-*l/ 34.9%

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

      3. associate-*r/ 34.9%

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

   7. Simplified 34.9%

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

      1. unpow2 34.9%

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

   9. Applied egg-rr 34.9%

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

   if 9.5999999999999995e129 < y

   1. Initial program 13.3%

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

      1. remove-double-neg 13.3%

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

      2. distribute-lft-neg-out 13.3%

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

      3. distribute-frac-neg2 13.3%

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

      4. distribute-frac-neg 13.3%

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

      5. neg-mul-1 13.3%

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

      6. distribute-lft-neg-out 13.3%

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

      7. *-commutative 13.3%

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

      8. distribute-lft-neg-in 13.3%

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

      9. times-frac 13.3%

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

      10. metadata-eval 13.3%

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

      11. metadata-eval 13.3%

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

      12. associate--l+ 13.3%

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

      13. fma-define 13.3%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in y around inf 84.9%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.6 \cdot 10^{+129}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

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 * (0.5d0 * (y_m + (x * (x / y_m))))
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 * (0.5 * (y_m + (x * (x / y_m))));
}

Python

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

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(0.5 * Float64(y_m + Float64(x * Float64(x / y_m)))))
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 * (0.5 * (y_m + (x * (x / y_m))));
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[(0.5 * N[(y$95$m + N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.5%

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

   1. remove-double-neg 69.5%

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

   2. distribute-lft-neg-out 69.5%

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

   3. distribute-frac-neg2 69.5%

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

   4. distribute-frac-neg 69.5%

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

   5. neg-mul-1 69.5%

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

   6. distribute-lft-neg-out 69.5%

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

   7. *-commutative 69.5%

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

   8. distribute-lft-neg-in 69.5%

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

   9. times-frac 69.5%

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

   10. metadata-eval 69.5%

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

   11. metadata-eval 69.5%

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

   12. associate--l+ 69.5%

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

   13. fma-define 71.0%

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

3. Simplified 71.0%

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

5. Taylor expanded in x around 0 80.9%

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

   1. associate--l+ 80.9%

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

   2. div-sub 85.6%

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

7. Simplified 85.6%

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

8. Taylor expanded in z around 0 56.4%

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

   1. +-commutative 56.4%

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

   2. unpow2 56.4%

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

   3. associate-*r/ 61.1%

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

   4. fma-define 61.1%

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

10. Simplified 61.1%

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

    1. fma-undefine 61.1%

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

12. Applied egg-rr 61.1%

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

13. Final simplification 61.1%

    \[\leadsto 0.5 \cdot \left(y + x \cdot \frac{x}{y}\right) \]
14. Add Preprocessing

Alternative 8: 33.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. remove-double-neg 69.5%

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

   2. distribute-lft-neg-out 69.5%

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

   3. distribute-frac-neg2 69.5%

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

   4. distribute-frac-neg 69.5%

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

   5. neg-mul-1 69.5%

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

   6. distribute-lft-neg-out 69.5%

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

   7. *-commutative 69.5%

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

   8. distribute-lft-neg-in 69.5%

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

   9. times-frac 69.5%

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

   10. metadata-eval 69.5%

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

   11. metadata-eval 69.5%

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

   12. associate--l+ 69.5%

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

   13. fma-define 71.0%

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

3. Simplified 71.0%

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

5. Taylor expanded in y around inf 31.0%

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

6. Final simplification 31.0%

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

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024191 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

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

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