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

Percentage Accurate: 69.0% → 95.2%

Time: 5.1s

Alternatives: 8

Speedup: 0.6×

• 41.9% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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: 69.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.2% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(y$95$m - N[(N[(z$95$m / y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] * x$95$m + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(0.5 / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * N[(x$95$m - z$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m + 0.5), $MachinePrecision] * y$95$m), $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 81.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 65.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 67.6

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

   7. Applied rewrites 67.6%

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

   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 76.9%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.2%

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

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 37.9

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

   7. Applied rewrites 37.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(\frac{\frac{1}{2}}{y \cdot y}, \left(x - z\right) \cdot x, \frac{1}{2}\right) \cdot y \]
   9. Step-by-step derivation

   10. Applied rewrites 62.9%

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

       1. lift-fma.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-*.f64 70.9

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

   12. Applied rewrites 70.9%

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

Alternative 2: 69.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(z\_m \cdot \frac{z\_m}{y\_m}\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+153} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot \frac{x\_m}{y\_m}\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* (* z_m (/ z_m y_m)) -0.5)
      (if (or (<= t_0 5e+153) (not (<= t_0 INFINITY)))
        (* 0.5 y_m)
        (* (* x_m (/ x_m y_m)) 0.5))))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (z_m * (z_m / y_m)) * -0.5;
	} else if ((t_0 <= 5e+153) || !(t_0 <= ((double) INFINITY))) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * (x_m / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (z_m * (z_m / y_m)) * -0.5;
	} else if ((t_0 <= 5e+153) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * (x_m / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = (z_m * (z_m / y_m)) * -0.5
	elif (t_0 <= 5e+153) or not (t_0 <= math.inf):
		tmp = 0.5 * y_m
	else:
		tmp = (x_m * (x_m / y_m)) * 0.5
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(z_m * Float64(z_m / y_m)) * -0.5);
	elseif ((t_0 <= 5e+153) || !(t_0 <= Inf))
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x_m * Float64(x_m / y_m)) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (z_m * (z_m / y_m)) * -0.5;
	elseif ((t_0 <= 5e+153) || ~((t_0 <= Inf)))
		tmp = 0.5 * y_m;
	else
		tmp = (x_m * (x_m / y_m)) * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(z$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], If[Or[LessEqual[t$95$0, 5e+153], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x$95$m * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $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 81.1%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 84.0%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 29.1

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

   8. Applied rewrites 29.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 31.3

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

   10. Applied rewrites 31.3%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.00000000000000018e153 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 50.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 48.6

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

   5. Applied rewrites 48.6%

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

   if 5.00000000000000018e153 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 66.3%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 40.7

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

   8. Applied rewrites 40.7%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 0:\\ \;\;\;\;\left(z \cdot \frac{z}{y}\right) \cdot -0.5\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 5 \cdot 10^{+153} \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-103}:\\ \;\;\;\;-0.5 \cdot \frac{z\_m \cdot z\_m}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+153} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot \frac{x\_m}{y\_m}\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 -5e-103)
      (* -0.5 (/ (* z_m z_m) y_m))
      (if (or (<= t_0 5e+153) (not (<= t_0 INFINITY)))
        (* 0.5 y_m)
        (* (* x_m (/ x_m y_m)) 0.5))))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -5e-103) {
		tmp = -0.5 * ((z_m * z_m) / y_m);
	} else if ((t_0 <= 5e+153) || !(t_0 <= ((double) INFINITY))) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * (x_m / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -5e-103) {
		tmp = -0.5 * ((z_m * z_m) / y_m);
	} else if ((t_0 <= 5e+153) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * (x_m / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= -5e-103:
		tmp = -0.5 * ((z_m * z_m) / y_m)
	elif (t_0 <= 5e+153) or not (t_0 <= math.inf):
		tmp = 0.5 * y_m
	else:
		tmp = (x_m * (x_m / y_m)) * 0.5
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-103)
		tmp = Float64(-0.5 * Float64(Float64(z_m * z_m) / y_m));
	elseif ((t_0 <= 5e+153) || !(t_0 <= Inf))
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x_m * Float64(x_m / y_m)) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= -5e-103)
		tmp = -0.5 * ((z_m * z_m) / y_m);
	elseif ((t_0 <= 5e+153) || ~((t_0 <= Inf)))
		tmp = 0.5 * y_m;
	else
		tmp = (x_m * (x_m / y_m)) * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -5e-103], N[(-0.5 * N[(N[(z$95$m * z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 5e+153], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x$95$m * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $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))) < -4.99999999999999966e-103

   1. Initial program 83.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 30.1

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

   5. Applied rewrites 30.1%

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

   if -4.99999999999999966e-103 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.00000000000000018e153 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 49.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 48.2

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

   5. Applied rewrites 48.2%

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

   if 5.00000000000000018e153 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 66.3%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 40.7

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

   8. Applied rewrites 40.7%

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

4. Final simplification 38.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-103}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 5 \cdot 10^{+153} \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{x}{y}\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-103}:\\ \;\;\;\;-0.5 \cdot \frac{z\_m \cdot z\_m}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+153} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 -5e-103)
      (* -0.5 (/ (* z_m z_m) y_m))
      (if (or (<= t_0 5e+153) (not (<= t_0 INFINITY)))
        (* 0.5 y_m)
        (/ (* x_m x_m) (+ y_m y_m)))))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -5e-103) {
		tmp = -0.5 * ((z_m * z_m) / y_m);
	} else if ((t_0 <= 5e+153) || !(t_0 <= ((double) INFINITY))) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * x_m) / (y_m + y_m);
	}
	return y_s * tmp;
}

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -5e-103) {
		tmp = -0.5 * ((z_m * z_m) / y_m);
	} else if ((t_0 <= 5e+153) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * x_m) / (y_m + y_m);
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= -5e-103:
		tmp = -0.5 * ((z_m * z_m) / y_m)
	elif (t_0 <= 5e+153) or not (t_0 <= math.inf):
		tmp = 0.5 * y_m
	else:
		tmp = (x_m * x_m) / (y_m + y_m)
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-103)
		tmp = Float64(-0.5 * Float64(Float64(z_m * z_m) / y_m));
	elseif ((t_0 <= 5e+153) || !(t_0 <= Inf))
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x_m * x_m) / Float64(y_m + y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= -5e-103)
		tmp = -0.5 * ((z_m * z_m) / y_m);
	elseif ((t_0 <= 5e+153) || ~((t_0 <= Inf)))
		tmp = 0.5 * y_m;
	else
		tmp = (x_m * x_m) / (y_m + y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -5e-103], N[(-0.5 * N[(N[(z$95$m * z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 5e+153], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y$95$m + y$95$m), $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))) < -4.99999999999999966e-103

   1. Initial program 83.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 30.1

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

   5. Applied rewrites 30.1%

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

   if -4.99999999999999966e-103 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.00000000000000018e153 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 49.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 48.2

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

   5. Applied rewrites 48.2%

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

   if 5.00000000000000018e153 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 66.3%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 34.3

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

   5. Applied rewrites 34.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 34.3

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

   7. Applied rewrites 34.3%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-103}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 5 \cdot 10^{+153} \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.3% accurate, 0.3× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (or (<= t_0 0.0) (not (<= t_0 INFINITY)))
      (* (- y_m (* (/ z_m y_m) z_m)) 0.5)
      (* (fma (/ x_m y_m) x_m y_m) 0.5)))))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(y$95$m - N[(N[(z$95$m / y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] * x$95$m + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 64.0%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 59.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 69.3

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

   7. Applied rewrites 69.3%

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

   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 76.9%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 73.2%

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

4. Final simplification 70.9%

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

Alternative 6: 93.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2} \leq -5 \cdot 10^{-103}:\\ \;\;\;\;\left(z\_m \cdot \frac{z\_m}{y\_m}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{y\_m}, x\_m, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0)) -5e-103)
    (* (* z_m (/ z_m y_m)) -0.5)
    (* (fma (/ x_m y_m) x_m y_m) 0.5))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (((((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)) <= -5e-103) {
		tmp = (z_m * (z_m / y_m)) * -0.5;
	} else {
		tmp = fma((x_m / y_m), x_m, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0)) <= -5e-103)
		tmp = Float64(Float64(z_m * Float64(z_m / y_m)) * -0.5);
	else
		tmp = Float64(fma(Float64(x_m / y_m), x_m, y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -5e-103], N[(N[(z$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] * x$95$m + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.99999999999999966e-103

   1. Initial program 83.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 86.8%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 30.1

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

   8. Applied rewrites 30.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lower-*.f64 31.7

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

   10. Applied rewrites 31.7%

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

   if -4.99999999999999966e-103 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 58.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} + {y}^{2}}{y}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 70.9%

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

Alternative 7: 50.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 260000000000:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (* y_s (if (<= x_m 260000000000.0) (* 0.5 y_m) (/ (* x_m x_m) (+ y_m y_m)))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 260000000000.0) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * x_m) / (y_m + y_m);
	}
	return y_s * tmp;
}

Fortran

x_m =     private
z_m =     private
y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(y_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 260000000000.0d0) then
        tmp = 0.5d0 * y_m
    else
        tmp = (x_m * x_m) / (y_m + y_m)
    end if
    code = y_s * tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 260000000000.0) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * x_m) / (y_m + y_m);
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	tmp = 0
	if x_m <= 260000000000.0:
		tmp = 0.5 * y_m
	else:
		tmp = (x_m * x_m) / (y_m + y_m)
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 260000000000.0)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x_m * x_m) / Float64(y_m + y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (x_m <= 260000000000.0)
		tmp = 0.5 * y_m;
	else
		tmp = (x_m * x_m) / (y_m + y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[x$95$m, 260000000000.0], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.6e11

   1. Initial program 70.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 42.4

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

   5. Applied rewrites 42.4%

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

   if 2.6e11 < x

   1. Initial program 63.2%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 54.6

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

   7. Applied rewrites 54.6%

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

Alternative 8: 34.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

x_m =     private
z_m =     private
y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(y_s, x_m, y_m, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    code = y_s * (0.5d0 * y_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
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$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(0.5 * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.2%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 37.4

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

5. Applied rewrites 37.4%

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

Developer Target 1: 99.9% accurate, 1.1× 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    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 2025026 
(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)))