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

Percentage Accurate: 69.5% → 99.8%

Time: 3.8s

Alternatives: 6

Speedup: 1.1×

• 40.9% 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: 69.5% 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

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: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \mathsf{fma}\left(y, 0.5, \left(\frac{x\_m - z\_m}{y} \cdot \left(z\_m + x\_m\right)\right) \cdot 0.5\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (fma y 0.5 (* (* (/ (- x_m z_m) y) (+ z_m x_m)) 0.5)))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	return fma(y, 0.5, ((((x_m - z_m) / y) * (z_m + x_m)) * 0.5));
}

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	return fma(y, 0.5, Float64(Float64(Float64(Float64(x_m - z_m) / y) * Float64(z_m + x_m)) * 0.5))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := N[(y * 0.5 + N[(N[(N[(N[(x$95$m - z$95$m), $MachinePrecision] / y), $MachinePrecision] * N[(z$95$m + x$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 82.5%

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

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l* N/A

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

   6. sub-div N/A

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

   7. lower-*.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. sub-div N/A

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

   11. lower-/.f64 N/A

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

   12. lift--.f64 94.8

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

7. Applied rewrites 94.8%

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

9. Applied rewrites 93.2%

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

10. Taylor expanded in y around 0

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

    1. *-commutative N/A

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

    2. associate-/l* N/A

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

    3. +-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. lift-/.f64 N/A

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

    8. lift--.f64 N/A

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

    9. lift-+.f64 99.9

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

12. Applied rewrites 99.9%

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

Alternative 2: 35.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-9}:\\ \;\;\;\;-0.5 \cdot \frac{z\_m \cdot z\_m}{y}\\ \mathbf{elif}\;t\_0 \leq 10^{+152} \lor \neg \left(t\_0 \leq 10^{+262}\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y + y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -2e-9)
     (* -0.5 (/ (* z_m z_m) y))
     (if (or (<= t_0 1e+152) (not (<= t_0 1e+262)))
       (* 0.5 y)
       (/ (* x_m x_m) (+ y y))))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double t_0 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -2e-9) {
		tmp = -0.5 * ((z_m * z_m) / y);
	} else if ((t_0 <= 1e+152) || !(t_0 <= 1e+262)) {
		tmp = 0.5 * y;
	} else {
		tmp = (x_m * x_m) / (y + y);
	}
	return tmp;
}

Fortran

x_m =     private
z_m =     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(x_m, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)
    if (t_0 <= (-2d-9)) then
        tmp = (-0.5d0) * ((z_m * z_m) / y)
    else if ((t_0 <= 1d+152) .or. (.not. (t_0 <= 1d+262))) then
        tmp = 0.5d0 * y
    else
        tmp = (x_m * x_m) / (y + y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double t_0 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -2e-9) {
		tmp = -0.5 * ((z_m * z_m) / y);
	} else if ((t_0 <= 1e+152) || !(t_0 <= 1e+262)) {
		tmp = 0.5 * y;
	} else {
		tmp = (x_m * x_m) / (y + y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	t_0 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_0 <= -2e-9:
		tmp = -0.5 * ((z_m * z_m) / y)
	elif (t_0 <= 1e+152) or not (t_0 <= 1e+262):
		tmp = 0.5 * y
	else:
		tmp = (x_m * x_m) / (y + y)
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -2e-9)
		tmp = Float64(-0.5 * Float64(Float64(z_m * z_m) / y));
	elseif ((t_0 <= 1e+152) || !(t_0 <= 1e+262))
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x_m * x_m) / Float64(y + y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	t_0 = (((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -2e-9)
		tmp = -0.5 * ((z_m * z_m) / y);
	elseif ((t_0 <= 1e+152) || ~((t_0 <= 1e+262)))
		tmp = 0.5 * y;
	else
		tmp = (x_m * x_m) / (y + y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-9], N[(-0.5 * N[(N[(z$95$m * z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$0, 1e+152], N[Not[LessEqual[t$95$0, 1e+262]], $MachinePrecision]], N[(0.5 * y), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y + y), $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))) < -2.00000000000000012e-9

   1. Initial program 76.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. pow2 N/A

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

      4. lift-*.f64 32.3

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

   5. Applied rewrites 32.3%

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

   if -2.00000000000000012e-9 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1e152 or 1e262 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 56.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 47.9

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

   5. Applied rewrites 47.9%

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

   if 1e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1e262

   1. Initial program 99.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. pow2 N/A

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

      2. lift-*.f64 40.8

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

   5. Applied rewrites 40.8%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   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 40.8

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

   7. Applied rewrites 40.8%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -2 \cdot 10^{-9}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 10^{+152} \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 10^{+262}\right):\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq 0:\\ \;\;\;\;\left(\left(z\_m + x\_m\right) \cdot \frac{x\_m - z\_m}{y}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x\_m \cdot x\_m}{y} + y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0)) 0.0)
   (* (* (+ z_m x_m) (/ (- x_m z_m) y)) 0.5)
   (* 0.5 (+ (/ (* x_m x_m) y) y))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= 0.0) {
		tmp = ((z_m + x_m) * ((x_m - z_m) / y)) * 0.5;
	} else {
		tmp = 0.5 * (((x_m * x_m) / y) + y);
	}
	return tmp;
}

Fortran

x_m =     private
z_m =     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(x_m, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)) <= 0.0d0) then
        tmp = ((z_m + x_m) * ((x_m - z_m) / y)) * 0.5d0
    else
        tmp = 0.5d0 * (((x_m * x_m) / y) + y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= 0.0) {
		tmp = ((z_m + x_m) * ((x_m - z_m) / y)) * 0.5;
	} else {
		tmp = 0.5 * (((x_m * x_m) / y) + y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if ((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= 0.0:
		tmp = ((z_m + x_m) * ((x_m - z_m) / y)) * 0.5
	else:
		tmp = 0.5 * (((x_m * x_m) / y) + y)
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= 0.0)
		tmp = Float64(Float64(Float64(z_m + x_m) * Float64(Float64(x_m - z_m) / y)) * 0.5);
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(x_m * x_m) / y) + y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= 0.0)
		tmp = ((z_m + x_m) * ((x_m - z_m) / y)) * 0.5;
	else
		tmp = 0.5 * (((x_m * x_m) / y) + y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(z$95$m + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / y), $MachinePrecision] + y), $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

   1. Initial program 74.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. difference-of-squares-rev N/A

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 63.9

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

   8. Applied rewrites 63.9%

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

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

   1. Initial program 60.3%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 80.5%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. sub-div N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 94.4

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

   7. Applied rewrites 94.4%

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

   9. Applied rewrites 93.0%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 64.2%

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

Alternative 4: 49.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -2 \cdot 10^{-9}:\\ \;\;\;\;-0.5 \cdot \frac{z\_m \cdot z\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x\_m \cdot x\_m}{y} + y\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x_m x_m) (* y y)) (* z_m z_m)) (* y 2.0)) -2e-9)
   (* -0.5 (/ (* z_m z_m) y))
   (* 0.5 (+ (/ (* x_m x_m) y) y))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -2e-9) {
		tmp = -0.5 * ((z_m * z_m) / y);
	} else {
		tmp = 0.5 * (((x_m * x_m) / y) + y);
	}
	return tmp;
}

Fortran

x_m =     private
z_m =     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(x_m, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)) <= (-2d-9)) then
        tmp = (-0.5d0) * ((z_m * z_m) / y)
    else
        tmp = 0.5d0 * (((x_m * x_m) / y) + y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -2e-9) {
		tmp = -0.5 * ((z_m * z_m) / y);
	} else {
		tmp = 0.5 * (((x_m * x_m) / y) + y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if ((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -2e-9:
		tmp = -0.5 * ((z_m * z_m) / y)
	else:
		tmp = 0.5 * (((x_m * x_m) / y) + y)
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -2e-9)
		tmp = Float64(-0.5 * Float64(Float64(z_m * z_m) / y));
	else
		tmp = Float64(0.5 * Float64(Float64(Float64(x_m * x_m) / y) + y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (((((x_m * x_m) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -2e-9)
		tmp = -0.5 * ((z_m * z_m) / y);
	else
		tmp = 0.5 * (((x_m * x_m) / y) + y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -2e-9], N[(-0.5 * N[(N[(z$95$m * z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / y), $MachinePrecision] + y), $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))) < -2.00000000000000012e-9

   1. Initial program 76.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. pow2 N/A

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

      4. lift-*.f64 32.3

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

   5. Applied rewrites 32.3%

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

   if -2.00000000000000012e-9 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 59.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. sub-div N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. sub-div N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 94.6

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

   7. Applied rewrites 94.6%

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

   9. Applied rewrites 92.0%

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

   10. Taylor expanded in z around 0

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

   12. Applied rewrites 64.3%

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

Alternative 5: 50.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{+89}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y + y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m)
 :precision binary64
 (if (<= x_m 4e+89) (* 0.5 y) (/ (* x_m x_m) (+ y y))))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	double tmp;
	if (x_m <= 4e+89) {
		tmp = 0.5 * y;
	} else {
		tmp = (x_m * x_m) / (y + y);
	}
	return tmp;
}

Fortran

x_m =     private
z_m =     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(x_m, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x_m <= 4d+89) then
        tmp = 0.5d0 * y
    else
        tmp = (x_m * x_m) / (y + y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	double tmp;
	if (x_m <= 4e+89) {
		tmp = 0.5 * y;
	} else {
		tmp = (x_m * x_m) / (y + y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	tmp = 0
	if x_m <= 4e+89:
		tmp = 0.5 * y
	else:
		tmp = (x_m * x_m) / (y + y)
	return tmp

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	tmp = 0.0
	if (x_m <= 4e+89)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x_m * x_m) / Float64(y + y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp_2 = code(x_m, y, z_m)
	tmp = 0.0;
	if (x_m <= 4e+89)
		tmp = 0.5 * y;
	else
		tmp = (x_m * x_m) / (y + y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
code[x$95$m_, y_, z$95$m_] := If[LessEqual[x$95$m, 4e+89], N[(0.5 * y), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.99999999999999998e89

   1. Initial program 65.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 46.6

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

   5. Applied rewrites 46.6%

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

   if 3.99999999999999998e89 < x

   1. Initial program 73.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. pow2 N/A

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

      2. lift-*.f64 63.1

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

   5. Applied rewrites 63.1%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   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 63.1

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

   7. Applied rewrites 63.1%

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

Alternative 6: 34.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ 0.5 \cdot y \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
(FPCore (x_m y z_m) :precision binary64 (* 0.5 y))

C

x_m = fabs(x);
z_m = fabs(z);
double code(double x_m, double y, double z_m) {
	return 0.5 * y;
}

Fortran

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

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
public static double code(double x_m, double y, double z_m) {
	return 0.5 * y;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
def code(x_m, y, z_m):
	return 0.5 * y

Julia

x_m = abs(x)
z_m = abs(z)
function code(x_m, y, z_m)
	return Float64(0.5 * y)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
function tmp = code(x_m, y, z_m)
	tmp = 0.5 * y;
end

Wolfram

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

Derivation

1. Initial program 66.7%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 40.4

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

5. Applied rewrites 40.4%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
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 2025051 
(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)))