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

Percentage Accurate: 69.2% → 98.6%

Time: 4.4s

Alternatives: 14

Speedup: 0.6×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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: 69.2% 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: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;z\_m \leq 5.8 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(z\_m + y, \frac{y - z\_m}{2 \cdot y}, \frac{x}{2} \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z\_m + x, \frac{\left(x - z\_m\right) \cdot \frac{-0.5}{y}}{y}, -0.5\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= z_m 5.8e+133)
   (fma (+ z_m y) (/ (- y z_m) (* 2.0 y)) (* (/ x 2.0) (/ x y)))
   (* (fma (+ z_m x) (/ (* (- x z_m) (/ -0.5 y)) y) -0.5) (- y))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (z_m <= 5.8e+133) {
		tmp = fma((z_m + y), ((y - z_m) / (2.0 * y)), ((x / 2.0) * (x / y)));
	} else {
		tmp = fma((z_m + x), (((x - z_m) * (-0.5 / y)) / y), -0.5) * -y;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (z_m <= 5.8e+133)
		tmp = fma(Float64(z_m + y), Float64(Float64(y - z_m) / Float64(2.0 * y)), Float64(Float64(x / 2.0) * Float64(x / y)));
	else
		tmp = Float64(fma(Float64(z_m + x), Float64(Float64(Float64(x - z_m) * Float64(-0.5 / y)) / y), -0.5) * Float64(-y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[z$95$m, 5.8e+133], N[(N[(z$95$m + y), $MachinePrecision] * N[(N[(y - z$95$m), $MachinePrecision] / N[(2.0 * y), $MachinePrecision]), $MachinePrecision] + N[(N[(x / 2.0), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m + x), $MachinePrecision] * N[(N[(N[(x - z$95$m), $MachinePrecision] * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + -0.5), $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.8000000000000002e133

   1. Initial program 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. div-sub N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. sub-div N/A

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

      13. associate--l+ N/A

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

      14. div-add N/A

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

      15. lower-+.f64 N/A

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

   4. Applied rewrites 65.1%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. times-frac N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lower-/.f64 93.0

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

   6. Applied rewrites 93.0%

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

   if 5.8000000000000002e133 < z

   1. Initial program 48.7%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 75.5%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-/.f64 99.8

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

   9. Applied rewrites 99.8%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift--.f64 N/A

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

       9. lift-/.f64 99.9

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

   11. Applied rewrites 99.9%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.8 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(z + y, \frac{y - z}{2 \cdot y}, \frac{x}{2} \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + x, \frac{\left(x - z\right) \cdot \frac{-0.5}{y}}{y}, -0.5\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 37.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(-z\_m\right) \cdot z\_m}{y + y}\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -0.0002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (* (- z_m) z_m) (+ y y)))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -0.0002)
     t_0
     (if (<= t_1 2e+152)
       (* 0.5 y)
       (if (<= t_1 INFINITY) (/ (* x x) (+ y y)) t_0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (-z_m * z_m) / (y + y);
	double t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -0.0002) {
		tmp = t_0;
	} else if (t_1 <= 2e+152) {
		tmp = 0.5 * y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (-z_m * z_m) / (y + y);
	double t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -0.0002) {
		tmp = t_0;
	} else if (t_1 <= 2e+152) {
		tmp = 0.5 * y;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (-z_m * z_m) / (y + y)
	t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_1 <= -0.0002:
		tmp = t_0
	elif t_1 <= 2e+152:
		tmp = 0.5 * y
	elif t_1 <= math.inf:
		tmp = (x * x) / (y + y)
	else:
		tmp = t_0
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(-z_m) * z_m) / Float64(y + y))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -0.0002)
		tmp = t_0;
	elseif (t_1 <= 2e+152)
		tmp = Float64(0.5 * y);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * x) / Float64(y + y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (-z_m * z_m) / (y + y);
	t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -0.0002)
		tmp = t_0;
	elseif (t_1 <= 2e+152)
		tmp = 0.5 * y;
	elseif (t_1 <= Inf)
		tmp = (x * x) / (y + y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[((-z$95$m) * z$95$m), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $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$1, -0.0002], t$95$0, If[LessEqual[t$95$1, 2e+152], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * x), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

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))) < -2.0000000000000001e-4 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 59.5%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left({z}^{2}\right)}{y \cdot 2} \]

      2. pow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(z \cdot z\right)}{y \cdot 2} \]

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 37.4

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

   5. Applied rewrites 37.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 37.4

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

   7. Applied rewrites 37.4%

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

   if -2.0000000000000001e-4 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e152

   1. Initial program 93.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 77.5

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

   5. Applied rewrites 77.5%

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

   if 2.0000000000000001e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 64.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 34.0

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

   5. Applied rewrites 34.0%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   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 34.0

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

   7. Applied rewrites 34.0%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -0.0002:\\ \;\;\;\;\frac{\left(-z\right) \cdot z}{y + y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-z\right) \cdot z}{y + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{z\_m \cdot z\_m}{y}\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq -0.0002:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (* z_m z_m) y)))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_1 -0.0002)
     t_0
     (if (<= t_1 2e+152)
       (* 0.5 y)
       (if (<= t_1 INFINITY) (/ (* x x) (+ y y)) t_0)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = -0.5 * ((z_m * z_m) / y);
	double t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -0.0002) {
		tmp = t_0;
	} else if (t_1 <= 2e+152) {
		tmp = 0.5 * y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = -0.5 * ((z_m * z_m) / y);
	double t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_1 <= -0.0002) {
		tmp = t_0;
	} else if (t_1 <= 2e+152) {
		tmp = 0.5 * y;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = -0.5 * ((z_m * z_m) / y)
	t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_1 <= -0.0002:
		tmp = t_0
	elif t_1 <= 2e+152:
		tmp = 0.5 * y
	elif t_1 <= math.inf:
		tmp = (x * x) / (y + y)
	else:
		tmp = t_0
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(-0.5 * Float64(Float64(z_m * z_m) / y))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= -0.0002)
		tmp = t_0;
	elseif (t_1 <= 2e+152)
		tmp = Float64(0.5 * y);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * x) / Float64(y + y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = -0.5 * ((z_m * z_m) / y);
	t_1 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= -0.0002)
		tmp = t_0;
	elseif (t_1 <= 2e+152)
		tmp = 0.5 * y;
	elseif (t_1 <= Inf)
		tmp = (x * x) / (y + y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(-0.5 * N[(N[(z$95$m * z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $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$1, -0.0002], t$95$0, If[LessEqual[t$95$1, 2e+152], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * x), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

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))) < -2.0000000000000001e-4 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 59.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 36.8

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

   5. Applied rewrites 36.8%

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

   if -2.0000000000000001e-4 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e152

   1. Initial program 93.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 77.5

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

   5. Applied rewrites 77.5%

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

   if 2.0000000000000001e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 64.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 34.0

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

   5. Applied rewrites 34.0%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   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 34.0

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

   7. Applied rewrites 34.0%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -0.0002:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 2 \cdot 10^{+152}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.0% accurate, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (or (<= t_0 0.0) (not (<= t_0 INFINITY)))
     (* (+ z_m x) (* (/ (- x z_m) y) 0.5))
     (* (fma x (/ x y) y) 0.5))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= ((double) INFINITY))) {
		tmp = (z_m + x) * (((x - z_m) / y) * 0.5);
	} else {
		tmp = fma(x, (x / y), y) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= Inf))
		tmp = Float64(Float64(z_m + x) * Float64(Float64(Float64(x - z_m) / y) * 0.5));
	else
		tmp = Float64(fma(x, Float64(x / y), y) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(z$95$m + x), $MachinePrecision] * N[(N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision] * 0.5), $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 59.9%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 72.1

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

   8. Applied rewrites 72.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 73.2

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

   10. Applied rewrites 73.2%

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

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

   1. Initial program 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. div-sub N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. sub-div N/A

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

      13. associate--l+ N/A

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

      14. div-add N/A

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

      15. lower-+.f64 N/A

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 67.8

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

   7. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 72.7

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

   9. Applied rewrites 72.7%

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

4. Final simplification 73.0%

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

Alternative 5: 67.8% accurate, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 0.0)
     (* (* (+ z_m y) (/ (- y z_m) y)) 0.5)
     (if (<= t_0 INFINITY)
       (* (fma x (/ x y) y) 0.5)
       (* (* (+ z_m x) (/ (- x z_m) y)) 0.5)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z_m + y) * ((y - z_m) / y)) * 0.5;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x, (x / y), y) * 0.5;
	} else {
		tmp = ((z_m + x) * ((x - z_m) / y)) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(z_m + y) * Float64(Float64(y - z_m) / y)) * 0.5);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(x, Float64(x / y), y) * 0.5);
	else
		tmp = Float64(Float64(Float64(z_m + x) * Float64(Float64(x - z_m) / y)) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $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, 0.0], N[(N[(N[(z$95$m + y), $MachinePrecision] * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(z$95$m + x), $MachinePrecision] * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 78.6%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. div-add-rev N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

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

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 67.4%

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

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

   1. Initial program 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. div-sub N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. sub-div N/A

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

      13. associate--l+ N/A

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

      14. div-add N/A

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

      15. lower-+.f64 N/A

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 67.8

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

   7. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 72.7

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

   9. Applied rewrites 72.7%

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

   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

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 79.3

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

   8. Applied rewrites 79.3%

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

4. Final simplification 71.3%

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

Alternative 6: 49.1% accurate, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (or (<= t_0 0.0) (not (<= t_0 INFINITY)))
     (* (* z_m (/ (- y z_m) y)) 0.5)
     (* (fma x (/ x y) y) 0.5))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= ((double) INFINITY))) {
		tmp = (z_m * ((y - z_m) / y)) * 0.5;
	} else {
		tmp = fma(x, (x / y), y) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if ((t_0 <= 0.0) || !(t_0 <= Inf))
		tmp = Float64(Float64(z_m * Float64(Float64(y - z_m) / y)) * 0.5);
	else
		tmp = Float64(fma(x, Float64(x / y), y) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(z$95$m * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision] * 0.5), $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 59.9%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. div-add-rev N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

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

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 68.4%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(z \cdot \frac{y - z}{y}\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.6%

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

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

   1. Initial program 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. div-sub N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. sub-div N/A

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

      13. associate--l+ N/A

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

      14. div-add N/A

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

      15. lower-+.f64 N/A

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 67.8

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

   7. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 72.7

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

   9. Applied rewrites 72.7%

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

4. Final simplification 55.4%

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

Alternative 7: 66.3% accurate, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 0.0)
     (* (* (+ z_m y) (/ (- y z_m) y)) 0.5)
     (if (<= t_0 INFINITY)
       (* (fma x (/ x y) y) 0.5)
       (* (* z_m (/ (- x z_m) y)) 0.5)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z_m + y) * ((y - z_m) / y)) * 0.5;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x, (x / y), y) * 0.5;
	} else {
		tmp = (z_m * ((x - z_m) / y)) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(z_m + y) * Float64(Float64(y - z_m) / y)) * 0.5);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(x, Float64(x / y), y) * 0.5);
	else
		tmp = Float64(Float64(z_m * Float64(Float64(x - z_m) / y)) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $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, 0.0], N[(N[(N[(z$95$m + y), $MachinePrecision] * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(z$95$m * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 78.6%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. div-add-rev N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

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

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 67.4%

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

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

   1. Initial program 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. div-sub N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. sub-div N/A

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

      13. associate--l+ N/A

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

      14. div-add N/A

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

      15. lower-+.f64 N/A

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 67.8

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

   7. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 72.7

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

   9. Applied rewrites 72.7%

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

   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

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 79.3

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

   8. Applied rewrites 79.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.7%

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

4. Final simplification 68.9%

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

Alternative 8: 50.5% accurate, 0.3× speedup

Math

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

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 0.0)
     (* (* z_m (/ (- y z_m) y)) 0.5)
     (if (<= t_0 INFINITY)
       (* (fma x (/ x y) y) 0.5)
       (* (* z_m (/ (- x z_m) y)) 0.5)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (z_m * ((y - z_m) / y)) * 0.5;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x, (x / y), y) * 0.5;
	} else {
		tmp = (z_m * ((x - z_m) / y)) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(z_m * Float64(Float64(y - z_m) / y)) * 0.5);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(x, Float64(x / y), y) * 0.5);
	else
		tmp = Float64(Float64(z_m * Float64(Float64(x - z_m) / y)) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $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, 0.0], N[(N[(z$95$m * N[(N[(y - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(z$95$m * N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 78.6%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. div-add-rev N/A

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

      10. pow2 N/A

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

      11. *-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

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

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

      15. lower-*.f64 N/A

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

   8. Applied rewrites 67.4%

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

   9. Taylor expanded in y around 0

      \[\leadsto \left(z \cdot \frac{y - z}{y}\right) \cdot \frac{1}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 37.6%

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

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

   1. Initial program 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. div-sub N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. sub-div N/A

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

      13. associate--l+ N/A

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

      14. div-add N/A

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

      15. lower-+.f64 N/A

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

   4. Applied rewrites 68.3%

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

   5. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 67.8

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

   7. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 72.7

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

   9. Applied rewrites 72.7%

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

   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

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 79.3

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

   8. Applied rewrites 79.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.7%

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

4. Final simplification 56.2%

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

Alternative 9: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -0.0002:\\ \;\;\;\;\left(z\_m + x\right) \cdot \left(\frac{x - z\_m}{y} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z\_m + x, \frac{\left(x - z\_m\right) \cdot \frac{-0.5}{y}}{y}, -0.5\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) -0.0002)
   (* (+ z_m x) (* (/ (- x z_m) y) 0.5))
   (* (fma (+ z_m x) (/ (* (- x z_m) (/ -0.5 y)) y) -0.5) (- y))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -0.0002) {
		tmp = (z_m + x) * (((x - z_m) / y) * 0.5);
	} else {
		tmp = fma((z_m + x), (((x - z_m) * (-0.5 / y)) / y), -0.5) * -y;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -0.0002)
		tmp = Float64(Float64(z_m + x) * Float64(Float64(Float64(x - z_m) / y) * 0.5));
	else
		tmp = Float64(fma(Float64(z_m + x), Float64(Float64(Float64(x - z_m) * Float64(-0.5 / y)) / y), -0.5) * Float64(-y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -0.0002], N[(N[(z$95$m + x), $MachinePrecision] * N[(N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m + x), $MachinePrecision] * N[(N[(N[(x - z$95$m), $MachinePrecision] * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + -0.5), $MachinePrecision] * (-y)), $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))) < -2.0000000000000001e-4

   1. Initial program 79.2%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 72.7

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

   8. Applied rewrites 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 73.5

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

   10. Applied rewrites 73.5%

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

   if -2.0000000000000001e-4 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 55.7%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 79.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 93.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-/.f64 94.4

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

   9. Applied rewrites 94.4%

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

       1. lift-*.f64 N/A

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

       2. lift--.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-*l/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lift--.f64 N/A

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

       9. lift-/.f64 94.4

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

   11. Applied rewrites 94.4%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -0.0002:\\ \;\;\;\;\left(z + x\right) \cdot \left(\frac{x - z}{y} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + x, \frac{\left(x - z\right) \cdot \frac{-0.5}{y}}{y}, -0.5\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{x - z\_m}{y}\\ \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -0.0002:\\ \;\;\;\;\left(z\_m + x\right) \cdot \left(t\_0 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z\_m + x, t\_0 \cdot \frac{-0.5}{y}, -0.5\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- x z_m) y)))
   (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) -0.0002)
     (* (+ z_m x) (* t_0 0.5))
     (* (fma (+ z_m x) (* t_0 (/ -0.5 y)) -0.5) (- y)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (x - z_m) / y;
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -0.0002) {
		tmp = (z_m + x) * (t_0 * 0.5);
	} else {
		tmp = fma((z_m + x), (t_0 * (-0.5 / y)), -0.5) * -y;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(x - z_m) / y)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -0.0002)
		tmp = Float64(Float64(z_m + x) * Float64(t_0 * 0.5));
	else
		tmp = Float64(fma(Float64(z_m + x), Float64(t_0 * Float64(-0.5 / y)), -0.5) * Float64(-y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -0.0002], N[(N[(z$95$m + x), $MachinePrecision] * N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m + x), $MachinePrecision] * N[(t$95$0 * N[(-0.5 / y), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * (-y)), $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))) < -2.0000000000000001e-4

   1. Initial program 79.2%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 72.7

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

   8. Applied rewrites 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 73.5

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

   10. Applied rewrites 73.5%

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

   if -2.0000000000000001e-4 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 55.7%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 79.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 93.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-/.f64 94.4

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

   9. Applied rewrites 94.4%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -0.0002:\\ \;\;\;\;\left(z + x\right) \cdot \left(\frac{x - z}{y} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + x, \frac{x - z}{y} \cdot \frac{-0.5}{y}, -0.5\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{x - z\_m}{y}\\ \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -4 \cdot 10^{+65}:\\ \;\;\;\;\left(z\_m + x\right) \cdot \left(t\_0 \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z\_m + x\right) \cdot t\_0, \frac{-0.5}{y}, -0.5\right) \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- x z_m) y)))
   (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) -4e+65)
     (* (+ z_m x) (* t_0 0.5))
     (* (fma (* (+ z_m x) t_0) (/ -0.5 y) -0.5) (- y)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (x - z_m) / y;
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -4e+65) {
		tmp = (z_m + x) * (t_0 * 0.5);
	} else {
		tmp = fma(((z_m + x) * t_0), (-0.5 / y), -0.5) * -y;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(x - z_m) / y)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -4e+65)
		tmp = Float64(Float64(z_m + x) * Float64(t_0 * 0.5));
	else
		tmp = Float64(fma(Float64(Float64(z_m + x) * t_0), Float64(-0.5 / y), -0.5) * Float64(-y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(x - z$95$m), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -4e+65], N[(N[(z$95$m + x), $MachinePrecision] * N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z$95$m + x), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(-0.5 / y), $MachinePrecision] + -0.5), $MachinePrecision] * (-y)), $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))) < -4e65

   1. Initial program 78.6%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 85.3%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 73.9

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

   8. Applied rewrites 73.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot \frac{1}{2} \]

      6. associate-*l* N/A

         \[\leadsto \left(z + x\right) \cdot \left(\frac{x - z}{y} \cdot \color{blue}{\frac{1}{2}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(z + x\right) \cdot \left(\frac{x - z}{y} \cdot \color{blue}{\frac{1}{2}}\right) \]

      8. lift-+.f64 N/A

         \[\leadsto \left(z + x\right) \cdot \left(\frac{x - z}{y} \cdot \frac{1}{2}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(z + x\right) \cdot \left(\frac{x - z}{y} \cdot \frac{1}{2}\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \left(z + x\right) \cdot \left(\frac{x - z}{y} \cdot \frac{1}{2}\right) \]

      11. lift--.f64 74.7

          \[\leadsto \left(z + x\right) \cdot \left(\frac{x - z}{y} \cdot 0.5\right) \]

   10. Applied rewrites 74.7%

       \[\leadsto \left(z + x\right) \cdot \color{blue}{\left(\frac{x - z}{y} \cdot 0.5\right)} \]

   if -4e65 < (/.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

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} - \frac{1}{2}\right)\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} - \frac{1}{2}\right)\right) \]

      2. distribute-lft-neg-in N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} - \frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} - \frac{1}{2}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \left(-y\right) \cdot \left(\color{blue}{\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}} - \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \left(-y\right) \cdot \left(\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} - \frac{1}{2} \cdot \color{blue}{1}\right) \]

      6. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(-y\right) \cdot \left(\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \left(-y\right) \cdot \left(\frac{\frac{-1}{2} \cdot \left({x}^{2} - {z}^{2}\right)}{{y}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot 1\right) \]

      8. pow2 N/A

         \[\leadsto \left(-y\right) \cdot \left(\frac{\frac{-1}{2} \cdot \left({x}^{2} - {z}^{2}\right)}{y \cdot y} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1\right) \]

      9. times-frac N/A

         \[\leadsto \left(-y\right) \cdot \left(\frac{\frac{-1}{2}}{y} \cdot \frac{{x}^{2} - {z}^{2}}{y} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot 1\right) \]

      10. metadata-eval N/A

          \[\leadsto \left(-y\right) \cdot \left(\frac{\frac{-1}{2} \cdot 1}{y} \cdot \frac{{x}^{2} - {z}^{2}}{y} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1\right) \]

      11. associate-*r/ N/A

          \[\leadsto \left(-y\right) \cdot \left(\left(\frac{-1}{2} \cdot \frac{1}{y}\right) \cdot \frac{{x}^{2} - {z}^{2}}{y} + \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right) \cdot 1\right) \]

      12. metadata-eval N/A

          \[\leadsto \left(-y\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{y}\right) \cdot \frac{{x}^{2} - {z}^{2}}{y} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot 1\right) \]

      13. metadata-eval N/A

          \[\leadsto \left(-y\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{y}\right) \cdot \frac{{x}^{2} - {z}^{2}}{y} + \frac{-1}{2} \cdot 1\right) \]

      14. metadata-eval N/A

          \[\leadsto \left(-y\right) \cdot \left(\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{y}\right) \cdot \frac{{x}^{2} - {z}^{2}}{y} + \frac{-1}{2}\right) \]

      15. lower-fma.f64 N/A

          \[\leadsto \left(-y\right) \cdot \mathsf{fma}\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{y}, \color{blue}{\frac{{x}^{2} - {z}^{2}}{y}}, \frac{-1}{2}\right) \]

   5. Applied rewrites 80.3%

      \[\leadsto \color{blue}{\left(-y\right) \cdot \mathsf{fma}\left(\frac{-0.5}{y}, \frac{\left(x + z\right) \cdot \left(x - z\right)}{y}, -0.5\right)} \]
   6. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{-1}{2}}{y}}, \frac{\left(x + z\right) \cdot \left(x - z\right)}{y}, \frac{-1}{2}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{\frac{-1}{2}}{y}, \frac{\left(x + z\right) \cdot \left(x - z\right)}{y}, \frac{-1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \mathsf{fma}\left(\frac{\frac{-1}{2}}{y}, \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}, \frac{-1}{2}\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \left(\frac{\frac{-1}{2}}{y} \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} + \color{blue}{\frac{-1}{2}}\right) \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \left(\frac{\frac{-1}{2}}{y} \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} + \frac{-1}{2}\right) \]

      6. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \left(\frac{\frac{-1}{2}}{y} \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} + \frac{-1}{2}\right) \]

      7. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \left(\frac{\frac{-1}{2}}{y} \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} + \frac{-1}{2}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(y\right)\right) \cdot \left(\frac{\frac{-1}{2}}{y} \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} + \frac{-1}{2}\right) \]

      9. *-commutative N/A

         \[\leadsto \left(\frac{\frac{-1}{2}}{y} \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} + \frac{-1}{2}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\frac{\frac{-1}{2}}{y} \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{y} + \frac{-1}{2}\right) \cdot \color{blue}{\left(\mathsf{neg}\left(y\right)\right)} \]

   7. Applied rewrites 94.0%

      \[\leadsto \mathsf{fma}\left(\left(z + x\right) \cdot \frac{x - z}{y}, \frac{-0.5}{y}, -0.5\right) \cdot \color{blue}{\left(-y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -4 \cdot 10^{+65}:\\ \;\;\;\;\left(z + x\right) \cdot \left(\frac{x - z}{y} \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z + x\right) \cdot \frac{x - z}{y}, \frac{-0.5}{y}, -0.5\right) \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -0.0002:\\ \;\;\;\;\frac{\left(-z\_m\right) \cdot z\_m}{y + y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y}, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) -0.0002)
   (/ (* (- z_m) z_m) (+ y y))
   (* (fma x (/ x y) y) 0.5)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -0.0002) {
		tmp = (-z_m * z_m) / (y + y);
	} else {
		tmp = fma(x, (x / y), y) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -0.0002)
		tmp = Float64(Float64(Float64(-z_m) * z_m) / Float64(y + y));
	else
		tmp = Float64(fma(x, Float64(x / y), y) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -0.0002], N[(N[((-z$95$m) * z$95$m), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision] * 0.5), $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))) < -2.0000000000000001e-4

   1. Initial program 79.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{-1 \cdot {z}^{2}}}{y \cdot 2} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left({z}^{2}\right)}{y \cdot 2} \]

      2. pow2 N/A

         \[\leadsto \frac{\mathsf{neg}\left(z \cdot z\right)}{y \cdot 2} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{z}}{y \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{z}}{y \cdot 2} \]

      5. lower-neg.f64 37.3

         \[\leadsto \frac{\left(-z\right) \cdot z}{y \cdot 2} \]

   5. Applied rewrites 37.3%

      \[\leadsto \frac{\color{blue}{\left(-z\right) \cdot z}}{y \cdot 2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(-z\right) \cdot z}{\color{blue}{y \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(-z\right) \cdot z}{\color{blue}{2 \cdot y}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{\left(-z\right) \cdot z}{\color{blue}{y + y}} \]

      4. lower-+.f64 37.3

         \[\leadsto \frac{\left(-z\right) \cdot z}{\color{blue}{y + y}} \]

   7. Applied rewrites 37.3%

      \[\leadsto \frac{\left(-z\right) \cdot z}{\color{blue}{y + y}} \]

   if -2.0000000000000001e-4 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 55.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{y \cdot 2}} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) - z \cdot z}}{y \cdot 2} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot x + y \cdot y\right)} - z \cdot z}{y \cdot 2} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{x \cdot x} + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + \color{blue}{y \cdot y}\right) - z \cdot z}{y \cdot 2} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{z \cdot z}}{y \cdot 2} \]

      8. pow2 N/A

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - \color{blue}{{z}^{2}}}{y \cdot 2} \]

      9. div-sub N/A

         \[\leadsto \color{blue}{\frac{x \cdot x + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2}} \]

      10. pow2 N/A

          \[\leadsto \frac{\color{blue}{{x}^{2}} + y \cdot y}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      11. pow2 N/A

          \[\leadsto \frac{{x}^{2} + \color{blue}{{y}^{2}}}{y \cdot 2} - \frac{{z}^{2}}{y \cdot 2} \]

      12. sub-div N/A

          \[\leadsto \color{blue}{\frac{\left({x}^{2} + {y}^{2}\right) - {z}^{2}}{y \cdot 2}} \]

      13. associate--l+ N/A

          \[\leadsto \frac{\color{blue}{{x}^{2} + \left({y}^{2} - {z}^{2}\right)}}{y \cdot 2} \]

      14. div-add N/A

          \[\leadsto \color{blue}{\frac{{x}^{2}}{y \cdot 2} + \frac{{y}^{2} - {z}^{2}}{y \cdot 2}} \]

      15. lower-+.f64 N/A

          \[\leadsto \color{blue}{\frac{{x}^{2}}{y \cdot 2} + \frac{{y}^{2} - {z}^{2}}{y \cdot 2}} \]

   4. Applied rewrites 56.4%

      \[\leadsto \color{blue}{\frac{x \cdot x}{2 \cdot y} + \frac{\left(y + z\right) \cdot \left(y - z\right)}{2 \cdot y}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y + \frac{1}{2} \cdot \frac{{x}^{2}}{y}} \]
   6. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(y + \frac{{x}^{2}}{y}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(y + \frac{{x}^{2}}{y}\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{{x}^{2}}{y} + \color{blue}{y}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{{x}^{2}}{y} + \color{blue}{y}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{{x}^{2}}{y} + y\right) \]

      6. pow2 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{x \cdot x}{y} + y\right) \]

      7. lift-*.f64 60.4

         \[\leadsto 0.5 \cdot \left(\frac{x \cdot x}{y} + y\right) \]

   7. Applied rewrites 60.4%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{x \cdot x}{y} + y\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{x \cdot x}{y} + y\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{x \cdot x}{y} + \color{blue}{y}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{x \cdot x}{y} + y\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\frac{x \cdot x}{y} + y\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\frac{x \cdot x}{y} + y\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\frac{x \cdot x}{y} + y\right) \cdot \color{blue}{\frac{1}{2}} \]

      7. associate-/l* N/A

         \[\leadsto \left(x \cdot \frac{x}{y} + y\right) \cdot \frac{1}{2} \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \frac{x}{y}, y\right) \cdot \frac{1}{2} \]

      9. lift-/.f64 65.8

         \[\leadsto \mathsf{fma}\left(x, \frac{x}{y}, y\right) \cdot 0.5 \]

   9. Applied rewrites 65.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{x}{y}, y\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 42.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= x 5.6e+23) (* 0.5 y) (/ (* x x) (+ y y))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (x <= 5.6e+23) {
		tmp = 0.5 * y;
	} else {
		tmp = (x * x) / (y + y);
	}
	return tmp;
}

Fortran

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, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 5.6d+23) then
        tmp = 0.5d0 * y
    else
        tmp = (x * x) / (y + y)
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (x <= 5.6e+23) {
		tmp = 0.5 * y;
	} else {
		tmp = (x * x) / (y + y);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if x <= 5.6e+23:
		tmp = 0.5 * y
	else:
		tmp = (x * x) / (y + y)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (x <= 5.6e+23)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x * x) / Float64(y + y));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (x <= 5.6e+23)
		tmp = 0.5 * y;
	else
		tmp = (x * x) / (y + y);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[x, 5.6e+23], N[(0.5 * y), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.6e23

   1. Initial program 65.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 40.4

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   5. Applied rewrites 40.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]

   if 5.6e23 < x

   1. Initial program 63.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 54.8

         \[\leadsto \frac{x \cdot \color{blue}{x}}{y \cdot 2} \]

   5. Applied rewrites 54.8%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   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 54.8

         \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

   7. Applied rewrites 54.8%

      \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 34.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ 0.5 \cdot y \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m) :precision binary64 (* 0.5 y))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	return 0.5 * y;
}

Fortran

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, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = 0.5d0 * y
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	return 0.5 * y;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	return 0.5 * y

Julia

z_m = abs(z)
function code(x, y, z_m)
	return Float64(0.5 * y)
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m)
	tmp = 0.5 * y;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 65.2%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 33.7

      \[\leadsto 0.5 \cdot \color{blue}{y} \]

5. Applied rewrites 33.7%

   \[\leadsto \color{blue}{0.5 \cdot y} \]

6. Final simplification 33.7%

   \[\leadsto 0.5 \cdot y \]
7. 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 2025064 
(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)))