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

Percentage Accurate: 68.0% → 96.0%

Time: 2.4s

Alternatives: 4

Speedup: 0.7×

• 41.0% 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: 68.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y\_m + z\_m\right) \cdot \left(y\_m - z\_m\right)\right)}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m - z\_m, \frac{z\_m + y\_m}{y\_m + y\_m}, \frac{x}{y\_m + y\_m} \cdot x\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= y_m 5e-27)
    (/ (fma x x (* (+ y_m z_m) (- y_m z_m))) (* y_m 2.0))
    (fma (- y_m z_m) (/ (+ z_m y_m) (+ y_m y_m)) (* (/ x (+ y_m y_m)) x)))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double tmp;
	if (y_m <= 5e-27) {
		tmp = fma(x, x, ((y_m + z_m) * (y_m - z_m))) / (y_m * 2.0);
	} else {
		tmp = fma((y_m - z_m), ((z_m + y_m) / (y_m + y_m)), ((x / (y_m + y_m)) * x));
	}
	return y_s * tmp;
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	tmp = 0.0
	if (y_m <= 5e-27)
		tmp = Float64(fma(x, x, Float64(Float64(y_m + z_m) * Float64(y_m - z_m))) / Float64(y_m * 2.0));
	else
		tmp = fma(Float64(y_m - z_m), Float64(Float64(z_m + y_m) / Float64(y_m + y_m)), Float64(Float64(x / Float64(y_m + y_m)) * x));
	end
	return Float64(y_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[y$95$m, 5e-27], N[(N[(x * x + N[(N[(y$95$m + z$95$m), $MachinePrecision] * N[(y$95$m - z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m - z$95$m), $MachinePrecision] * N[(N[(z$95$m + y$95$m), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(x / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000002e-27

   1. Initial program 68.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 72.5

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

   3. Applied rewrites 72.5%

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

   if 5.0000000000000002e-27 < y

   1. Initial program 68.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 72.5

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

   3. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-/.f64 69.8

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 69.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 69.8

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

   5. Applied rewrites 69.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.5

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.5

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

   7. Applied rewrites 93.5%

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

Alternative 2: 95.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := 0.5 \cdot y\_m - z\_m \cdot \frac{z\_m}{y\_m + y\_m}\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y\_m - z\_m, 0.5, \frac{x}{y\_m + y\_m} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z_m)
 :precision binary64
 (let* ((t_0 (- (* 0.5 y_m) (* z_m (/ z_m (+ y_m y_m)))))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -1e-113)
      t_0
      (if (<= t_1 INFINITY)
        (fma (- y_m z_m) 0.5 (* (/ x (+ y_m y_m)) x))
        t_0)))))

C

z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z_m) {
	double t_0 = (0.5 * y_m) - (z_m * (z_m / (y_m + y_m)));
	double t_1 = (((x * x) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -1e-113) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((y_m - z_m), 0.5, ((x / (y_m + y_m)) * x));
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Julia

z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z_m)
	t_0 = Float64(Float64(0.5 * y_m) - Float64(z_m * Float64(z_m / Float64(y_m + y_m))))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= -1e-113)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = fma(Float64(y_m - z_m), 0.5, Float64(Float64(x / Float64(y_m + y_m)) * x));
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(0.5 * y$95$m), $MachinePrecision] - N[(z$95$m * N[(z$95$m / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, -1e-113], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(y$95$m - z$95$m), $MachinePrecision] * 0.5 + N[(N[(x / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$0]]), $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))) < -9.99999999999999979e-114 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 66.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 66.7

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

   3. Applied rewrites 66.7%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 68.0

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

   6. Applied rewrites 68.0%

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

   if -9.99999999999999979e-114 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 68.0%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. difference-of-squares N/A

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

      9. lower-*.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 72.5

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

   3. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. count-2 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2 N/A

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

      16. lift-+.f64 N/A

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

      17. lower-/.f64 69.8

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 69.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 69.8

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

   5. Applied rewrites 69.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.5

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 93.5

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

   7. Applied rewrites 93.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 67.1%

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

Alternative 3: 68.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. lower--.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. count-2-rev N/A

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

   13. lower-+.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-/l* N/A

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

   16. lower-*.f64 N/A

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

   17. lower-/.f64 66.7

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. count-2-rev N/A

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

   21. lower-+.f64 66.7

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

3. Applied rewrites 66.7%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 68.0

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

6. Applied rewrites 68.0%

   \[\leadsto \color{blue}{0.5 \cdot y} - z \cdot \frac{z}{y + y} \]
7. Add Preprocessing

Alternative 4: 34.7% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.0%

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

2. Taylor expanded in y around inf

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

   1. lower-*.f64 34.7

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

4. Applied rewrites 34.7%

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

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64
  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))