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

Percentage Accurate: 69.1% → 98.7%

Time: 3.8s

Alternatives: 12

Speedup: 1.3×

• 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: 69.1% 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: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-77}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z + x\right) \cdot \frac{\frac{x - z}{y\_m}}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 5e-77)
      (* 0.5 (- y_m (* z (/ z y_m))))
      (if (<= t_0 2e+300)
        t_0
        (* (fma (* (+ z x) (/ (/ (- x z) y_m) y_m)) 0.5 0.5) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 5e-77) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else if (t_0 <= 2e+300) {
		tmp = t_0;
	} else {
		tmp = fma(((z + x) * (((x - z) / y_m) / y_m)), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 5e-77)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	elseif (t_0 <= 2e+300)
		tmp = t_0;
	else
		tmp = Float64(fma(Float64(Float64(z + x) * Float64(Float64(Float64(x - z) / y_m) / y_m)), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 5e-77], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+300], t$95$0, N[(N[(N[(N[(z + x), $MachinePrecision] * N[(N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.99999999999999963e-77

   1. Initial program 91.2%

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

      1. lift-*.f64 N/A

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

      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

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

      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

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

      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. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   3. Applied rewrites 89.6%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 91.5

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

   6. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if 4.99999999999999963e-77 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e300

   1. Initial program 99.7%

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

   if 2.0000000000000001e300 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 41.2%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 73.9

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

   4. Applied rewrites 73.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 87.2

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

   6. Applied rewrites 87.2%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 99.6

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

   8. Applied rewrites 99.6%

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

Alternative 2: 94.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-77}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{x - z}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{\frac{x - z}{y\_m}}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 5e-77)
      (* 0.5 (- y_m (* z (/ z y_m))))
      (if (<= t_0 2e+300)
        t_0
        (if (<= t_0 INFINITY)
          (* (fma (* x (/ (- x z) (* y_m y_m))) 0.5 0.5) y_m)
          (* (fma (* z (/ (/ (- x z) y_m) y_m)) 0.5 0.5) y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 5e-77) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else if (t_0 <= 2e+300) {
		tmp = t_0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x * ((x - z) / (y_m * y_m))), 0.5, 0.5) * y_m;
	} else {
		tmp = fma((z * (((x - z) / y_m) / y_m)), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 5e-77)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	elseif (t_0 <= 2e+300)
		tmp = t_0;
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x * Float64(Float64(x - z) / Float64(y_m * y_m))), 0.5, 0.5) * y_m);
	else
		tmp = Float64(fma(Float64(z * Float64(Float64(Float64(x - z) / y_m) / y_m)), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 5e-77], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+300], t$95$0, If[LessEqual[t$95$0, Infinity], N[(N[(N[(x * N[(N[(x - z), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(z * N[(N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.99999999999999963e-77

   1. Initial program 91.2%

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

      1. lift-*.f64 N/A

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

      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

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

      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

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

      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. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   3. Applied rewrites 89.6%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 91.5

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

   6. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if 4.99999999999999963e-77 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e300

   1. Initial program 99.7%

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

   if 2.0000000000000001e300 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 53.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 93.3%

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

   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. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 39.0

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

   4. Applied rewrites 39.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 67.5

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

   6. Applied rewrites 67.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.7%

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

Alternative 3: 94.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-77}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, y\_m, x \cdot x\right)}{y\_m + y\_m}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{x - z}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{\frac{x - z}{y\_m}}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 5e-77)
      (* 0.5 (- y_m (* z (/ z y_m))))
      (if (<= t_0 2e+300)
        (/ (fma y_m y_m (* x x)) (+ y_m y_m))
        (if (<= t_0 INFINITY)
          (* (fma (* x (/ (- x z) (* y_m y_m))) 0.5 0.5) y_m)
          (* (fma (* z (/ (/ (- x z) y_m) y_m)) 0.5 0.5) y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 5e-77) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else if (t_0 <= 2e+300) {
		tmp = fma(y_m, y_m, (x * x)) / (y_m + y_m);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x * ((x - z) / (y_m * y_m))), 0.5, 0.5) * y_m;
	} else {
		tmp = fma((z * (((x - z) / y_m) / y_m)), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 5e-77)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	elseif (t_0 <= 2e+300)
		tmp = Float64(fma(y_m, y_m, Float64(x * x)) / Float64(y_m + y_m));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x * Float64(Float64(x - z) / Float64(y_m * y_m))), 0.5, 0.5) * y_m);
	else
		tmp = Float64(fma(Float64(z * Float64(Float64(Float64(x - z) / y_m) / y_m)), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 5e-77], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+300], N[(N[(y$95$m * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x * N[(N[(x - z), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(z * N[(N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.99999999999999963e-77

   1. Initial program 91.2%

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

      1. lift-*.f64 N/A

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

      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

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

      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

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

      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. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   3. Applied rewrites 89.6%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 91.5

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

   6. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if 4.99999999999999963e-77 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e300

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 98.6

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

   6. Applied rewrites 98.6%

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

   if 2.0000000000000001e300 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 53.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 93.3%

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

   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. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 39.0

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

   4. Applied rewrites 39.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 67.5

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

   6. Applied rewrites 67.5%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.7%

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

Alternative 4: 92.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-77}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, y\_m, x \cdot x\right)}{y\_m + y\_m}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{x}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 5e-77)
      (* 0.5 (- y_m (* z (/ z y_m))))
      (if (<= t_0 2e+300)
        (/ (fma y_m y_m (* x x)) (+ y_m y_m))
        (if (<= t_0 INFINITY)
          (* (fma (* x (/ x (* y_m y_m))) 0.5 0.5) y_m)
          (* (* (+ z x) (/ (- x z) y_m)) 0.5)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 5e-77) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else if (t_0 <= 2e+300) {
		tmp = fma(y_m, y_m, (x * x)) / (y_m + y_m);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x * (x / (y_m * y_m))), 0.5, 0.5) * y_m;
	} else {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 5e-77)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	elseif (t_0 <= 2e+300)
		tmp = Float64(fma(y_m, y_m, Float64(x * x)) / Float64(y_m + y_m));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x * Float64(x / Float64(y_m * y_m))), 0.5, 0.5) * y_m);
	else
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 5e-77], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+300], N[(N[(y$95$m * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x * N[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.99999999999999963e-77

   1. Initial program 91.2%

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

      1. lift-*.f64 N/A

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

      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

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

      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

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

      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. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   3. Applied rewrites 89.6%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 91.5

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

   6. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if 4.99999999999999963e-77 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e300

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 98.6

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

   6. Applied rewrites 98.6%

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

   if 2.0000000000000001e300 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 53.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 93.3%

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

   10. Taylor expanded in x around inf

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

   12. Applied rewrites 93.3%

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

   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. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 39.0

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

   4. Applied rewrites 39.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   6. 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 70.1

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

   7. Applied rewrites 70.1%

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

Alternative 5: 92.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-77}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, y\_m, x \cdot x\right)}{y\_m + y\_m}\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{x - z}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 5e-77)
      (* 0.5 (- y_m (* z (/ z y_m))))
      (if (<= t_0 2e+300)
        (/ (fma y_m y_m (* x x)) (+ y_m y_m))
        (if (<= t_0 INFINITY)
          (* (fma (* x (/ (- x z) (* y_m y_m))) 0.5 0.5) y_m)
          (* (* (+ z x) (/ (- x z) y_m)) 0.5)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 5e-77) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else if (t_0 <= 2e+300) {
		tmp = fma(y_m, y_m, (x * x)) / (y_m + y_m);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x * ((x - z) / (y_m * y_m))), 0.5, 0.5) * y_m;
	} else {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 5e-77)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	elseif (t_0 <= 2e+300)
		tmp = Float64(fma(y_m, y_m, Float64(x * x)) / Float64(y_m + y_m));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(x * Float64(Float64(x - z) / Float64(y_m * y_m))), 0.5, 0.5) * y_m);
	else
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 5e-77], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+300], N[(N[(y$95$m * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x * N[(N[(x - z), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.99999999999999963e-77

   1. Initial program 91.2%

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

      1. lift-*.f64 N/A

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

      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

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

      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

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

      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. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   3. Applied rewrites 89.6%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 91.5

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

   6. Applied rewrites 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 96.7

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

   8. Applied rewrites 96.7%

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

   if 4.99999999999999963e-77 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.0000000000000001e300

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 98.6

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

   4. Applied rewrites 98.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 98.6

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

   6. Applied rewrites 98.6%

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

   if 2.0000000000000001e300 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 53.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.6

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

   4. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 93.3

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

   6. Applied rewrites 93.3%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 93.3%

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

   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. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 39.0

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

   4. Applied rewrites 39.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   6. 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 70.1

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

   7. Applied rewrites 70.1%

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

Alternative 6: 84.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.7 \cdot 10^{-48}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 2.7e-48)
    (* (* (+ z x) (/ (- x z) y_m)) 0.5)
    (* 0.5 (- y_m (* z (/ z y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.7e-48) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Fortran

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)
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
    real(8) :: tmp
    if (y_m <= 2.7d-48) then
        tmp = ((z + x) * ((x - z) / y_m)) * 0.5d0
    else
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.7e-48) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 2.7e-48:
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5
	else:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 2.7e-48)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 2.7e-48)
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	else
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 2.7e-48], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.70000000000000011e-48

   1. Initial program 90.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.6

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

   4. Applied rewrites 77.6%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   6. 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 93.9

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

   7. Applied rewrites 93.9%

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

   if 2.70000000000000011e-48 < y

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      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

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

      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

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

      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. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   3. Applied rewrites 56.5%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 67.4

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

   6. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 77.4

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

   8. Applied rewrites 77.4%

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

Alternative 7: 83.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{\left(x - z\right) \cdot \left(z + x\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 2.7e-48)
    (/ (* (- x z) (+ z x)) (+ y_m y_m))
    (* 0.5 (- y_m (* z (/ z y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.7e-48) {
		tmp = ((x - z) * (z + x)) / (y_m + y_m);
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Fortran

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)
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
    real(8) :: tmp
    if (y_m <= 2.7d-48) then
        tmp = ((x - z) * (z + x)) / (y_m + y_m)
    else
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.7e-48) {
		tmp = ((x - z) * (z + x)) / (y_m + y_m);
	} else {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 2.7e-48:
		tmp = ((x - z) * (z + x)) / (y_m + y_m)
	else:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 2.7e-48)
		tmp = Float64(Float64(Float64(x - z) * Float64(z + x)) / Float64(y_m + y_m));
	else
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 2.7e-48)
		tmp = ((x - z) * (z + x)) / (y_m + y_m);
	else
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 2.7e-48], N[(N[(N[(x - z), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.70000000000000011e-48

   1. Initial program 90.3%

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

   2. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 91.4

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

   4. Applied rewrites 91.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 91.4

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

      9. lower-+.f64 N/A

         \[\leadsto \frac{\left(x - z\right) \cdot \left(z + \color{blue}{x}\right)}{\mathsf{Rewrite=>}\left(lift-*.f64, \left(y \cdot 2\right)\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \frac{\left(x - z\right) \cdot \left(z + \color{blue}{x}\right)}{\mathsf{Rewrite<=}\left(*-commutative, \left(2 \cdot y\right)\right)} \]

      11. lower-+.f64 N/A

          \[\leadsto \frac{\left(x - z\right) \cdot \left(z + \color{blue}{x}\right)}{\mathsf{Rewrite=>}\left(count-2-rev, \left(y + y\right)\right)} \]

      12. lower-+.f64 N/A

          \[\leadsto \frac{\left(x - z\right) \cdot \left(z + \color{blue}{x}\right)}{\mathsf{Rewrite<=}\left(lift-+.f64, \left(y + y\right)\right)} \]

   6. Applied rewrites 91.4%

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

   if 2.70000000000000011e-48 < y

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      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

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

      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

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

      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. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   3. Applied rewrites 56.5%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 67.4

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

   6. Applied rewrites 67.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 77.4

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

   8. Applied rewrites 77.4%

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

Alternative 8: 74.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+157}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 7.5e+157)
    (* 0.5 (- y_m (* z (/ z y_m))))
    (* (* x (/ (- x z) y_m)) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 7.5e+157) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * ((x - z) / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Fortran

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)
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
    real(8) :: tmp
    if (x <= 7.5d+157) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = (x * ((x - z) / y_m)) * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 7.5e+157) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * ((x - z) / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 7.5e+157:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = (x * ((x - z) / y_m)) * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 7.5e+157)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(Float64(x * Float64(Float64(x - z) / y_m)) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 7.5e+157)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = (x * ((x - z) / y_m)) * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 7.5e+157], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 7.5e157

   1. Initial program 70.7%

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

      1. 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

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

      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

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

      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. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   3. Applied rewrites 68.9%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 66.1

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

   6. Applied rewrites 66.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 72.8

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

   8. Applied rewrites 72.8%

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

   if 7.5e157 < x

   1. Initial program 57.7%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in y around 0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   6. 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 89.8

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

   7. Applied rewrites 89.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 83.0%

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

Alternative 9: 72.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+192}:\\ \;\;\;\;0.5 \cdot \left(y\_m - z \cdot \frac{z}{y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 3.7e+192)
    (* 0.5 (- y_m (* z (/ z y_m))))
    (/ (* x x) (+ y_m y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 3.7e+192) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * x) / (y_m + y_m);
	}
	return y_s * tmp;
}

Fortran

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)
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
    real(8) :: tmp
    if (x <= 3.7d+192) then
        tmp = 0.5d0 * (y_m - (z * (z / y_m)))
    else
        tmp = (x * x) / (y_m + y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 3.7e+192) {
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	} else {
		tmp = (x * x) / (y_m + y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if x <= 3.7e+192:
		tmp = 0.5 * (y_m - (z * (z / y_m)))
	else:
		tmp = (x * x) / (y_m + y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 3.7e+192)
		tmp = Float64(0.5 * Float64(y_m - Float64(z * Float64(z / y_m))));
	else
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (x <= 3.7e+192)
		tmp = 0.5 * (y_m - (z * (z / y_m)));
	else
		tmp = (x * x) / (y_m + y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[x, 3.7e+192], N[(0.5 * N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.7000000000000001e192

   1. Initial program 69.8%

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

      1. lift-*.f64 N/A

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

      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

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

      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

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

      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. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. +-commutative N/A

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

      15. pow2 N/A

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

      16. lower-fma.f64 N/A

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

      17. pow2 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2-rev N/A

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

      21. lower-+.f64 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in x around 0

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

      1. distribute-lft-out-- N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 64.9

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

   6. Applied rewrites 64.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 71.5

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

   8. Applied rewrites 71.5%

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

   if 3.7000000000000001e192 < x

   1. Initial program 61.6%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 77.1

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

   4. Applied rewrites 77.1%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   5. 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. lift-+.f64 77.1

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

   6. Applied rewrites 77.1%

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

Alternative 10: 69.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(\frac{z}{y\_m} \cdot z\right) \cdot -0.5\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+143}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (* (/ z y_m) z) -0.5))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 2e+143)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (/ (* x x) (+ y_m y_m)) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = ((z / y_m) * z) * -0.5;
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+143) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = ((z / y_m) * z) * -0.5;
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+143) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = ((z / y_m) * z) * -0.5
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 2e+143:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = (x * x) / (y_m + y_m)
	else:
		tmp = t_0
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(z / y_m) * z) * -0.5)
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+143)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = ((z / y_m) * z) * -0.5;
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+143)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = (x * x) / (y_m + y_m);
	else
		tmp = t_0;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 2e+143], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $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 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 65.4%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 66.5

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

   4. Applied rewrites 66.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. pow2 N/A

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

      7. associate-/l* N/A

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

      8. sub-div N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 77.3

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

   6. Applied rewrites 77.3%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 90.0

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

   8. Applied rewrites 90.0%

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

   9. Taylor expanded in z around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. lift-/.f64 81.4

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

   11. Applied rewrites 81.4%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2e143

   1. Initial program 99.3%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 72.9

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

   4. Applied rewrites 72.9%

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

   if 2e143 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 60.4%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 56.5

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

   4. Applied rewrites 56.5%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   5. 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. lift-+.f64 56.5

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

   6. Applied rewrites 56.5%

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

Alternative 11: 50.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 2.7e-48) (/ (* x x) (+ y_m y_m)) (* 0.5 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.7e-48) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

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)
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
    real(8) :: tmp
    if (y_m <= 2.7d-48) then
        tmp = (x * x) / (y_m + y_m)
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 2.7e-48) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 2.7e-48:
		tmp = (x * x) / (y_m + y_m)
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 2.7e-48)
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 2.7e-48)
		tmp = (x * x) / (y_m + y_m);
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 2.7e-48], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.70000000000000011e-48

   1. Initial program 90.3%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 47.1

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

   4. Applied rewrites 47.1%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   5. 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. lift-+.f64 47.1

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

   6. Applied rewrites 47.1%

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

   if 2.70000000000000011e-48 < y

   1. Initial program 53.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 53.7

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

   4. Applied rewrites 53.7%

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

Alternative 12: 34.2% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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)
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
    code = y_s * (0.5d0 * y_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.1%

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

2. Taylor expanded in y around inf

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

   1. lower-*.f64 34.2

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

4. Applied rewrites 34.2%

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

Reproduce

herbie shell --seed 2025114 
(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)))