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

Percentage Accurate: 69.4% → 97.8%

Time: 4.1s

Alternatives: 10

Speedup: 1.0×

• 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: 69.4% 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: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z + x\_m\right) \cdot \frac{x\_m - z}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8 \cdot 10^{-98}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (+ z x_m) (/ (- x_m z) y_m))))
   (* y_s (if (<= y_m 8e-98) (* t_0 0.5) (* (fma (/ t_0 y_m) 0.5 0.5) y_m)))))

C

x_m = fabs(x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z) {
	double t_0 = (z + x_m) * ((x_m - z) / y_m);
	double tmp;
	if (y_m <= 8e-98) {
		tmp = t_0 * 0.5;
	} else {
		tmp = fma((t_0 / y_m), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z)
	t_0 = Float64(Float64(z + x_m) * Float64(Float64(x_m - z) / y_m))
	tmp = 0.0
	if (y_m <= 8e-98)
		tmp = Float64(t_0 * 0.5);
	else
		tmp = Float64(fma(Float64(t_0 / y_m), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 8e-98], N[(t$95$0 * 0.5), $MachinePrecision], N[(N[(N[(t$95$0 / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.99999999999999951e-98

   1. Initial program 69.4%

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

   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 77.2

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

   4. Applied rewrites 77.2%

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

   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. difference-of-squares-rev N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. sub-flip N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 66.8%

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

   if 7.99999999999999951e-98 < y

   1. Initial program 69.4%

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

   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 77.2

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

   4. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y} \]
   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. associate-/r* N/A

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

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

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. sub-flip N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 94.0%

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

Alternative 2: 93.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(z + x\_m\right) \cdot \frac{x\_m - z}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, y\_m, x\_m \cdot x\_m\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{y\_m} \cdot \frac{x\_m}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* (* (+ z x_m) (/ (- x_m z) y_m)) 0.5)
      (if (<= t_0 1e+303)
        (/ (fma y_m y_m (* x_m x_m)) (+ y_m y_m))
        (* (fma (* (/ x_m y_m) (/ x_m y_m)) 0.5 0.5) y_m))))))

C

x_m = fabs(x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z + x_m) * ((x_m - z) / y_m)) * 0.5;
	} else if (t_0 <= 1e+303) {
		tmp = fma(y_m, y_m, (x_m * x_m)) / (y_m + y_m);
	} else {
		tmp = fma(((x_m / y_m) * (x_m / y_m)), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(z + x_m) * Float64(Float64(x_m - z) / y_m)) * 0.5);
	elseif (t_0 <= 1e+303)
		tmp = Float64(fma(y_m, y_m, Float64(x_m * x_m)) / Float64(y_m + y_m));
	else
		tmp = Float64(fma(Float64(Float64(x_m / y_m) * Float64(x_m / y_m)), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(N[(z + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 1e+303], N[(N[(y$95$m * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(x$95$m / 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))) < 0.0

   1. Initial program 69.4%

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

   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 77.2

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

   4. Applied rewrites 77.2%

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

   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. difference-of-squares-rev N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. sub-flip N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 66.8%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1e303

   1. Initial program 69.4%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. mult-flip N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   if 1e303 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 69.4%

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

   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 77.2

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

   4. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\left(x + z\right) \cdot \left(x - z\right)}{y \cdot y}, 0.5, 0.5\right) \cdot y} \]
   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. associate-/r* N/A

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

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

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. sub-flip N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. lower-/.f64 N/A

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 54.4

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

   9. Applied rewrites 54.4%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-/.f64 N/A

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

       4. times-frac N/A

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

       5. lower-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-/.f64 63.3

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

   11. Applied rewrites 63.3%

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

Alternative 3: 86.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(z + x\_m\right) \cdot \frac{x\_m - z}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq 10^{+303}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, y\_m, x\_m \cdot x\_m\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m \cdot x\_m}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* (* (+ z x_m) (/ (- x_m z) y_m)) 0.5)
      (if (<= t_0 1e+303)
        (/ (fma y_m y_m (* x_m x_m)) (+ y_m y_m))
        (* (fma (/ (* x_m x_m) (* y_m y_m)) 0.5 0.5) y_m))))))

C

x_m = fabs(x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((z + x_m) * ((x_m - z) / y_m)) * 0.5;
	} else if (t_0 <= 1e+303) {
		tmp = fma(y_m, y_m, (x_m * x_m)) / (y_m + y_m);
	} else {
		tmp = fma(((x_m * x_m) / (y_m * y_m)), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(z + x_m) * Float64(Float64(x_m - z) / y_m)) * 0.5);
	elseif (t_0 <= 1e+303)
		tmp = Float64(fma(y_m, y_m, Float64(x_m * x_m)) / Float64(y_m + y_m));
	else
		tmp = Float64(fma(Float64(Float64(x_m * x_m) / Float64(y_m * y_m)), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(N[(z + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, 1e+303], N[(N[(y$95$m * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y$95$m * 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))) < 0.0

   1. Initial program 69.4%

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

   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 77.2

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

   4. Applied rewrites 77.2%

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

   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. difference-of-squares-rev N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. sub-flip N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 66.8%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1e303

   1. Initial program 69.4%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. mult-flip N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   if 1e303 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 69.4%

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

   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 77.2

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

   4. Applied rewrites 77.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 54.4

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

   7. Applied rewrites 54.4%

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

Alternative 4: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.38 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(z + x\_m\right) \cdot \frac{x\_m - z}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 1.1 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, y\_m, x\_m \cdot x\_m\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.38e+73)
    (* (* (+ z x_m) (/ (- x_m z) y_m)) 0.5)
    (if (<= y_m 1.1e+142)
      (/ (fma y_m y_m (* x_m x_m)) (+ y_m y_m))
      (* 0.5 y_m)))))

C

x_m = fabs(x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1.38e+73) {
		tmp = ((z + x_m) * ((x_m - z) / y_m)) * 0.5;
	} else if (y_m <= 1.1e+142) {
		tmp = fma(y_m, y_m, (x_m * x_m)) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Julia

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1.38e+73)
		tmp = Float64(Float64(Float64(z + x_m) * Float64(Float64(x_m - z) / y_m)) * 0.5);
	elseif (y_m <= 1.1e+142)
		tmp = Float64(fma(y_m, y_m, Float64(x_m * x_m)) / Float64(y_m + y_m));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.38e+73], N[(N[(N[(z + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$95$m, 1.1e+142], N[(N[(y$95$m * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.38000000000000007e73

   1. Initial program 69.4%

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

   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 77.2

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

   4. Applied rewrites 77.2%

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

   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. difference-of-squares-rev N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. sub-flip N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. lower-*.f64 N/A

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

   7. Applied rewrites 66.8%

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

   if 1.38000000000000007e73 < y < 1.09999999999999993e142

   1. Initial program 69.4%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. mult-flip N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   if 1.09999999999999993e142 < y

   1. Initial program 69.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.85 \cdot 10^{+61}:\\ \;\;\;\;\frac{\left(x\_m + z\right) \cdot \left(x\_m - z\right)}{y\_m + y\_m}\\ \mathbf{elif}\;y\_m \leq 1.1 \cdot 10^{+142}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y\_m, y\_m, x\_m \cdot x\_m\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.85e+61)
    (/ (* (+ x_m z) (- x_m z)) (+ y_m y_m))
    (if (<= y_m 1.1e+142)
      (/ (fma y_m y_m (* x_m x_m)) (+ y_m y_m))
      (* 0.5 y_m)))))

C

x_m = fabs(x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1.85e+61) {
		tmp = ((x_m + z) * (x_m - z)) / (y_m + y_m);
	} else if (y_m <= 1.1e+142) {
		tmp = fma(y_m, y_m, (x_m * x_m)) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Julia

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1.85e+61)
		tmp = Float64(Float64(Float64(x_m + z) * Float64(x_m - z)) / Float64(y_m + y_m));
	elseif (y_m <= 1.1e+142)
		tmp = Float64(fma(y_m, y_m, Float64(x_m * x_m)) / Float64(y_m + y_m));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.85e+61], N[(N[(N[(x$95$m + z), $MachinePrecision] * N[(x$95$m - z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.1e+142], N[(N[(y$95$m * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.85000000000000001e61

   1. Initial program 69.4%

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

   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 61.6

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

   4. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 61.6

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

   6. Applied rewrites 61.6%

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

   if 1.85000000000000001e61 < y < 1.09999999999999993e142

   1. Initial program 69.4%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. mult-flip N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

   if 1.09999999999999993e142 < y

   1. Initial program 69.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(-0.5 * N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(y$95$m * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 31.5

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

   4. Applied rewrites 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 34.1

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

   6. Applied rewrites 34.1%

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

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

   1. Initial program 69.4%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. mult-flip N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

      5. times-frac N/A

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

      6. metadata-eval N/A

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

      7. mult-flip N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. pow2 N/A

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

      11. lower-fma.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f64 45.0

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

   4. Applied rewrites 45.0%

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

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

   1. Initial program 69.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\left(x\_m \cdot \frac{x\_m}{y\_m}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* -0.5 (* z (/ z y_m)))
      (if (<= t_0 5e+151)
        (* 0.5 y_m)
        (if (<= t_0 INFINITY) (* (* x_m (/ x_m y_m)) 0.5) (* 0.5 y_m)))))))

C

x_m = fabs(x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * (z * (z / y_m));
	} else if (t_0 <= 5e+151) {
		tmp = 0.5 * y_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_m * (x_m / y_m)) * 0.5;
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Java

x_m = Math.abs(x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * (z * (z / y_m));
	} else if (t_0 <= 5e+151) {
		tmp = 0.5 * y_m;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (x_m * (x_m / y_m)) * 0.5;
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z):
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = -0.5 * (z * (z / y_m))
	elif t_0 <= 5e+151:
		tmp = 0.5 * y_m
	elif t_0 <= math.inf:
		tmp = (x_m * (x_m / y_m)) * 0.5
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(-0.5 * Float64(z * Float64(z / y_m)));
	elseif (t_0 <= 5e+151)
		tmp = Float64(0.5 * y_m);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_m * Float64(x_m / y_m)) * 0.5);
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z)
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = -0.5 * (z * (z / y_m));
	elseif (t_0 <= 5e+151)
		tmp = 0.5 * y_m;
	elseif (t_0 <= Inf)
		tmp = (x_m * (x_m / y_m)) * 0.5;
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(-0.5 * N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+151], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$95$m * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 31.5

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

   4. Applied rewrites 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 34.1

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

   6. Applied rewrites 34.1%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.0000000000000002e151 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 69.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

   if 5.0000000000000002e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 69.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 31.4

         \[\leadsto \frac{x \cdot x}{y} \cdot 0.5 \]

   4. Applied rewrites 31.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 33.9

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

   6. Applied rewrites 33.9%

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

Alternative 8: 68.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y\_m}\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+151}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 0.0)
      (* -0.5 (* z (/ z y_m)))
      (if (<= t_0 5e+151)
        (* 0.5 y_m)
        (if (<= t_0 INFINITY) (/ (* x_m x_m) (+ y_m y_m)) (* 0.5 y_m)))))))

C

x_m = fabs(x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * (z * (z / y_m));
	} else if (t_0 <= 5e+151) {
		tmp = 0.5 * y_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x_m * x_m) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Java

x_m = Math.abs(x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = -0.5 * (z * (z / y_m));
	} else if (t_0 <= 5e+151) {
		tmp = 0.5 * y_m;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (x_m * x_m) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z):
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = -0.5 * (z * (z / y_m))
	elif t_0 <= 5e+151:
		tmp = 0.5 * y_m
	elif t_0 <= math.inf:
		tmp = (x_m * x_m) / (y_m + y_m)
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(-0.5 * Float64(z * Float64(z / y_m)));
	elseif (t_0 <= 5e+151)
		tmp = Float64(0.5 * y_m);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x_m * x_m) / Float64(y_m + y_m));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z)
	t_0 = (((x_m * x_m) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = -0.5 * (z * (z / y_m));
	elseif (t_0 <= 5e+151)
		tmp = 0.5 * y_m;
	elseif (t_0 <= Inf)
		tmp = (x_m * x_m) / (y_m + y_m);
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(-0.5 * N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+151], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.4%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 31.5

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

   4. Applied rewrites 31.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 34.1

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

   6. Applied rewrites 34.1%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.0000000000000002e151 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 69.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

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

   if 5.0000000000000002e151 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 69.4%

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

   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 61.6

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

   4. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 61.6

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

   6. Applied rewrites 61.6%

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

   7. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 31.4

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

   9. Applied rewrites 31.4%

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

Alternative 9: 52.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z)
 :precision binary64
 (* y_s (if (<= y_m 2.5e+35) (/ (* x_m x_m) (+ y_m y_m)) (* 0.5 y_m))))

C

x_m = fabs(x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2.5e+35) {
		tmp = (x_m * x_m) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x_m, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 2.5d+35) then
        tmp = (x_m * x_m) / (y_m + y_m)
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

x_m = Math.abs(x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2.5e+35) {
		tmp = (x_m * x_m) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z):
	tmp = 0
	if y_m <= 2.5e+35:
		tmp = (x_m * x_m) / (y_m + y_m)
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

x_m = abs(x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 2.5e+35)
		tmp = Float64(Float64(x_m * x_m) / Float64(y_m + y_m));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 2.5e+35)
		tmp = (x_m * x_m) / (y_m + y_m);
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 2.5e+35], N[(N[(x$95$m * x$95$m), $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 < 2.50000000000000011e35

   1. Initial program 69.4%

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

   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 61.6

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

   4. Applied rewrites 61.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 61.6

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

   6. Applied rewrites 61.6%

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

   7. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lower-*.f64 31.4

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

   9. Applied rewrites 31.4%

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

   if 2.50000000000000011e35 < y

   1. Initial program 69.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 34.6

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

   4. Applied rewrites 34.6%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 34.6% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.4%

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

2. Taylor expanded in y around inf

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

   1. lower-*.f64 34.6

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

4. Applied rewrites 34.6%

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

Reproduce

herbie shell --seed 2025134 
(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)))