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

Percentage Accurate: 69.3% → 98.7%

Time: 3.5s

Alternatives: 11

Speedup: 0.8×

• 41.4% 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: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(x$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], N[(N[(N[(z$95$m + x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x$95$m * N[(x$95$m / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(z$95$m / y$95$m), $MachinePrecision] * t$95$0), $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 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 66.7

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

   7. Applied rewrites 66.7%

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

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

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. div-add-rev N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in z around 0

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

      1. pow2 N/A

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

      2. div-add-rev N/A

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

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

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate--l+ N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. count-2-rev N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. distribute-lft-out N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 61.4%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 66.4

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

   12. Applied rewrites 66.4%

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

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

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 93.9

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

   6. Applied rewrites 93.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 78.1%

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

Alternative 2: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := 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$95$m * z$95$m), $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$95$m + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$95$m * N[(x$95$m / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(z$95$m + y$95$m), $MachinePrecision] * N[(N[(y$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $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 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 66.7

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

   7. Applied rewrites 66.7%

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

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

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. div-add-rev N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in z around 0

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

      1. pow2 N/A

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

      2. div-add-rev N/A

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

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

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate--l+ N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. count-2-rev N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. distribute-lft-out N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 61.4%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 66.4

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

   12. Applied rewrites 66.4%

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

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

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. div-add-rev N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 67.4%

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

Alternative 3: 98.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 8.5 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(z\_m + y\_m, \frac{y\_m - z\_m}{y\_m + y\_m}, x\_m \cdot \frac{x\_m}{y\_m + y\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z\_m + x\_m}{y\_m} \cdot \frac{x\_m - z\_m}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (*
  y_s
  (if (<= x_m 8.5e+81)
    (fma (+ z_m y_m) (/ (- y_m z_m) (+ y_m y_m)) (* x_m (/ x_m (+ y_m y_m))))
    (* (fma (* (/ (+ z_m x_m) y_m) (/ (- x_m z_m) y_m)) 0.5 0.5) y_m))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (x_m <= 8.5e+81) {
		tmp = fma((z_m + y_m), ((y_m - z_m) / (y_m + y_m)), (x_m * (x_m / (y_m + y_m))));
	} else {
		tmp = fma((((z_m + x_m) / y_m) * ((x_m - z_m) / y_m)), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (x_m <= 8.5e+81)
		tmp = fma(Float64(z_m + y_m), Float64(Float64(y_m - z_m) / Float64(y_m + y_m)), Float64(x_m * Float64(x_m / Float64(y_m + y_m))));
	else
		tmp = Float64(fma(Float64(Float64(Float64(z_m + x_m) / y_m) * Float64(Float64(x_m - z_m) / y_m)), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[x$95$m, 8.5e+81], N[(N[(z$95$m + y$95$m), $MachinePrecision] * N[(N[(y$95$m - z$95$m), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(x$95$m / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(z$95$m + x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(N[(x$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 8.49999999999999986e81

   1. Initial program 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. pow2 N/A

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

      9. pow2 N/A

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

      10. pow2 N/A

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

      11. associate--l+ N/A

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

      12. div-add N/A

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

      13. lower-+.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. count-2-rev N/A

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

      19. lower-+.f64 N/A

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

      20. lower-/.f64 N/A

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

   3. Applied rewrites 66.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift--.f64 N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift--.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 N/A

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

      21. lift-+.f64 93.4

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

   5. Applied rewrites 93.4%

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

   if 8.49999999999999986e81 < x

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 93.9

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

   6. Applied rewrites 93.9%

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

Alternative 4: 96.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < 0.5

   1. Initial program 69.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. difference-of-squares N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 74.5

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. count-2-rev N/A

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

      23. lower-+.f64 74.5

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

   3. Applied rewrites 74.5%

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

   if 0.5 < y

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 93.9

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

   6. Applied rewrites 93.9%

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

Alternative 5: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-120}:\\ \;\;\;\;\left(z\_m \cdot \frac{z\_m}{y\_m}\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{x\_m}{y\_m}, y\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z\_m + y\_m\right) \cdot \frac{y\_m - z\_m}{y\_m}\right) \cdot 0.5\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 -1e-120)
      (* (* z_m (/ z_m y_m)) -0.5)
      (if (<= t_0 INFINITY)
        (* (fma x_m (/ x_m y_m) y_m) 0.5)
        (* (* (+ z_m y_m) (/ (- y_m z_m) y_m)) 0.5))))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -1e-120) {
		tmp = (z_m * (z_m / y_m)) * -0.5;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x_m, (x_m / y_m), y_m) * 0.5;
	} else {
		tmp = ((z_m + y_m) * ((y_m - z_m) / y_m)) * 0.5;
	}
	return y_s * tmp;
}

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -1e-120)
		tmp = Float64(Float64(z_m * Float64(z_m / y_m)) * -0.5);
	elseif (t_0 <= Inf)
		tmp = Float64(fma(x_m, Float64(x_m / y_m), y_m) * 0.5);
	else
		tmp = Float64(Float64(Float64(z_m + y_m) * Float64(Float64(y_m - z_m) / y_m)) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := 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$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -1e-120], N[(N[(z$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x$95$m * N[(x$95$m / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(z$95$m + y$95$m), $MachinePrecision] * N[(N[(y$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $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))) < -9.99999999999999979e-121

   1. Initial program 69.3%

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

   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 31.1

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

   4. Applied rewrites 31.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 34.0

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

   6. Applied rewrites 34.0%

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

   if -9.99999999999999979e-121 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. div-add-rev N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in z around 0

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

      1. pow2 N/A

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

      2. div-add-rev N/A

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

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

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate--l+ N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. count-2-rev N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. distribute-lft-out N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 61.4%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 66.4

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

   12. Applied rewrites 66.4%

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

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

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. div-add-rev N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 67.4%

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

Alternative 6: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := z\_m \cdot \frac{z\_m}{y\_m}\\ t_1 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t\_0 \cdot -0.5\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{x\_m}{y\_m}, y\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -0.5, 0.5 \cdot y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* z_m (/ z_m y_m)))
        (t_1 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -1e-120)
      (* t_0 -0.5)
      (if (<= t_1 INFINITY)
        (* (fma x_m (/ x_m y_m) y_m) 0.5)
        (fma t_0 -0.5 (* 0.5 y_m)))))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = z_m * (z_m / y_m);
	double t_1 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -1e-120) {
		tmp = t_0 * -0.5;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(x_m, (x_m / y_m), y_m) * 0.5;
	} else {
		tmp = fma(t_0, -0.5, (0.5 * y_m));
	}
	return y_s * tmp;
}

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	t_0 = Float64(z_m * Float64(z_m / y_m))
	t_1 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= -1e-120)
		tmp = Float64(t_0 * -0.5);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(x_m, Float64(x_m / y_m), y_m) * 0.5);
	else
		tmp = fma(t_0, -0.5, Float64(0.5 * y_m));
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(z$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, -1e-120], N[(t$95$0 * -0.5), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x$95$m * N[(x$95$m / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(t$95$0 * -0.5 + N[(0.5 * y$95$m), $MachinePrecision]), $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))) < -9.99999999999999979e-121

   1. Initial program 69.3%

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

   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 31.1

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

   4. Applied rewrites 31.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 34.0

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

   6. Applied rewrites 34.0%

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

   if -9.99999999999999979e-121 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. div-add-rev N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in z around 0

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

      1. pow2 N/A

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

      2. div-add-rev N/A

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

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

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate--l+ N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. count-2-rev N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. distribute-lft-out N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 61.4%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 66.4

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

   12. Applied rewrites 66.4%

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

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

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. div-add-rev N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. pow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 67.4

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

   10. Applied rewrites 67.4%

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

Alternative 7: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z\_m \cdot \frac{z\_m}{y\_m}\right) \cdot -0.5\\ t_1 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{x\_m}{y\_m}, y\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* (* z_m (/ z_m y_m)) -0.5))
        (t_1 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -1e-120)
      t_0
      (if (<= t_1 INFINITY) (* (fma x_m (/ x_m y_m) y_m) 0.5) t_0)))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (z_m * (z_m / y_m)) * -0.5;
	double t_1 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -1e-120) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(x_m, (x_m / y_m), y_m) * 0.5;
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(z_m * Float64(z_m / y_m)) * -0.5)
	t_1 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= -1e-120)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(x_m, Float64(x_m / y_m), y_m) * 0.5);
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(z$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, -1e-120], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(x$95$m * N[(x$95$m / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -9.99999999999999979e-121 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 69.3%

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

   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 31.1

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

   4. Applied rewrites 31.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 34.0

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

   6. Applied rewrites 34.0%

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

   if -9.99999999999999979e-121 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. div-add-rev N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in z around 0

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

      1. pow2 N/A

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

      2. div-add-rev N/A

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

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

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. associate--l+ N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. count-2-rev N/A

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

      10. *-commutative N/A

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

      11. pow2 N/A

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

      12. distribute-lft-out N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

   10. Applied rewrites 61.4%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. associate-/l* N/A

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

       8. lower-fma.f64 N/A

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

       9. lower-/.f64 66.4

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

   12. Applied rewrites 66.4%

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

Alternative 8: 69.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z\_m \cdot \frac{z\_m}{y\_m}\right) \cdot -0.5\\ t_1 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(z\_m + y\_m\right) \cdot 1\right) \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* (* z_m (/ z_m y_m)) -0.5))
        (t_1 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 0.0)
      t_0
      (if (<= t_1 2e+153)
        (* (* (+ z_m y_m) 1.0) 0.5)
        (if (<= t_1 INFINITY) (/ (* x_m x_m) (+ y_m y_m)) t_0))))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (z_m * (z_m / y_m)) * -0.5;
	double t_1 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+153) {
		tmp = ((z_m + y_m) * 1.0) * 0.5;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_m * x_m) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = (z_m * (z_m / y_m)) * -0.5;
	double t_1 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 2e+153) {
		tmp = ((z_m + y_m) * 1.0) * 0.5;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x_m * x_m) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	t_0 = (z_m * (z_m / y_m)) * -0.5
	t_1 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 2e+153:
		tmp = ((z_m + y_m) * 1.0) * 0.5
	elif t_1 <= math.inf:
		tmp = (x_m * x_m) / (y_m + y_m)
	else:
		tmp = t_0
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	t_0 = Float64(Float64(z_m * Float64(z_m / y_m)) * -0.5)
	t_1 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+153)
		tmp = Float64(Float64(Float64(z_m + y_m) * 1.0) * 0.5);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_m * x_m) / Float64(y_m + y_m));
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	t_0 = (z_m * (z_m / y_m)) * -0.5;
	t_1 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+153)
		tmp = ((z_m + y_m) * 1.0) * 0.5;
	elseif (t_1 <= Inf)
		tmp = (x_m * x_m) / (y_m + y_m);
	else
		tmp = t_0;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(N[(z$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 2e+153], N[(N[(N[(z$95$m + y$95$m), $MachinePrecision] * 1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 69.3%

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

   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 31.1

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

   4. Applied rewrites 31.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-commutative N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 34.0

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

   6. Applied rewrites 34.0%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2e153

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. div-add-rev N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 34.4%

       \[\leadsto \left(\left(z + y\right) \cdot 1\right) \cdot 0.5 \]

   if 2e153 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 69.3%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 31.7

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

   4. Applied rewrites 31.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 31.7

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

   6. Applied rewrites 31.7%

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

Alternative 9: 67.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{z\_m \cdot z\_m}{y\_m}\\ t_1 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(z\_m + y\_m\right) \cdot 1\right) \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ (* z_m z_m) y_m)))
        (t_1 (/ (- (+ (* x_m x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -1e-120)
      t_0
      (if (<= t_1 2e+153)
        (* (* (+ z_m y_m) 1.0) 0.5)
        (if (<= t_1 INFINITY) (/ (* x_m x_m) (+ y_m y_m)) t_0))))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = -0.5 * ((z_m * z_m) / y_m);
	double t_1 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -1e-120) {
		tmp = t_0;
	} else if (t_1 <= 2e+153) {
		tmp = ((z_m + y_m) * 1.0) * 0.5;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x_m * x_m) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double t_0 = -0.5 * ((z_m * z_m) / y_m);
	double t_1 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -1e-120) {
		tmp = t_0;
	} else if (t_1 <= 2e+153) {
		tmp = ((z_m + y_m) * 1.0) * 0.5;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x_m * x_m) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	t_0 = -0.5 * ((z_m * z_m) / y_m)
	t_1 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= -1e-120:
		tmp = t_0
	elif t_1 <= 2e+153:
		tmp = ((z_m + y_m) * 1.0) * 0.5
	elif t_1 <= math.inf:
		tmp = (x_m * x_m) / (y_m + y_m)
	else:
		tmp = t_0
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	t_0 = Float64(-0.5 * Float64(Float64(z_m * z_m) / y_m))
	t_1 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y_m * y_m)) - Float64(z_m * z_m)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= -1e-120)
		tmp = t_0;
	elseif (t_1 <= 2e+153)
		tmp = Float64(Float64(Float64(z_m + y_m) * 1.0) * 0.5);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x_m * x_m) / Float64(y_m + y_m));
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	t_0 = -0.5 * ((z_m * z_m) / y_m);
	t_1 = (((x_m * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= -1e-120)
		tmp = t_0;
	elseif (t_1 <= 2e+153)
		tmp = ((z_m + y_m) * 1.0) * 0.5;
	elseif (t_1 <= Inf)
		tmp = (x_m * x_m) / (y_m + y_m);
	else
		tmp = t_0;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := Block[{t$95$0 = N[(-0.5 * N[(N[(z$95$m * z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, -1e-120], t$95$0, If[LessEqual[t$95$1, 2e+153], N[(N[(N[(z$95$m + y$95$m), $MachinePrecision] * 1.0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -9.99999999999999979e-121 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 69.3%

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

   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 31.1

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

   4. Applied rewrites 31.1%

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

   if -9.99999999999999979e-121 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2e153

   1. Initial program 69.3%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 77.5

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

   4. Applied rewrites 77.5%

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

   5. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. associate--l+ N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

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

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. div-add-rev N/A

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

      11. pow2 N/A

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

      12. *-commutative N/A

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 34.4%

       \[\leadsto \left(\left(z + y\right) \cdot 1\right) \cdot 0.5 \]

   if 2e153 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 69.3%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 31.7

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

   4. Applied rewrites 31.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 31.7

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

   6. Applied rewrites 31.7%

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

Alternative 10: 52.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{+82}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m)
 :precision binary64
 (* y_s (if (<= y_m 3e+82) (/ (* x_m x_m) (+ y_m y_m)) (* 0.5 y_m))))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 3e+82) {
		tmp = (x_m * x_m) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x_m, y_m, z_m)
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_m
    real(8) :: tmp
    if (y_m <= 3d+82) 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);
z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	double tmp;
	if (y_m <= 3e+82) {
		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)
z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	tmp = 0
	if y_m <= 3e+82:
		tmp = (x_m * x_m) / (y_m + y_m)
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	tmp = 0.0
	if (y_m <= 3e+82)
		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);
z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_m, y_m, z_m)
	tmp = 0.0;
	if (y_m <= 3e+82)
		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]
z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * If[LessEqual[y$95$m, 3e+82], 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 < 2.99999999999999989e82

   1. Initial program 69.3%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

         \[\leadsto \frac{x \cdot \color{blue}{x}}{y \cdot 2} \]

      2. lift-*.f64 31.7

         \[\leadsto \frac{x \cdot \color{blue}{x}}{y \cdot 2} \]

   4. Applied rewrites 31.7%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{2 \cdot y}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

      4. lift-+.f64 31.7

         \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

   6. Applied rewrites 31.7%

      \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

   if 2.99999999999999989e82 < y

   1. Initial program 69.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   3. Step-by-step derivation

      1. lower-*.f64 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 11: 34.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(0.5 \cdot y\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
z_m = (fabs.f64 z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x_m y_m z_m) :precision binary64 (* y_s (* 0.5 y_m)))

C

x_m = fabs(x);
z_m = fabs(z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x_m, double y_m, double z_m) {
	return y_s * (0.5 * y_m);
}

Fortran

x_m =     private
z_m =     private
y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(y_s, x_m, y_m, z_m)
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_m
    code = y_s * (0.5d0 * y_m)
end function

Java

x_m = Math.abs(x);
z_m = Math.abs(z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_m, double y_m, double z_m) {
	return y_s * (0.5 * y_m);
}

Python

x_m = math.fabs(x)
z_m = math.fabs(z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x_m, y_m, z_m):
	return y_s * (0.5 * y_m)

Julia

x_m = abs(x)
z_m = abs(z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x_m, y_m, z_m)
	return Float64(y_s * Float64(0.5 * y_m))
end

MATLAB

x_m = abs(x);
z_m = abs(z);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_m, y_m, z_m)
	tmp = y_s * (0.5 * y_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$m_, y$95$m_, z$95$m_] := N[(y$95$s * N[(0.5 * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.3%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]

2. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
3. Step-by-step derivation

   1. lower-*.f64 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 2025123 
(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)))