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

Percentage Accurate: 68.9% → 98.1%

Time: 4.2s

Alternatives: 14

Speedup: 1.0×

• 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: 68.9% 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: 98.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x\_m - z\_m}{y\_m}\\ t_1 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(z\_m + x\_m\right) \cdot t\_0\right) \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 10^{+295}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m, y\_m \cdot y\_m\right)}{y\_m \cdot 2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(z\_m + x\_m\right) \cdot \frac{x\_m}{y\_m}}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z\_m \cdot \frac{t\_0}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
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 1e+295)
        (/ (fma x_m x_m (* y_m y_m)) (* y_m 2.0))
        (if (<= t_1 INFINITY)
          (* (fma (/ (* (+ z_m x_m) (/ x_m y_m)) y_m) 0.5 0.5) y_m)
          (* (fma (* z_m (/ t_0 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 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 <= 1e+295) {
		tmp = fma(x_m, x_m, (y_m * y_m)) / (y_m * 2.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((((z_m + x_m) * (x_m / y_m)) / y_m), 0.5, 0.5) * y_m;
	} else {
		tmp = fma((z_m * (t_0 / 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)
	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 <= 1e+295)
		tmp = Float64(fma(x_m, x_m, Float64(y_m * y_m)) / Float64(y_m * 2.0));
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(Float64(Float64(z_m + x_m) * Float64(x_m / y_m)) / y_m), 0.5, 0.5) * y_m);
	else
		tmp = Float64(fma(Float64(z_m * Float64(t_0 / y_m)), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
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, 1e+295], N[(N[(x$95$m * x$95$m + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(N[(z$95$m + x$95$m), $MachinePrecision] * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(z$95$m * N[(t$95$0 / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 77.6%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 66.1

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

   8. Applied rewrites 66.1%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 9.9999999999999998e294

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 65.6

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

   7. Applied rewrites 65.6%

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

   if 9.9999999999999998e294 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 73.3%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 72.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 58.1%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 83.2%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift--.f64 83.2

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

   12. Applied rewrites 83.2%

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

Alternative 2: 93.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(\left(z\_m + x\_m\right) \cdot \frac{x\_m - z\_m}{y\_m}\right) \cdot 0.5\\ t_1 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+295}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, x\_m, y\_m \cdot y\_m\right)}{y\_m \cdot 2}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(z\_m + x\_m\right) \cdot \frac{x\_m}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 x_m) (/ (- x_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 1e+295)
        (/ (fma x_m x_m (* y_m y_m)) (* y_m 2.0))
        (if (<= t_1 INFINITY)
          (* (fma (* (+ z_m x_m) (/ x_m (* y_m y_m))) 0.5 0.5) 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 + x_m) * ((x_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 <= 1e+295) {
		tmp = fma(x_m, x_m, (y_m * y_m)) / (y_m * 2.0);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(((z_m + x_m) * (x_m / (y_m * y_m))), 0.5, 0.5) * 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(Float64(z_m + x_m) * Float64(Float64(x_m - 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 <= 1e+295)
		tmp = Float64(fma(x_m, x_m, Float64(y_m * y_m)) / Float64(y_m * 2.0));
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(Float64(z_m + x_m) * Float64(x_m / Float64(y_m * y_m))), 0.5, 0.5) * y_m);
	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[(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]}, 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, 1e+295], N[(N[(x$95$m * x$95$m + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(z$95$m + x$95$m), $MachinePrecision] * N[(x$95$m / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $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 61.8%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 9.9999999999999998e294

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 65.6

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

   7. Applied rewrites 65.6%

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

   if 9.9999999999999998e294 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 73.3%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 95.7

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

   7. Applied rewrites 95.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 68.5%

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

Alternative 3: 71.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z\_m \cdot \frac{z\_m}{y\_m}\right) \cdot -0.5\\ t_1 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+142}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{x\_m}{y\_m} \cdot \left(z\_m + x\_m\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 0.0)
      t_0
      (if (<= t_1 1e+142)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (* (* (/ x_m y_m) (+ z_m x_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 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+142) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x_m / y_m) * (z_m + x_m)) * 0.5;
	} 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 <= 1e+142) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((x_m / y_m) * (z_m + x_m)) * 0.5;
	} 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 <= 1e+142:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = ((x_m / y_m) * (z_m + x_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 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+142)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x_m / y_m) * Float64(z_m + x_m)) * 0.5);
	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 <= 1e+142)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = ((x_m / y_m) * (z_m + x_m)) * 0.5;
	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, 1e+142], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(z$95$m + x$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $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 61.8%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 42.5

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

   10. Applied rewrites 42.5%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.00000000000000005e142

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 50.0

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

   5. Applied rewrites 50.0%

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

   if 1.00000000000000005e142 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 75.8%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 89.9%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 93.7

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift-+.f64 77.2

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

   10. Applied rewrites 77.2%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 45.1%

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

Alternative 4: 69.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z\_m \cdot \frac{z\_m}{y\_m}\right) \cdot -0.5\\ t_1 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 4e+149)
        (* 0.5 y_m)
        (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 <= 4e+149) {
		tmp = 0.5 * y_m;
	} 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 <= 4e+149) {
		tmp = 0.5 * y_m;
	} 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 <= 4e+149:
		tmp = 0.5 * y_m
	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 <= 4e+149)
		tmp = Float64(0.5 * y_m);
	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 <= 4e+149)
		tmp = 0.5 * y_m;
	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, 4e+149], N[(0.5 * y$95$m), $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 61.8%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 42.5

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

   10. Applied rewrites 42.5%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.0000000000000002e149

   1. Initial program 99.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 51.3

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

   5. Applied rewrites 51.3%

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

   if 4.0000000000000002e149 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 75.5%

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

   3. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 42.8

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

   5. Applied rewrites 42.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 42.8

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

   7. Applied rewrites 42.8%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 38.6

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

   10. Applied rewrites 38.6%

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

Alternative 5: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{z\_m \cdot z\_m}{y\_m}\\ t_1 := \frac{\left(x\_m \cdot x\_m + y\_m \cdot y\_m\right) - z\_m \cdot z\_m}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+149}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 -4e-20)
      t_0
      (if (<= t_1 4e+149)
        (* 0.5 y_m)
        (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 <= -4e-20) {
		tmp = t_0;
	} else if (t_1 <= 4e+149) {
		tmp = 0.5 * y_m;
	} 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 <= -4e-20) {
		tmp = t_0;
	} else if (t_1 <= 4e+149) {
		tmp = 0.5 * y_m;
	} 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 <= -4e-20:
		tmp = t_0
	elif t_1 <= 4e+149:
		tmp = 0.5 * y_m
	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 <= -4e-20)
		tmp = t_0;
	elseif (t_1 <= 4e+149)
		tmp = Float64(0.5 * y_m);
	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 <= -4e-20)
		tmp = t_0;
	elseif (t_1 <= 4e+149)
		tmp = 0.5 * y_m;
	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, -4e-20], t$95$0, If[LessEqual[t$95$1, 4e+149], N[(0.5 * y$95$m), $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))) < -3.99999999999999978e-20 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 39.7

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

   5. Applied rewrites 39.7%

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

   if -3.99999999999999978e-20 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.0000000000000002e149

   1. Initial program 88.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 55.3

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

   5. Applied rewrites 55.3%

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

   if 4.0000000000000002e149 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 75.5%

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

   3. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 42.8

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

   5. Applied rewrites 42.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 42.8

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

   7. Applied rewrites 42.8%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 38.6

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

   10. Applied rewrites 38.6%

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

Alternative 6: 74.8% accurate, 0.3× speedup

Math

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

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 x_m) (* y_m y_m)) (* z_m z_m)) (* y_m 2.0))))
   (*
    y_s
    (if (or (<= t_0 0.0) (not (<= t_0 INFINITY)))
      (* (* z_m (/ z_m y_m)) -0.5)
      (/ (fma y_m y_m (* x_m x_m)) (+ y_m 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 * x_m) + (y_m * y_m)) - (z_m * z_m)) / (y_m * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= ((double) INFINITY))) {
		tmp = (z_m * (z_m / y_m)) * -0.5;
	} else {
		tmp = fma(y_m, y_m, (x_m * x_m)) / (y_m + 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(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) || !(t_0 <= Inf))
		tmp = Float64(Float64(z_m * Float64(z_m / y_m)) * -0.5);
	else
		tmp = Float64(fma(y_m, y_m, Float64(x_m * x_m)) / Float64(y_m + 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[(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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(z$95$m * N[(z$95$m / y$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[(y$95$m * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 61.8%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 36.3

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

   8. Applied rewrites 36.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 42.5

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

   10. Applied rewrites 42.5%

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

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

   1. Initial program 83.5%

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

   3. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 54.4

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

   5. Applied rewrites 54.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 54.4

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

   7. Applied rewrites 54.4%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 33.8

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

   10. Applied rewrites 33.8%

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

   11. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. pow2 N/A

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

       3. lower-fma.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 50.9

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

   13. Applied rewrites 50.9%

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

4. Final simplification 46.1%

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

Alternative 7: 96.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x\_m - z\_m}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-229}:\\ \;\;\;\;\left(\left(z\_m + x\_m\right) \cdot t\_0\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m + z\_m, x\_m - z\_m, y\_m \cdot y\_m\right)}{y\_m \cdot 2}\\ \mathbf{elif}\;y\_m \leq 1.65 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(z\_m + x\_m\right) \cdot \frac{x\_m - z\_m}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z\_m \cdot \frac{t\_0}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
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)))
   (*
    y_s
    (if (<= y_m 2e-229)
      (* (* (+ z_m x_m) t_0) 0.5)
      (if (<= y_m 4.8e-23)
        (/ (fma (+ x_m z_m) (- x_m z_m) (* y_m y_m)) (* y_m 2.0))
        (if (<= y_m 1.65e+153)
          (* (fma (* (+ z_m x_m) (/ (- x_m z_m) (* y_m y_m))) 0.5 0.5) y_m)
          (* (fma (* z_m (/ t_0 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 t_0 = (x_m - z_m) / y_m;
	double tmp;
	if (y_m <= 2e-229) {
		tmp = ((z_m + x_m) * t_0) * 0.5;
	} else if (y_m <= 4.8e-23) {
		tmp = fma((x_m + z_m), (x_m - z_m), (y_m * y_m)) / (y_m * 2.0);
	} else if (y_m <= 1.65e+153) {
		tmp = fma(((z_m + x_m) * ((x_m - z_m) / (y_m * y_m))), 0.5, 0.5) * y_m;
	} else {
		tmp = fma((z_m * (t_0 / 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)
	t_0 = Float64(Float64(x_m - z_m) / y_m)
	tmp = 0.0
	if (y_m <= 2e-229)
		tmp = Float64(Float64(Float64(z_m + x_m) * t_0) * 0.5);
	elseif (y_m <= 4.8e-23)
		tmp = Float64(fma(Float64(x_m + z_m), Float64(x_m - z_m), Float64(y_m * y_m)) / Float64(y_m * 2.0));
	elseif (y_m <= 1.65e+153)
		tmp = Float64(fma(Float64(Float64(z_m + x_m) * Float64(Float64(x_m - z_m) / Float64(y_m * y_m))), 0.5, 0.5) * y_m);
	else
		tmp = Float64(fma(Float64(z_m * Float64(t_0 / y_m)), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
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]}, N[(y$95$s * If[LessEqual[y$95$m, 2e-229], N[(N[(N[(z$95$m + x$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$95$m, 4.8e-23], 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 * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.65e+153], N[(N[(N[(N[(z$95$m + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z$95$m), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(z$95$m * N[(t$95$0 / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < 2.00000000000000014e-229

   1. Initial program 70.9%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 71.1

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

   8. Applied rewrites 71.1%

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

   if 2.00000000000000014e-229 < y < 4.79999999999999993e-23

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      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 97.6

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

   4. Applied rewrites 97.6%

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

   if 4.79999999999999993e-23 < y < 1.64999999999999997e153

   1. Initial program 83.1%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 93.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.64999999999999997e153 < y

   1. Initial program 14.7%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 79.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 86.1%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 N/A

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

       8. lift-/.f64 N/A

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

       9. lift--.f64 86.1

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

   12. Applied rewrites 86.1%

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

Alternative 8: 96.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-229}:\\ \;\;\;\;\left(\left(z\_m + x\_m\right) \cdot \frac{x\_m - z\_m}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m + z\_m, x\_m - z\_m, y\_m \cdot y\_m\right)}{y\_m \cdot 2}\\ \mathbf{elif}\;y\_m \leq 1.65 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\left(z\_m + x\_m\right) \cdot \frac{x\_m - z\_m}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z\_m \cdot \frac{-z\_m}{y\_m}}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

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 2e-229)
    (* (* (+ z_m x_m) (/ (- x_m z_m) y_m)) 0.5)
    (if (<= y_m 4.8e-23)
      (/ (fma (+ x_m z_m) (- x_m z_m) (* y_m y_m)) (* y_m 2.0))
      (if (<= y_m 1.65e+153)
        (* (fma (* (+ z_m x_m) (/ (- x_m z_m) (* y_m y_m))) 0.5 0.5) y_m)
        (* (fma (/ (* z_m (/ (- z_m) y_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 <= 2e-229) {
		tmp = ((z_m + x_m) * ((x_m - z_m) / y_m)) * 0.5;
	} else if (y_m <= 4.8e-23) {
		tmp = fma((x_m + z_m), (x_m - z_m), (y_m * y_m)) / (y_m * 2.0);
	} else if (y_m <= 1.65e+153) {
		tmp = fma(((z_m + x_m) * ((x_m - z_m) / (y_m * y_m))), 0.5, 0.5) * y_m;
	} else {
		tmp = fma(((z_m * (-z_m / y_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 <= 2e-229)
		tmp = Float64(Float64(Float64(z_m + x_m) * Float64(Float64(x_m - z_m) / y_m)) * 0.5);
	elseif (y_m <= 4.8e-23)
		tmp = Float64(fma(Float64(x_m + z_m), Float64(x_m - z_m), Float64(y_m * y_m)) / Float64(y_m * 2.0));
	elseif (y_m <= 1.65e+153)
		tmp = Float64(fma(Float64(Float64(z_m + x_m) * Float64(Float64(x_m - z_m) / Float64(y_m * y_m))), 0.5, 0.5) * y_m);
	else
		tmp = Float64(fma(Float64(Float64(z_m * Float64(Float64(-z_m) / y_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, 2e-229], 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[y$95$m, 4.8e-23], 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 * 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.65e+153], N[(N[(N[(N[(z$95$m + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z$95$m), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(z$95$m * N[((-z$95$m) / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if y < 2.00000000000000014e-229

   1. Initial program 70.9%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 71.1

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

   8. Applied rewrites 71.1%

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

   if 2.00000000000000014e-229 < y < 4.79999999999999993e-23

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      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 97.6

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

   4. Applied rewrites 97.6%

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

   if 4.79999999999999993e-23 < y < 1.64999999999999997e153

   1. Initial program 83.1%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 93.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.64999999999999997e153 < y

   1. Initial program 14.7%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 79.3%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 86.1%

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

   11. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 86.6

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

   13. Applied rewrites 86.6%

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

Alternative 9: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-229}:\\ \;\;\;\;\left(\left(z\_m + x\_m\right) \cdot \frac{x\_m - z\_m}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m + z\_m, x\_m - z\_m, y\_m \cdot y\_m\right)}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(z\_m + x\_m\right) \cdot \frac{x\_m - z\_m}{y\_m \cdot y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

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 2e-229)
    (* (* (+ z_m x_m) (/ (- x_m z_m) y_m)) 0.5)
    (if (<= y_m 4.8e-23)
      (/ (fma (+ x_m z_m) (- x_m z_m) (* y_m y_m)) (* y_m 2.0))
      (* (fma (* (+ z_m x_m) (/ (- x_m z_m) (* y_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 <= 2e-229) {
		tmp = ((z_m + x_m) * ((x_m - z_m) / y_m)) * 0.5;
	} else if (y_m <= 4.8e-23) {
		tmp = fma((x_m + z_m), (x_m - z_m), (y_m * y_m)) / (y_m * 2.0);
	} else {
		tmp = fma(((z_m + x_m) * ((x_m - z_m) / (y_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 <= 2e-229)
		tmp = Float64(Float64(Float64(z_m + x_m) * Float64(Float64(x_m - z_m) / y_m)) * 0.5);
	elseif (y_m <= 4.8e-23)
		tmp = Float64(fma(Float64(x_m + z_m), Float64(x_m - z_m), Float64(y_m * y_m)) / Float64(y_m * 2.0));
	else
		tmp = Float64(fma(Float64(Float64(z_m + x_m) * Float64(Float64(x_m - z_m) / Float64(y_m * y_m))), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
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, 2e-229], 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[y$95$m, 4.8e-23], 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 * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z$95$m + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z$95$m), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 2.00000000000000014e-229

   1. Initial program 70.9%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 71.1

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

   8. Applied rewrites 71.1%

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

   if 2.00000000000000014e-229 < y < 4.79999999999999993e-23

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      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 97.6

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

   4. Applied rewrites 97.6%

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

   if 4.79999999999999993e-23 < y

   1. Initial program 55.0%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 87.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. pow2 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 N/A

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

      14. pow2 N/A

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

      15. lower-*.f64 88.6

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

   7. Applied rewrites 88.6%

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

Alternative 10: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z\_m + x\_m\right) \cdot \frac{x\_m - z\_m}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-197}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
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 x_m) (/ (- x_m z_m) y_m))))
   (*
    y_s
    (if (<= y_m 1.3e-197) (* t_0 0.5) (* (fma (/ t_0 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 t_0 = (z_m + x_m) * ((x_m - z_m) / y_m);
	double tmp;
	if (y_m <= 1.3e-197) {
		tmp = t_0 * 0.5;
	} else {
		tmp = fma((t_0 / y_m), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

x_m = abs(x)
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 + x_m) * Float64(Float64(x_m - z_m) / y_m))
	tmp = 0.0
	if (y_m <= 1.3e-197)
		tmp = Float64(t_0 * 0.5);
	else
		tmp = Float64(fma(Float64(t_0 / y_m), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
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 + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 1.3e-197], N[(t$95$0 * 0.5), $MachinePrecision], N[(N[(N[(t$95$0 / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3000000000000001e-197

   1. Initial program 72.6%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 72.8

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

   8. Applied rewrites 72.8%

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

   if 1.3000000000000001e-197 < y

   1. Initial program 68.7%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 97.1

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

   7. Applied rewrites 97.1%

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

Alternative 11: 90.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{-229}:\\ \;\;\;\;\left(\left(z\_m + x\_m\right) \cdot \frac{x\_m - z\_m}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 1.35 \cdot 10^{+157}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m + z\_m, x\_m - z\_m, y\_m \cdot y\_m\right)}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 2e-229)
    (* (* (+ z_m x_m) (/ (- x_m z_m) y_m)) 0.5)
    (if (<= y_m 1.35e+157)
      (/ (fma (+ x_m z_m) (- x_m z_m) (* y_m y_m)) (* y_m 2.0))
      (* 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 <= 2e-229) {
		tmp = ((z_m + x_m) * ((x_m - z_m) / y_m)) * 0.5;
	} else if (y_m <= 1.35e+157) {
		tmp = fma((x_m + z_m), (x_m - z_m), (y_m * y_m)) / (y_m * 2.0);
	} 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 <= 2e-229)
		tmp = Float64(Float64(Float64(z_m + x_m) * Float64(Float64(x_m - z_m) / y_m)) * 0.5);
	elseif (y_m <= 1.35e+157)
		tmp = Float64(fma(Float64(x_m + z_m), Float64(x_m - z_m), Float64(y_m * y_m)) / Float64(y_m * 2.0));
	else
		tmp = 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_] := N[(y$95$s * If[LessEqual[y$95$m, 2e-229], 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[y$95$m, 1.35e+157], 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 * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 2.00000000000000014e-229

   1. Initial program 70.9%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 71.1

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

   8. Applied rewrites 71.1%

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

   if 2.00000000000000014e-229 < y < 1.35e157

   1. Initial program 90.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      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 95.4

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

   4. Applied rewrites 95.4%

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

   if 1.35e157 < y

   1. Initial program 14.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 72.4

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

   5. Applied rewrites 72.4%

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

Alternative 12: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.6 \cdot 10^{+78}:\\ \;\;\;\;\left(\left(z\_m + x\_m\right) \cdot \frac{x\_m - z\_m}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 2.2 \cdot 10^{+194}:\\ \;\;\;\;\frac{\left(y\_m + z\_m\right) \cdot \left(y\_m - z\_m\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 5.6e+78)
    (* (* (+ z_m x_m) (/ (- x_m z_m) y_m)) 0.5)
    (if (<= y_m 2.2e+194)
      (/ (* (+ y_m z_m) (- y_m z_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 <= 5.6e+78) {
		tmp = ((z_m + x_m) * ((x_m - z_m) / y_m)) * 0.5;
	} else if (y_m <= 2.2e+194) {
		tmp = ((y_m + z_m) * (y_m - z_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 <= 5.6d+78) then
        tmp = ((z_m + x_m) * ((x_m - z_m) / y_m)) * 0.5d0
    else if (y_m <= 2.2d+194) then
        tmp = ((y_m + z_m) * (y_m - z_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 <= 5.6e+78) {
		tmp = ((z_m + x_m) * ((x_m - z_m) / y_m)) * 0.5;
	} else if (y_m <= 2.2e+194) {
		tmp = ((y_m + z_m) * (y_m - z_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 <= 5.6e+78:
		tmp = ((z_m + x_m) * ((x_m - z_m) / y_m)) * 0.5
	elif y_m <= 2.2e+194:
		tmp = ((y_m + z_m) * (y_m - z_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 <= 5.6e+78)
		tmp = Float64(Float64(Float64(z_m + x_m) * Float64(Float64(x_m - z_m) / y_m)) * 0.5);
	elseif (y_m <= 2.2e+194)
		tmp = Float64(Float64(Float64(y_m + z_m) * Float64(y_m - z_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 <= 5.6e+78)
		tmp = ((z_m + x_m) * ((x_m - z_m) / y_m)) * 0.5;
	elseif (y_m <= 2.2e+194)
		tmp = ((y_m + z_m) * (y_m - z_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, 5.6e+78], 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[y$95$m, 2.2e+194], N[(N[(N[(y$95$m + z$95$m), $MachinePrecision] * N[(y$95$m - z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 5.6000000000000002e78

   1. Initial program 77.9%

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

   3. Taylor expanded in y around inf

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

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 76.3

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

   8. Applied rewrites 76.3%

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

   if 5.6000000000000002e78 < y < 2.2000000000000001e194

   1. Initial program 65.2%

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

   3. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 81.2

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

   5. Applied rewrites 81.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 81.2

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

   7. Applied rewrites 81.2%

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

   if 2.2000000000000001e194 < y

   1. Initial program 9.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 81.8

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

   5. Applied rewrites 81.8%

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

Alternative 13: 50.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ z_m = \left|z\right| \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1 \cdot 10^{+84}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y\_m + y\_m}\\ \end{array} \end{array} \]

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 1.1e+84) (* 0.5 y_m) (/ (* x_m x_m) (+ y_m 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 <= 1.1e+84) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * x_m) / (y_m + 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 (x_m <= 1.1d+84) then
        tmp = 0.5d0 * y_m
    else
        tmp = (x_m * x_m) / (y_m + 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 (x_m <= 1.1e+84) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x_m * x_m) / (y_m + 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 x_m <= 1.1e+84:
		tmp = 0.5 * y_m
	else:
		tmp = (x_m * x_m) / (y_m + 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 <= 1.1e+84)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x_m * x_m) / Float64(y_m + 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 (x_m <= 1.1e+84)
		tmp = 0.5 * y_m;
	else
		tmp = (x_m * x_m) / (y_m + 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[x$95$m, 1.1e+84], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.0999999999999999e84

   1. Initial program 72.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 36.3

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

   5. Applied rewrites 36.3%

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

   if 1.0999999999999999e84 < x

   1. Initial program 63.7%

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

   3. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 21.3

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

   5. Applied rewrites 21.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 21.3

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

   7. Applied rewrites 21.3%

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

   8. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 64.0

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

   10. Applied rewrites 64.0%

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

Alternative 14: 34.9% accurate, 6.3× speedup

Math

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

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 71.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 33.4

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

5. Applied rewrites 33.4%

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

Reproduce

herbie shell --seed 2025085 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y 1/2) (* (* (/ 1/2 y) (+ z x)) (- z x))))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))