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

Percentage Accurate: 70.0% → 97.9%

Time: 4.4s

Alternatives: 11

Speedup: 0.7×

• 41.8% 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: 70.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x - z}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.2 \cdot 10^{-102}:\\ \;\;\;\;\left(\left(z + x\right) \cdot t\_0\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y\_m, 0.5, \left(\left(z + x\right) \cdot \left(\frac{t\_0}{y\_m} \cdot 0.5\right)\right) \cdot y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- x z) y_m)))
   (*
    y_s
    (if (<= y_m 5.2e-102)
      (* (* (+ z x) t_0) 0.5)
      (fma y_m 0.5 (* (* (+ z x) (* (/ t_0 y_m) 0.5)) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (x - z) / y_m;
	double tmp;
	if (y_m <= 5.2e-102) {
		tmp = ((z + x) * t_0) * 0.5;
	} else {
		tmp = fma(y_m, 0.5, (((z + x) * ((t_0 / y_m) * 0.5)) * y_m));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(x - z) / y_m)
	tmp = 0.0
	if (y_m <= 5.2e-102)
		tmp = Float64(Float64(Float64(z + x) * t_0) * 0.5);
	else
		tmp = fma(y_m, 0.5, Float64(Float64(Float64(z + x) * Float64(Float64(t_0 / y_m) * 0.5)) * y_m));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 5.2e-102], N[(N[(N[(z + x), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], N[(y$95$m * 0.5 + N[(N[(N[(z + x), $MachinePrecision] * N[(N[(t$95$0 / y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.19999999999999973e-102

   1. Initial program 90.4%

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

   2. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
   6. 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 96.3

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

   7. Applied rewrites 96.3%

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

   if 5.19999999999999973e-102 < y

   1. Initial program 59.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}}{y}, 0.5, 0.5\right) \cdot y} \]
   5. 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 90.4

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

   6. Applied rewrites 90.4%

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

   8. Applied rewrites 98.7%

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

Alternative 2: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x - z}{y\_m}\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(z + x\right) \cdot t\_0\right) \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y\_m}, y\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \left(\frac{t\_0}{y\_m} \cdot 0.5\right)\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- x z) y_m))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -2e+16)
      (* (* (+ z x) t_0) 0.5)
      (if (<= t_1 INFINITY)
        (* (fma x (/ x y_m) y_m) 0.5)
        (* (* (+ z x) (* (/ t_0 y_m) 0.5)) y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (x - z) / y_m;
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -2e+16) {
		tmp = ((z + x) * t_0) * 0.5;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(x, (x / y_m), y_m) * 0.5;
	} else {
		tmp = ((z + x) * ((t_0 / y_m) * 0.5)) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(x - z) / y_m)
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= -2e+16)
		tmp = Float64(Float64(Float64(z + x) * t_0) * 0.5);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(x, Float64(x / y_m), y_m) * 0.5);
	else
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(t_0 / y_m) * 0.5)) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, -2e+16], N[(N[(N[(z + x), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * N[(x / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(t$95$0 / y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2e16

   1. Initial program 95.7%

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

   2. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
   6. 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 99.5

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

   7. Applied rewrites 99.5%

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

   if -2e16 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 70.9%

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

   2. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 86.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 95.4%

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

   7. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 88.7

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

   9. Applied rewrites 88.7%

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

       1. *-commutative 88.7

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

       2. count-2-rev 88.7

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

       3. lift-*.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. associate-/l* N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-/.f64 96.7

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

   11. Applied rewrites 96.7%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}}{y}, 0.5, 0.5\right) \cdot y} \]
   5. 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 69.9

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

   6. Applied rewrites 69.9%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 71.3

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

   9. Applied rewrites 71.3%

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

Alternative 3: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y\_m}, y\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (* (+ z x) (/ (- x z) y_m)) 0.5))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -2e+16)
      t_0
      (if (<= t_1 INFINITY) (* (fma x (/ x y_m) y_m) 0.5) t_0)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = ((z + x) * ((x - z) / y_m)) * 0.5;
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -2e+16) {
		tmp = t_0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(x, (x / y_m), y_m) * 0.5;
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5)
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= -2e+16)
		tmp = t_0;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(x, Float64(x / y_m), y_m) * 0.5);
	else
		tmp = t_0;
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, -2e+16], t$95$0, If[LessEqual[t$95$1, Infinity], N[(N[(x * N[(x / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2e16 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 68.5%

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

   2. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
   6. 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 91.5

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

   7. Applied rewrites 91.5%

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

   if -2e16 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 70.9%

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

   2. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 86.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 95.4%

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

   7. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 88.7

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

   9. Applied rewrites 88.7%

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

       1. *-commutative 88.7

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

       2. count-2-rev 88.7

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

       3. lift-*.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. associate-/l* N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-/.f64 96.7

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

   11. Applied rewrites 96.7%

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

Alternative 4: 67.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{y\_m} \cdot x\right) \cdot x\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 -2e+16)
      (* -0.5 (/ (* z z) y_m))
      (if (<= t_0 5e+136) (* 0.5 y_m) (* (* (/ 0.5 y_m) x) x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -2e+16) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 5e+136) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((0.5 / y_m) * x) * x;
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    if (t_0 <= (-2d+16)) then
        tmp = (-0.5d0) * ((z * z) / y_m)
    else if (t_0 <= 5d+136) then
        tmp = 0.5d0 * y_m
    else
        tmp = ((0.5d0 / y_m) * x) * x
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -2e+16) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 5e+136) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((0.5 / y_m) * x) * x;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= -2e+16:
		tmp = -0.5 * ((z * z) / y_m)
	elif t_0 <= 5e+136:
		tmp = 0.5 * y_m
	else:
		tmp = ((0.5 / y_m) * x) * x
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -2e+16)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y_m));
	elseif (t_0 <= 5e+136)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(Float64(0.5 / y_m) * x) * x);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= -2e+16)
		tmp = -0.5 * ((z * z) / y_m);
	elseif (t_0 <= 5e+136)
		tmp = 0.5 * y_m;
	else
		tmp = ((0.5 / y_m) * x) * x;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e+16], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+136], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(N[(0.5 / y$95$m), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2e16

   1. Initial program 95.7%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if -2e16 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.0000000000000002e136

   1. Initial program 93.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 68.4

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

   4. Applied rewrites 68.4%

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

   if 5.0000000000000002e136 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 49.4%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 58.6%

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

   5. Taylor expanded in x around inf

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

      1. lift-/.f64 49.7

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

   7. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 54.3

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

   9. Applied rewrites 54.3%

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

Alternative 5: 65.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+16}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+136}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 -2e+16)
      (* -0.5 (/ (* z z) y_m))
      (if (<= t_0 5e+136) (* 0.5 y_m) (/ (* x x) (+ y_m y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -2e+16) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 5e+136) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x * x) / (y_m + y_m);
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    if (t_0 <= (-2d+16)) then
        tmp = (-0.5d0) * ((z * z) / y_m)
    else if (t_0 <= 5d+136) then
        tmp = 0.5d0 * y_m
    else
        tmp = (x * x) / (y_m + y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -2e+16) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 5e+136) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x * x) / (y_m + y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= -2e+16:
		tmp = -0.5 * ((z * z) / y_m)
	elif t_0 <= 5e+136:
		tmp = 0.5 * y_m
	else:
		tmp = (x * x) / (y_m + y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -2e+16)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y_m));
	elseif (t_0 <= 5e+136)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= -2e+16)
		tmp = -0.5 * ((z * z) / y_m);
	elseif (t_0 <= 5e+136)
		tmp = 0.5 * y_m;
	else
		tmp = (x * x) / (y_m + y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e+16], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+136], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2e16

   1. Initial program 95.7%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if -2e16 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.0000000000000002e136

   1. Initial program 93.7%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 68.4

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

   4. Applied rewrites 68.4%

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

   if 5.0000000000000002e136 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 49.4%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 49.8

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

   4. Applied rewrites 49.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 49.8

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

   6. Applied rewrites 49.8%

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

Alternative 6: 91.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\left(x + z\right) \cdot \left(x - z\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y\_m}, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -2e+16)
    (/ (* (+ x z) (- x z)) (+ y_m y_m))
    (* (fma x (/ x y_m) y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e+16) {
		tmp = ((x + z) * (x - z)) / (y_m + y_m);
	} else {
		tmp = fma(x, (x / y_m), y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -2e+16)
		tmp = Float64(Float64(Float64(x + z) * Float64(x - z)) / Float64(y_m + y_m));
	else
		tmp = Float64(fma(x, Float64(x / y_m), y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -2e+16], N[(N[(N[(x + z), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $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))) < -2e16

   1. Initial program 95.7%

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

   2. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 95.3

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

   4. Applied rewrites 95.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 95.3

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

   6. Applied rewrites 95.3%

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

   if -2e16 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.0%

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

   2. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 82.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 82.6

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

   9. Applied rewrites 82.6%

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

       1. *-commutative 82.6

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

       2. count-2-rev 82.6

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

       3. lift-*.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. associate-/l* N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-/.f64 90.3

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

   11. Applied rewrites 90.3%

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

Alternative 7: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -2 \cdot 10^{+16}:\\ \;\;\;\;\left(z \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y\_m}, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -2e+16)
    (* (* z (/ (- x z) y_m)) 0.5)
    (* (fma x (/ x y_m) y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e+16) {
		tmp = (z * ((x - z) / y_m)) * 0.5;
	} else {
		tmp = fma(x, (x / y_m), y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -2e+16)
		tmp = Float64(Float64(z * Float64(Float64(x - z) / y_m)) * 0.5);
	else
		tmp = Float64(fma(x, Float64(x / y_m), y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -2e+16], N[(N[(z * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(x * N[(x / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $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))) < -2e16

   1. Initial program 95.7%

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

   2. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 84.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
   6. 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 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 98.7%

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

   if -2e16 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.0%

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

   2. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 82.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 82.6

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

   9. Applied rewrites 82.6%

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

       1. *-commutative 82.6

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

       2. count-2-rev 82.6

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

       3. lift-*.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. associate-/l* N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-/.f64 90.3

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

   11. Applied rewrites 90.3%

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

Alternative 8: 91.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -2 \cdot 10^{+16}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y\_m}, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -2e+16)
    (* -0.5 (/ (* z z) y_m))
    (* (fma x (/ x y_m) y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e+16) {
		tmp = -0.5 * ((z * z) / y_m);
	} else {
		tmp = fma(x, (x / y_m), y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -2e+16)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y_m));
	else
		tmp = Float64(fma(x, Float64(x / y_m), y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -2e+16], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x / y$95$m), $MachinePrecision] + y$95$m), $MachinePrecision] * 0.5), $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))) < -2e16

   1. Initial program 95.7%

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

   2. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if -2e16 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 61.0%

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

   2. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 82.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 96.0%

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

   7. Taylor expanded in z around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 82.6

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

   9. Applied rewrites 82.6%

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

       1. *-commutative 82.6

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

       2. count-2-rev 82.6

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

       3. lift-*.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 N/A

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

       9. associate-/l* N/A

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

       10. lower-fma.f64 N/A

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

       11. lower-/.f64 90.3

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

   11. Applied rewrites 90.3%

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

Alternative 9: 97.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z + x\right) \cdot \frac{x - z}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5.4 \cdot 10^{-102}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t\_0}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (+ z x) (/ (- x z) y_m))))
   (*
    y_s
    (if (<= y_m 5.4e-102) (* t_0 0.5) (* (fma (/ t_0 y_m) 0.5 0.5) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (z + x) * ((x - z) / y_m);
	double tmp;
	if (y_m <= 5.4e-102) {
		tmp = t_0 * 0.5;
	} else {
		tmp = fma((t_0 / y_m), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(z + x) * Float64(Float64(x - z) / y_m))
	tmp = 0.0
	if (y_m <= 5.4e-102)
		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

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 5.4e-102], 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 < 5.4e-102

   1. Initial program 90.4%

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

   2. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-lft-neg-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   4. Applied rewrites 82.9%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
   6. 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 96.3

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

   7. Applied rewrites 96.3%

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

   if 5.4e-102 < y

   1. Initial program 59.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 83.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{\left(x + z\right) \cdot \left(x - z\right)}{y}}{y}, 0.5, 0.5\right) \cdot y} \]
   5. 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 98.7

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

   6. Applied rewrites 98.7%

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

Alternative 10: 52.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.7 \cdot 10^{+31}:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (* y_s (if (<= y_m 1.7e+31) (/ (* x x) (+ y_m y_m)) (* 0.5 y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.7e+31) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.7d+31) then
        tmp = (x * x) / (y_m + y_m)
    else
        tmp = 0.5d0 * y_m
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.7e+31) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = 0.5 * y_m;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 1.7e+31:
		tmp = (x * x) / (y_m + y_m)
	else:
		tmp = 0.5 * y_m
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.7e+31)
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	else
		tmp = Float64(0.5 * y_m);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.7e+31)
		tmp = (x * x) / (y_m + y_m);
	else
		tmp = 0.5 * y_m;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.7e+31], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.6999999999999999e31

   1. Initial program 91.6%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 44.7

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

   4. Applied rewrites 44.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 44.7

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

   6. Applied rewrites 44.7%

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

   if 1.6999999999999999e31 < y

   1. Initial program 44.0%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 62.2

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

   4. Applied rewrites 62.2%

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

Alternative 11: 34.7% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

2. Taylor expanded in y around inf

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

   1. lower-*.f64 34.7

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

4. Applied rewrites 34.7%

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

Developer Target 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

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 = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Python

def code(x, y, z):
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x))

Julia

function code(x, y, z)
	return Float64(Float64(y * 0.5) - Float64(Float64(Float64(0.5 / y) * Float64(z + x)) * Float64(z - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025093 
(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)))