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

Percentage Accurate: 69.2% → 98.6%

Time: 3.5s

Alternatives: 11

Speedup: 1.1×

• 41.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 50:\\ \;\;\;\;\frac{\mathsf{fma}\left(x + z, x - z, y\_m \cdot y\_m\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\left(z + x\right) \cdot \frac{x - z}{y\_m}}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

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 50.0)
    (/ (fma (+ x z) (- x z) (* y_m y_m)) (+ y_m y_m))
    (* (fma (/ (* (+ z x) (/ (- x z) y_m)) y_m) 0.5 0.5) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 50.0) {
		tmp = fma((x + z), (x - z), (y_m * y_m)) / (y_m + y_m);
	} else {
		tmp = fma((((z + x) * ((x - z) / y_m)) / y_m), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 50.0)
		tmp = Float64(fma(Float64(x + z), Float64(x - z), Float64(y_m * y_m)) / Float64(y_m + y_m));
	else
		tmp = Float64(fma(Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) / y_m), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 50.0], N[(N[(N[(x + z), $MachinePrecision] * N[(x - z), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 50

   1. Initial program 90.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. difference-of-squares N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 97.2

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. count-2-rev N/A

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

      23. lower-+.f64 97.2

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

   3. Applied rewrites 97.2%

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

   if 50 < y

   1. Initial program 47.8%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 73.9

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

   4. Applied rewrites 73.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 99.9

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

   6. Applied rewrites 99.9%

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

Alternative 2: 95.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x - z}{y\_m}\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x + z, x - z, y\_m \cdot y\_m\right)}{y\_m + y\_m}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot t\_0}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot t\_0}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \end{array} \]

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 (/ (- x z) y_m))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 2e+277)
      (/ (fma (+ x z) (- x z) (* y_m y_m)) (+ y_m y_m))
      (if (<= t_1 INFINITY)
        (* (fma (/ (* x t_0) y_m) 0.5 0.5) y_m)
        (* (fma (/ (* z 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 = (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+277) {
		tmp = fma((x + z), (x - z), (y_m * y_m)) / (y_m + y_m);
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma(((x * t_0) / y_m), 0.5, 0.5) * y_m;
	} else {
		tmp = fma(((z * 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(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+277)
		tmp = Float64(fma(Float64(x + z), Float64(x - z), Float64(y_m * y_m)) / Float64(y_m + y_m));
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(Float64(x * t_0) / y_m), 0.5, 0.5) * y_m);
	else
		tmp = Float64(fma(Float64(Float64(z * 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[(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+277], N[(N[(N[(x + z), $MachinePrecision] * N[(x - z), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(N[(x * t$95$0), $MachinePrecision] / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(z * t$95$0), $MachinePrecision] / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2.00000000000000001e277

   1. Initial program 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. difference-of-squares N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 95.0

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. count-2-rev N/A

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

      23. lower-+.f64 95.0

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

   3. Applied rewrites 95.0%

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

   if 2.00000000000000001e277 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 54.6%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 83.5

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

   4. Applied rewrites 83.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 98.4

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

   6. Applied rewrites 98.4%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 98.4%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 43.6

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

   4. Applied rewrites 43.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 84.1%

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

Alternative 3: 93.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq 2 \cdot 10^{+277}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x + z, x - z, y\_m \cdot y\_m\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x \cdot \frac{x - z}{y\_m}}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

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 (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) 2e+277)
    (/ (fma (+ x z) (- x z) (* y_m y_m)) (+ y_m y_m))
    (* (fma (/ (* x (/ (- x z) y_m)) y_m) 0.5 0.5) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= 2e+277) {
		tmp = fma((x + z), (x - z), (y_m * y_m)) / (y_m + y_m);
	} else {
		tmp = fma(((x * ((x - z) / y_m)) / y_m), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= 2e+277)
		tmp = Float64(fma(Float64(x + z), Float64(x - z), Float64(y_m * y_m)) / Float64(y_m + y_m));
	else
		tmp = Float64(fma(Float64(Float64(x * Float64(Float64(x - z) / y_m)) / y_m), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := 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+277], N[(N[(N[(x + z), $MachinePrecision] * N[(x - z), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(x * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $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))) < 2.00000000000000001e277

   1. Initial program 95.0%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. difference-of-squares N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 95.0

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. count-2-rev N/A

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

      23. lower-+.f64 95.0

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

   3. Applied rewrites 95.0%

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

   if 2.00000000000000001e277 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 42.1%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.3

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

   4. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 98.8

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

   6. Applied rewrites 98.8%

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

   7. Taylor expanded in x around inf

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

   9. Applied rewrites 90.7%

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

Alternative 4: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.5 \cdot 10^{+150}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x + z, x - z, y\_m \cdot y\_m\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot \frac{-z}{y\_m}}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

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 4.5e+150)
    (/ (fma (+ x z) (- x z) (* y_m y_m)) (+ y_m y_m))
    (* (fma (/ (* z (/ (- z) y_m)) y_m) 0.5 0.5) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 4.5e+150) {
		tmp = fma((x + z), (x - z), (y_m * y_m)) / (y_m + y_m);
	} else {
		tmp = fma(((z * (-z / y_m)) / y_m), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 4.5e+150)
		tmp = Float64(fma(Float64(x + z), Float64(x - z), Float64(y_m * y_m)) / Float64(y_m + y_m));
	else
		tmp = Float64(fma(Float64(Float64(z * Float64(Float64(-z) / y_m)) / y_m), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 4.5e+150], N[(N[(N[(x + z), $MachinePrecision] * N[(x - z), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z * N[((-z) / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.5e150

   1. Initial program 88.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. difference-of-squares N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 95.0

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. count-2-rev N/A

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

      23. lower-+.f64 95.0

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

   3. Applied rewrites 95.0%

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

   if 4.5e150 < y

   1. Initial program 11.2%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 57.1

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

   4. Applied rewrites 57.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 88.5%

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

   10. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 88.3

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

   12. Applied rewrites 88.3%

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

Alternative 5: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.2 \cdot 10^{+95}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z \cdot \frac{-z}{y\_m}}{y\_m}, 0.5, 0.5\right) \cdot y\_m\\ \end{array} \end{array} \]

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 1.2e+95)
    (* (* (+ z x) (/ (- x z) y_m)) 0.5)
    (* (fma (/ (* z (/ (- z) y_m)) y_m) 0.5 0.5) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.2e+95) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else {
		tmp = fma(((z * (-z / y_m)) / y_m), 0.5, 0.5) * y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.2e+95)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	else
		tmp = Float64(fma(Float64(Float64(z * Float64(Float64(-z) / y_m)) / y_m), 0.5, 0.5) * y_m);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.2e+95], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(N[(z * N[((-z) / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision] * 0.5 + 0.5), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.2e95

   1. Initial program 89.7%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 84.2

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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
   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 86.0

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

   7. Applied rewrites 86.0%

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

   if 1.2e95 < y

   1. Initial program 29.9%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 64.7

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

   4. Applied rewrites 64.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift--.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 85.2%

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

   10. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 84.9

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

   12. Applied rewrites 84.9%

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

Alternative 6: 80.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y\_m \cdot y\_m\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (* (+ z 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 0.0)
      t_0
      (if (<= t_1 1e+152) (/ (fma x x (* y_m y_m)) (+ y_m y_m)) t_0)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = ((z + 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 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+152) {
		tmp = fma(x, x, (y_m * y_m)) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(z + 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 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 1e+152)
		tmp = Float64(fma(x, x, Float64(y_m * y_m)) / Float64(y_m + y_m));
	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, 0.0], t$95$0, If[LessEqual[t$95$1, 1e+152], N[(N[(x * x + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $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))) < 0.0 or 1e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.5%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. pow2 N/A

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

      9. difference-of-squares N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower--.f64 N/A

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

      13. pow2 N/A

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

      14. lift-*.f64 74.5

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

   4. Applied rewrites 74.5%

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

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]
   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 76.2

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

   7. Applied rewrites 76.2%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1e152

   1. Initial program 99.7%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 98.4

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

   6. Applied rewrites 98.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 98.4

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

   8. Applied rewrites 98.4%

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

Alternative 7: 78.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 8 \cdot 10^{+97}:\\ \;\;\;\;\frac{\left(x - z\right) \cdot \left(z + x\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 8e+97) (/ (* (- x z) (+ z 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 <= 8e+97) {
		tmp = ((x - z) * (z + 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 <= 8d+97) then
        tmp = ((x - z) * (z + 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 <= 8e+97) {
		tmp = ((x - z) * (z + 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 <= 8e+97:
		tmp = ((x - z) * (z + 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 <= 8e+97)
		tmp = Float64(Float64(Float64(x - z) * Float64(z + 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 <= 8e+97)
		tmp = ((x - z) * (z + 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, 8e+97], N[(N[(N[(x - z), $MachinePrecision] * N[(z + x), $MachinePrecision]), $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 < 8.0000000000000006e97

   1. Initial program 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. difference-of-squares N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 96.3

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. count-2-rev N/A

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

      23. lower-+.f64 96.3

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 83.1

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

   6. Applied rewrites 83.1%

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

   if 8.0000000000000006e97 < y

   1. Initial program 29.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 70.0

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

   4. Applied rewrites 70.0%

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

Alternative 8: 71.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -4 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y\_m \cdot y\_m\right)}{y\_m + y\_m}\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -4e-59)
    (* -0.5 (/ (* z z) y_m))
    (/ (fma x x (* y_m y_m)) (+ y_m y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -4e-59) {
		tmp = -0.5 * ((z * z) / y_m);
	} else {
		tmp = fma(x, x, (y_m * y_m)) / (y_m + y_m);
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -4e-59)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y_m));
	else
		tmp = Float64(fma(x, x, Float64(y_m * y_m)) / Float64(y_m + 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_] := 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], -4e-59], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x * x + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -4.0000000000000001e-59

   1. Initial program 95.6%

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

   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 94.6

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

   4. Applied rewrites 94.6%

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

   if -4.0000000000000001e-59 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 59.3%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 62.4

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

   4. Applied rewrites 62.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lift-+.f64 62.4

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

   6. Applied rewrites 62.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. pow2 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. pow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

Alternative 9: 66.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-59}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 10^{+152}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \end{array} \end{array} \end{array} \]

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 -4e-59)
      (* -0.5 (/ (* z z) y_m))
      (if (<= t_0 1e+152) (* 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 <= -4e-59) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 1e+152) {
		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 <= (-4d-59)) then
        tmp = (-0.5d0) * ((z * z) / y_m)
    else if (t_0 <= 1d+152) 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 <= -4e-59) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 1e+152) {
		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 <= -4e-59:
		tmp = -0.5 * ((z * z) / y_m)
	elif t_0 <= 1e+152:
		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 <= -4e-59)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y_m));
	elseif (t_0 <= 1e+152)
		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 <= -4e-59)
		tmp = -0.5 * ((z * z) / y_m);
	elseif (t_0 <= 1e+152)
		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, -4e-59], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+152], 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))) < -4.0000000000000001e-59

   1. Initial program 95.6%

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

   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 94.6

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

   4. Applied rewrites 94.6%

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

   if -4.0000000000000001e-59 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1e152

   1. Initial program 93.2%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 70.2

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

   4. Applied rewrites 70.2%

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

   if 1e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 46.5%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. difference-of-squares N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 56.6

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. count-2-rev N/A

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

      23. lower-+.f64 56.6

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

   3. Applied rewrites 56.6%

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

   4. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 50.3

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

   6. Applied rewrites 50.3%

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

Alternative 10: 51.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 1.2e+95) (/ (* 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 <= 1.2e+95) {
		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 <= 1.2d+95) 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 <= 1.2e+95) {
		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 <= 1.2e+95:
		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 <= 1.2e+95)
		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 <= 1.2e+95)
		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, 1.2e+95], 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 < 1.2e95

   1. Initial program 89.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. pow2 N/A

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

      10. associate--l+ N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. pow2 N/A

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

      14. difference-of-squares N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower--.f64 N/A

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

      18. pow2 N/A

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

      19. lift-*.f64 96.3

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

      20. lift-*.f64 N/A

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

      21. *-commutative N/A

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

      22. count-2-rev N/A

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

      23. lower-+.f64 96.3

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

   3. Applied rewrites 96.3%

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

   4. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 42.4

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

   6. Applied rewrites 42.4%

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

   if 1.2e95 < y

   1. Initial program 29.9%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 69.7

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

   4. Applied rewrites 69.7%

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

Alternative 11: 34.1% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.2%

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

2. Taylor expanded in y around inf

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

   1. lower-*.f64 34.1

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

4. Applied rewrites 34.1%

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

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64
  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))