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

Percentage Accurate: 69.2% → 97.4%

Time: 4.5s

Alternatives: 14

Speedup: 1.0×

• 42.2% 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: 97.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z + x\right) \cdot \frac{x - z}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 7.6 \cdot 10^{-168}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0 \cdot \frac{-0.5}{y\_m}, -y\_m, -0.5 \cdot \left(-y\_m\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (+ z x) (/ (- x z) y_m))))
   (*
    y_s
    (if (<= y_m 7.6e-168)
      (* t_0 0.5)
      (fma (* t_0 (/ -0.5 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 t_0 = (z + x) * ((x - z) / y_m);
	double tmp;
	if (y_m <= 7.6e-168) {
		tmp = t_0 * 0.5;
	} else {
		tmp = fma((t_0 * (-0.5 / y_m)), -y_m, (-0.5 * -y_m));
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(z + x) * Float64(Float64(x - z) / y_m))
	tmp = 0.0
	if (y_m <= 7.6e-168)
		tmp = Float64(t_0 * 0.5);
	else
		tmp = fma(Float64(t_0 * Float64(-0.5 / y_m)), Float64(-y_m), Float64(-0.5 * Float64(-y_m)));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 7.6e-168], N[(t$95$0 * 0.5), $MachinePrecision], N[(N[(t$95$0 * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision] * (-y$95$m) + N[(-0.5 * (-y$95$m)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.6000000000000001e-168

   1. Initial program 74.0%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 75.4

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

   8. Applied rewrites 75.4%

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

   if 7.6000000000000001e-168 < y

   1. Initial program 60.8%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 80.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-in N/A

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

      10. lower-fma.f64 N/A

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

   7. Applied rewrites 99.9%

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

Alternative 2: 97.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x - z}{y\_m}\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(z + x\right) \cdot t\_0\right) \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{x}{y\_m}, \frac{-0.5}{y\_m}, -0.5\right) \cdot \left(-y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, t\_0 \cdot \frac{-0.5}{y\_m}, -0.5\right) \cdot \left(-y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- x z) y_m))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 0.0)
      (* (* (+ z x) t_0) 0.5)
      (if (<= t_1 2e+303)
        t_1
        (if (<= t_1 INFINITY)
          (* (fma (* x (/ x y_m)) (/ -0.5 y_m) -0.5) (- y_m))
          (* (fma z (* t_0 (/ -0.5 y_m)) -0.5) (- y_m))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (x - z) / y_m;
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = ((z + x) * t_0) * 0.5;
	} else if (t_1 <= 2e+303) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x * (x / y_m)), (-0.5 / y_m), -0.5) * -y_m;
	} else {
		tmp = fma(z, (t_0 * (-0.5 / y_m)), -0.5) * -y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(x - z) / y_m)
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(Float64(z + x) * t_0) * 0.5);
	elseif (t_1 <= 2e+303)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x * Float64(x / y_m)), Float64(-0.5 / y_m), -0.5) * Float64(-y_m));
	else
		tmp = Float64(fma(z, Float64(t_0 * Float64(-0.5 / y_m)), -0.5) * Float64(-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, 0.0], N[(N[(N[(z + x), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, 2e+303], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / y$95$m), $MachinePrecision] + -0.5), $MachinePrecision] * (-y$95$m)), $MachinePrecision], N[(N[(z * N[(t$95$0 * N[(-0.5 / y$95$m), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision] * (-y$95$m)), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 78.8%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 71.8

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

   8. Applied rewrites 71.8%

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

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

   1. Initial program 99.6%

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

   if 2e303 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 70.5%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 90.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 80.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 72.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 51.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 94.8%

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

Alternative 3: 96.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+303}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{x}{y\_m}, \frac{-0.5}{y\_m}, -0.5\right) \cdot \left(-y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 2e+303)
        t_1
        (if (<= t_1 INFINITY)
          (* (fma (* x (/ x y_m)) (/ -0.5 y_m) -0.5) (- 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 <= 2e+303) {
		tmp = t_1;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x * (x / y_m)), (-0.5 / y_m), -0.5) * -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 <= 2e+303)
		tmp = t_1;
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x * Float64(x / y_m)), Float64(-0.5 / y_m), -0.5) * Float64(-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, 2e+303], t$95$1, If[LessEqual[t$95$1, Infinity], N[(N[(N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / y$95$m), $MachinePrecision] + -0.5), $MachinePrecision] * (-y$95$m)), $MachinePrecision], t$95$0]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 59.5%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 69.7

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

   8. Applied rewrites 69.7%

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

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

   1. Initial program 99.6%

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

   if 2e303 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 70.5%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 90.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 80.9%

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

   11. Taylor expanded in x around inf

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

   13. Applied rewrites 72.0%

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

Alternative 4: 73.3% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

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 = (z * ((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 <= 2e+153) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((z + x) * (x / y_m)) * 0.5;
	} else {
		tmp = t_0;
	}
	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 = (z * ((x - z) / y_m)) * 0.5
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 2e+153:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = ((z + x) * (x / y_m)) * 0.5
	else:
		tmp = t_0
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(z * 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 <= 2e+153)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(z + x) * Float64(x / y_m)) * 0.5);
	else
		tmp = t_0;
	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 = (z * ((x - z) / y_m)) * 0.5;
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+153)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = ((z + x) * (x / y_m)) * 0.5;
	else
		tmp = t_0;
	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[(z * 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, 2e+153], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(z + x), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 59.5%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 69.7

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

   8. Applied rewrites 69.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 46.5%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 62.5

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

   5. Applied rewrites 62.5%

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

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

   1. Initial program 74.1%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 76.2

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 52.2%

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

Alternative 5: 71.1% accurate, 0.2× speedup

Math

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

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 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 2e+153)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (/ (* x x) (+ 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 - 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 <= 2e+153) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

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 = (z * ((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 <= 2e+153) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	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 = (z * ((x - z) / y_m)) * 0.5
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 2e+153:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = (x * x) / (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(z * 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 <= 2e+153)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	else
		tmp = t_0;
	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 = (z * ((x - z) / y_m)) * 0.5;
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 2e+153)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = (x * x) / (y_m + y_m);
	else
		tmp = t_0;
	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[(z * 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, 2e+153], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 59.5%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 69.7

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

   8. Applied rewrites 69.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 46.5%

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

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

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 62.5

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

   5. Applied rewrites 62.5%

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

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

   1. Initial program 74.1%

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

   3. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 44.4

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

   7. Applied rewrites 44.4%

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

Alternative 6: 67.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-65}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{y\_m} \cdot \left(z \cdot z\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 -2e-65)
      (* -0.5 (/ (* z z) y_m))
      (if (<= t_0 2e+153)
        (* 0.5 y_m)
        (if (<= t_0 INFINITY)
          (/ (* x x) (+ y_m y_m))
          (* (/ -0.5 y_m) (* z z))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -2e-65) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 2e+153) {
		tmp = 0.5 * y_m;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = (-0.5 / y_m) * (z * z);
	}
	return y_s * tmp;
}

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -2e-65) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 2e+153) {
		tmp = 0.5 * y_m;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = (-0.5 / y_m) * (z * z);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= -2e-65:
		tmp = -0.5 * ((z * z) / y_m)
	elif t_0 <= 2e+153:
		tmp = 0.5 * y_m
	elif t_0 <= math.inf:
		tmp = (x * x) / (y_m + y_m)
	else:
		tmp = (-0.5 / y_m) * (z * z)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -2e-65)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y_m));
	elseif (t_0 <= 2e+153)
		tmp = Float64(0.5 * y_m);
	elseif (t_0 <= Inf)
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	else
		tmp = Float64(Float64(-0.5 / y_m) * Float64(z * z));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= -2e-65)
		tmp = -0.5 * ((z * z) / y_m);
	elseif (t_0 <= 2e+153)
		tmp = 0.5 * y_m;
	elseif (t_0 <= Inf)
		tmp = (x * x) / (y_m + y_m);
	else
		tmp = (-0.5 / y_m) * (z * z);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -2e-65], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+153], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 / y$95$m), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -1.99999999999999985e-65

   1. Initial program 81.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 33.6

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

   5. Applied rewrites 33.6%

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

   if -1.99999999999999985e-65 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2e153

   1. Initial program 89.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

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

   1. Initial program 74.1%

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

   3. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 44.4

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

   7. Applied rewrites 44.4%

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

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

   1. Initial program 0.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. div-sub N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. frac-sub N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 0.0%

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

   8. Taylor expanded in z around inf

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

      1. lift-/.f64 40.4

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

   10. Applied rewrites 40.4%

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

Alternative 7: 67.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{-0.5}{y\_m} \cdot \left(z \cdot z\right)\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* (/ -0.5 y_m) (* z z)))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -2e-65)
      t_0
      (if (<= t_1 2e+153)
        (* 0.5 y_m)
        (if (<= t_1 INFINITY) (/ (* x x) (+ 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 = (-0.5 / y_m) * (z * z);
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -2e-65) {
		tmp = t_0;
	} else if (t_1 <= 2e+153) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	return y_s * tmp;
}

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 = (-0.5 / y_m) * (z * z);
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -2e-65) {
		tmp = t_0;
	} else if (t_1 <= 2e+153) {
		tmp = 0.5 * y_m;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (x * x) / (y_m + y_m);
	} else {
		tmp = t_0;
	}
	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 = (-0.5 / y_m) * (z * z)
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_1 <= -2e-65:
		tmp = t_0
	elif t_1 <= 2e+153:
		tmp = 0.5 * y_m
	elif t_1 <= math.inf:
		tmp = (x * x) / (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(-0.5 / y_m) * Float64(z * z))
	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-65)
		tmp = t_0;
	elseif (t_1 <= 2e+153)
		tmp = Float64(0.5 * y_m);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	else
		tmp = t_0;
	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 = (-0.5 / y_m) * (z * z);
	t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_1 <= -2e-65)
		tmp = t_0;
	elseif (t_1 <= 2e+153)
		tmp = 0.5 * y_m;
	elseif (t_1 <= Inf)
		tmp = (x * x) / (y_m + y_m);
	else
		tmp = t_0;
	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[(-0.5 / y$95$m), $MachinePrecision] * N[(z * z), $MachinePrecision]), $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-65], t$95$0, If[LessEqual[t$95$1, 2e+153], N[(0.5 * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -1.99999999999999985e-65 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 60.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. div-sub N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. frac-sub N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 41.5%

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

   8. Taylor expanded in z around inf

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

      1. lift-/.f64 35.3

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

   10. Applied rewrites 35.3%

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

   if -1.99999999999999985e-65 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 2e153

   1. Initial program 89.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

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

   1. Initial program 74.1%

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

   3. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 44.4

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

   5. Applied rewrites 44.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 44.4

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

   7. Applied rewrites 44.4%

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

Alternative 8: 97.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z + x\right) \cdot \frac{x - z}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 7.6 \cdot 10^{-168}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{-0.5}{y\_m}, -0.5\right) \cdot \left(-y\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (* (+ z x) (/ (- x z) y_m))))
   (*
    y_s
    (if (<= y_m 7.6e-168)
      (* t_0 0.5)
      (* (fma t_0 (/ -0.5 y_m) -0.5) (- y_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (z + x) * ((x - z) / y_m);
	double tmp;
	if (y_m <= 7.6e-168) {
		tmp = t_0 * 0.5;
	} else {
		tmp = fma(t_0, (-0.5 / y_m), -0.5) * -y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(z + x) * Float64(Float64(x - z) / y_m))
	tmp = 0.0
	if (y_m <= 7.6e-168)
		tmp = Float64(t_0 * 0.5);
	else
		tmp = Float64(fma(t_0, Float64(-0.5 / y_m), -0.5) * Float64(-y_m));
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[y$95$m, 7.6e-168], N[(t$95$0 * 0.5), $MachinePrecision], N[(N[(t$95$0 * N[(-0.5 / y$95$m), $MachinePrecision] + -0.5), $MachinePrecision] * (-y$95$m)), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.6000000000000001e-168

   1. Initial program 74.0%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 75.4

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

   8. Applied rewrites 75.4%

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

   if 7.6000000000000001e-168 < y

   1. Initial program 60.8%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 80.1%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 99.9%

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

Alternative 9: 88.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.25 \cdot 10^{-146}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 4.7 \cdot 10^{+131}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + x, \frac{z}{y\_m \cdot y\_m} \cdot 0.5, -0.5\right) \cdot \left(-y\_m\right)\\ \end{array} \end{array} \]

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 1.25e-146)
    (* (* (+ z x) (/ (- x z) y_m)) 0.5)
    (if (<= y_m 4.7e+131)
      (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
      (* (fma (+ z 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 <= 1.25e-146) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else if (y_m <= 4.7e+131) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = fma((z + 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 <= 1.25e-146)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	elseif (y_m <= 4.7e+131)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(fma(Float64(z + x), Float64(Float64(z / Float64(y_m * y_m)) * 0.5), -0.5) * Float64(-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, 1.25e-146], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$95$m, 4.7e+131], 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[(N[(N[(z + x), $MachinePrecision] * N[(N[(z / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + -0.5), $MachinePrecision] * (-y$95$m)), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.24999999999999989e-146

   1. Initial program 73.6%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 80.7%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 75.6

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

   8. Applied rewrites 75.6%

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

   if 1.24999999999999989e-146 < y < 4.7e131

   1. Initial program 92.6%

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

   if 4.7e131 < y

   1. Initial program 21.6%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 57.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 72.7

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

   12. Applied rewrites 72.7%

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

Alternative 10: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + x, \frac{z}{y\_m \cdot y\_m} \cdot 0.5, -0.5\right) \cdot \left(-y\_m\right)\\ \end{array} \end{array} \]

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 1.1e+14)
    (* (* (+ z x) (/ (- x z) y_m)) 0.5)
    (* (fma (+ z 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 <= 1.1e+14) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else {
		tmp = fma((z + 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 <= 1.1e+14)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	else
		tmp = Float64(fma(Float64(z + x), Float64(Float64(z / Float64(y_m * y_m)) * 0.5), -0.5) * Float64(-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, 1.1e+14], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(z / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + -0.5), $MachinePrecision] * (-y$95$m)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e14

   1. Initial program 77.3%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 75.9

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

   8. Applied rewrites 75.9%

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

   if 1.1e14 < y

   1. Initial program 46.3%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 71.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift-/.f64 99.9

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 76.3

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

   12. Applied rewrites 76.3%

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

Alternative 11: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10200000:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 1.15 \cdot 10^{+178}:\\ \;\;\;\;\frac{\left(y\_m + z\right) \cdot \left(y\_m - z\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 10200000.0)
    (* (* (+ z x) (/ (- x z) y_m)) 0.5)
    (if (<= y_m 1.15e+178)
      (/ (* (+ y_m z) (- y_m z)) (+ 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 <= 10200000.0) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else if (y_m <= 1.15e+178) {
		tmp = ((y_m + z) * (y_m - z)) / (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 <= 10200000.0d0) then
        tmp = ((z + x) * ((x - z) / y_m)) * 0.5d0
    else if (y_m <= 1.15d+178) then
        tmp = ((y_m + z) * (y_m - z)) / (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 <= 10200000.0) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else if (y_m <= 1.15e+178) {
		tmp = ((y_m + z) * (y_m - z)) / (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 <= 10200000.0:
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5
	elif y_m <= 1.15e+178:
		tmp = ((y_m + z) * (y_m - z)) / (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 <= 10200000.0)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	elseif (y_m <= 1.15e+178)
		tmp = Float64(Float64(Float64(y_m + z) * Float64(y_m - z)) / 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 <= 10200000.0)
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	elseif (y_m <= 1.15e+178)
		tmp = ((y_m + z) * (y_m - z)) / (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, 10200000.0], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$95$m, 1.15e+178], N[(N[(N[(y$95$m + z), $MachinePrecision] * N[(y$95$m - z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.02e7

   1. Initial program 77.3%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. metadata-eval N/A

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

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

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

      7. associate-*r/ N/A

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

      8. pow2 N/A

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

      9. times-frac N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 75.9

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

   8. Applied rewrites 75.9%

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

   if 1.02e7 < y < 1.15e178

   1. Initial program 69.9%

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

   3. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 70.7

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

   7. Applied rewrites 70.7%

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

   if 1.15e178 < y

   1. Initial program 10.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 79.2

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

   5. Applied rewrites 79.2%

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

Alternative 12: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10200000:\\ \;\;\;\;\frac{\left(z + x\right) \cdot \left(x - z\right)}{y\_m + y\_m}\\ \mathbf{elif}\;y\_m \leq 1.15 \cdot 10^{+178}:\\ \;\;\;\;\frac{\left(y\_m + z\right) \cdot \left(y\_m - z\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 10200000.0)
    (/ (* (+ z x) (- x z)) (+ y_m y_m))
    (if (<= y_m 1.15e+178)
      (/ (* (+ y_m z) (- y_m z)) (+ 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 <= 10200000.0) {
		tmp = ((z + x) * (x - z)) / (y_m + y_m);
	} else if (y_m <= 1.15e+178) {
		tmp = ((y_m + z) * (y_m - z)) / (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 <= 10200000.0d0) then
        tmp = ((z + x) * (x - z)) / (y_m + y_m)
    else if (y_m <= 1.15d+178) then
        tmp = ((y_m + z) * (y_m - z)) / (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 <= 10200000.0) {
		tmp = ((z + x) * (x - z)) / (y_m + y_m);
	} else if (y_m <= 1.15e+178) {
		tmp = ((y_m + z) * (y_m - z)) / (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 <= 10200000.0:
		tmp = ((z + x) * (x - z)) / (y_m + y_m)
	elif y_m <= 1.15e+178:
		tmp = ((y_m + z) * (y_m - z)) / (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 <= 10200000.0)
		tmp = Float64(Float64(Float64(z + x) * Float64(x - z)) / Float64(y_m + y_m));
	elseif (y_m <= 1.15e+178)
		tmp = Float64(Float64(Float64(y_m + z) * Float64(y_m - z)) / 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 <= 10200000.0)
		tmp = ((z + x) * (x - z)) / (y_m + y_m);
	elseif (y_m <= 1.15e+178)
		tmp = ((y_m + z) * (y_m - z)) / (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, 10200000.0], N[(N[(N[(z + x), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.15e+178], N[(N[(N[(y$95$m + z), $MachinePrecision] * N[(y$95$m - z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.02e7

   1. Initial program 77.3%

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

   3. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 39.0

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

   5. Applied rewrites 39.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 39.0

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

   7. Applied rewrites 39.0%

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

   8. Taylor expanded in y around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

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

         \[\leadsto \frac{\left(x + z\right) \cdot \color{blue}{\left(x - z\right)}}{y + y} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(z + x\right) \cdot \left(\color{blue}{x} - z\right)}{y + y} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(z + x\right) \cdot \color{blue}{\left(x - z\right)}}{y + y} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\left(z + x\right) \cdot \left(\color{blue}{x} - z\right)}{y + y} \]

      7. lift--.f64 73.1

         \[\leadsto \frac{\left(z + x\right) \cdot \left(x - \color{blue}{z}\right)}{y + y} \]

   10. Applied rewrites 73.1%

       \[\leadsto \frac{\color{blue}{\left(z + x\right) \cdot \left(x - z\right)}}{y + y} \]

   if 1.02e7 < y < 1.15e178

   1. Initial program 69.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{{y}^{2} - {z}^{2}}}{y \cdot 2} \]
   4. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{y \cdot y - {\color{blue}{z}}^{2}}{y \cdot 2} \]

      2. pow2 N/A

         \[\leadsto \frac{y \cdot y - z \cdot \color{blue}{z}}{y \cdot 2} \]

      3. difference-of-squares N/A

         \[\leadsto \frac{\left(y + z\right) \cdot \color{blue}{\left(y - z\right)}}{y \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(y + z\right) \cdot \color{blue}{\left(y - z\right)}}{y \cdot 2} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\left(y + z\right) \cdot \left(\color{blue}{y} - z\right)}{y \cdot 2} \]

      6. lower--.f64 70.7

         \[\leadsto \frac{\left(y + z\right) \cdot \left(y - \color{blue}{z}\right)}{y \cdot 2} \]

   5. Applied rewrites 70.7%

      \[\leadsto \frac{\color{blue}{\left(y + z\right) \cdot \left(y - z\right)}}{y \cdot 2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(y + z\right) \cdot \left(y - z\right)}{\color{blue}{y \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(y + z\right) \cdot \left(y - z\right)}{\color{blue}{2 \cdot y}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{\left(y + z\right) \cdot \left(y - z\right)}{\color{blue}{y + y}} \]

      4. lower-+.f64 70.7

         \[\leadsto \frac{\left(y + z\right) \cdot \left(y - z\right)}{\color{blue}{y + y}} \]

   7. Applied rewrites 70.7%

      \[\leadsto \frac{\left(y + z\right) \cdot \left(y - z\right)}{\color{blue}{y + y}} \]

   if 1.15e178 < y

   1. Initial program 10.9%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 79.2

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   5. Applied rewrites 79.2%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 10200000:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 10200000.0) (/ (* 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 <= 10200000.0) {
		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 <= 10200000.0d0) 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 <= 10200000.0) {
		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 <= 10200000.0:
		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 <= 10200000.0)
		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 <= 10200000.0)
		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, 10200000.0], 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 < 1.02e7

   1. Initial program 77.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{{x}^{2}}}{y \cdot 2} \]
   4. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{x \cdot \color{blue}{x}}{y \cdot 2} \]

      2. lift-*.f64 39.0

         \[\leadsto \frac{x \cdot \color{blue}{x}}{y \cdot 2} \]

   5. Applied rewrites 39.0%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y \cdot 2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{2 \cdot y}} \]

      3. count-2-rev N/A

         \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

      4. lower-+.f64 39.0

         \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

   7. Applied rewrites 39.0%

      \[\leadsto \frac{x \cdot x}{\color{blue}{y + y}} \]

   if 1.02e7 < y

   1. Initial program 46.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. lower-*.f64 58.0

         \[\leadsto 0.5 \cdot \color{blue}{y} \]

   5. Applied rewrites 58.0%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 33.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(0.5 \cdot y\_m\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z) :precision binary64 (* y_s (* 0.5 y_m)))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	return y_s * (0.5 * y_m);
}

Fortran

y\_m =     private
y\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (0.5d0 * y_m)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	return y_s * (0.5 * y_m);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	return y_s * (0.5 * y_m)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	return Float64(y_s * Float64(0.5 * y_m))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp = code(y_s, x, y_m, z)
	tmp = y_s * (0.5 * y_m);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * N[(0.5 * y$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.8%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot y} \]
4. Step-by-step derivation

   1. lower-*.f64 34.9

      \[\leadsto 0.5 \cdot \color{blue}{y} \]

5. Applied rewrites 34.9%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
6. Add Preprocessing

Developer Target 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Python

def code(x, y, z):
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x))

Julia

function code(x, y, z)
	return Float64(Float64(y * 0.5) - Float64(Float64(Float64(0.5 / y) * Float64(z + x)) * Float64(z - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025082 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* y 1/2) (* (* (/ 1/2 y) (+ z x)) (- z x))))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))