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

Percentage Accurate: 68.6% → 96.8%

Time: 3.9s

Alternatives: 11

Speedup: 1.1×

• 41.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.6% 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: 96.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z + x\right) \cdot \frac{x - z}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.65 \cdot 10^{-47}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{-0.5}{y\_m}, -0.5\right) \cdot \left(-y\_m\right)\\ \end{array} \end{array} \end{array} \]

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 1.65e-47)
      (* 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 <= 1.65e-47) {
		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 <= 1.65e-47)
		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, 1.65e-47], 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 < 1.65000000000000002e-47

   1. Initial program 90.4%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} - \frac{1}{2}\right)\right)} \]
   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 85.3%

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

   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 93.2

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

   8. Applied rewrites 93.2%

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

   if 1.65000000000000002e-47 < y

   1. Initial program 52.4%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} - \frac{1}{2}\right)\right)} \]
   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 81.5%

      \[\leadsto \color{blue}{\left(-y\right) \cdot \mathsf{fma}\left(\frac{-0.5}{y}, \frac{\left(x + z\right) \cdot \left(x - z\right)}{y}, -0.5\right)} \]
   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 99.5%

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

Alternative 2: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 10^{+286}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\left(z + x\right) \cdot \frac{x}{y\_m}, \frac{-0.5}{y\_m}, -0.5\right) \cdot \left(-y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot \frac{x - z}{y\_m}, \frac{-0.5}{y\_m}, -0.5\right) \cdot \left(-y\_m\right)\\ \end{array} \end{array} \end{array} \]

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 x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 1e+286)
      t_0
      (if (<= t_0 INFINITY)
        (* (fma (* (+ z x) (/ x y_m)) (/ -0.5 y_m) -0.5) (- y_m))
        (* (fma (* z (/ (- x z) y_m)) (/ -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 * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= 1e+286) {
		tmp = t_0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(((z + x) * (x / y_m)), (-0.5 / y_m), -0.5) * -y_m;
	} else {
		tmp = fma((z * ((x - z) / y_m)), (-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(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= 1e+286)
		tmp = t_0;
	elseif (t_0 <= Inf)
		tmp = Float64(fma(Float64(Float64(z + x) * Float64(x / y_m)), Float64(-0.5 / y_m), -0.5) * Float64(-y_m));
	else
		tmp = Float64(fma(Float64(z * Float64(Float64(x - z) / y_m)), 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[(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, 1e+286], t$95$0, If[LessEqual[t$95$0, Infinity], N[(N[(N[(N[(z + x), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / y$95$m), $MachinePrecision] + -0.5), $MachinePrecision] * (-y$95$m)), $MachinePrecision], N[(N[(N[(z * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / y$95$m), $MachinePrecision] + -0.5), $MachinePrecision] * (-y$95$m)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 1.00000000000000003e286

   1. Initial program 94.6%

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

   if 1.00000000000000003e286 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 53.7%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} - \frac{1}{2}\right)\right)} \]
   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 86.6%

      \[\leadsto \color{blue}{\left(-y\right) \cdot \mathsf{fma}\left(\frac{-0.5}{y}, \frac{\left(x + z\right) \cdot \left(x - z\right)}{y}, -0.5\right)} \]
   6. 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 98.6%

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

   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 98.6%

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

   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 59.6%

      \[\leadsto \color{blue}{\left(-y\right) \cdot \mathsf{fma}\left(\frac{-0.5}{y}, \frac{\left(x + z\right) \cdot \left(x - z\right)}{y}, -0.5\right)} \]
   6. 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 99.8%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 82.9%

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

Alternative 3: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -100000000000:\\ \;\;\;\;\left(z \cdot \frac{z}{y\_m}\right) \cdot -0.5\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 -100000000000.0)
      (* (* z (/ z y_m)) -0.5)
      (if (<= t_0 5e+144) (* 0.5 y_m) (* (* (/ x y_m) 0.5) x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = (z * (z / y_m)) * -0.5;
	} else if (t_0 <= 5e+144) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((x / y_m) * 0.5) * x;
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    if (t_0 <= (-100000000000.0d0)) then
        tmp = (z * (z / y_m)) * (-0.5d0)
    else if (t_0 <= 5d+144) then
        tmp = 0.5d0 * y_m
    else
        tmp = ((x / y_m) * 0.5d0) * x
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = (z * (z / y_m)) * -0.5;
	} else if (t_0 <= 5e+144) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((x / y_m) * 0.5) * x;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= -100000000000.0:
		tmp = (z * (z / y_m)) * -0.5
	elif t_0 <= 5e+144:
		tmp = 0.5 * y_m
	else:
		tmp = ((x / y_m) * 0.5) * x
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -100000000000.0)
		tmp = Float64(Float64(z * Float64(z / y_m)) * -0.5);
	elseif (t_0 <= 5e+144)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(Float64(x / y_m) * 0.5) * x);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= -100000000000.0)
		tmp = (z * (z / y_m)) * -0.5;
	elseif (t_0 <= 5e+144)
		tmp = 0.5 * y_m;
	else
		tmp = ((x / y_m) * 0.5) * x;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -100000000000.0], N[(N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], If[LessEqual[t$95$0, 5e+144], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -1e11

   1. Initial program 94.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 84.4%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 94.1

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

   8. Applied rewrites 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 99.3

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

   10. Applied rewrites 99.3%

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

   if -1e11 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.9999999999999999e144

   1. Initial program 93.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   if 4.9999999999999999e144 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 47.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in x around inf

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

      1. lift-/.f64 49.6

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

   8. Applied rewrites 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 54.5

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

   10. Applied rewrites 54.5%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 54.5

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

   13. Applied rewrites 54.5%

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

Alternative 4: 67.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -100000000000:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 -100000000000.0)
      (* -0.5 (/ (* z z) y_m))
      (if (<= t_0 5e+144) (* 0.5 y_m) (* (* (/ x y_m) 0.5) x))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 5e+144) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((x / y_m) * 0.5) * x;
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    if (t_0 <= (-100000000000.0d0)) then
        tmp = (-0.5d0) * ((z * z) / y_m)
    else if (t_0 <= 5d+144) then
        tmp = 0.5d0 * y_m
    else
        tmp = ((x / y_m) * 0.5d0) * x
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 5e+144) {
		tmp = 0.5 * y_m;
	} else {
		tmp = ((x / y_m) * 0.5) * x;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= -100000000000.0:
		tmp = -0.5 * ((z * z) / y_m)
	elif t_0 <= 5e+144:
		tmp = 0.5 * y_m
	else:
		tmp = ((x / y_m) * 0.5) * x
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -100000000000.0)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y_m));
	elseif (t_0 <= 5e+144)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(Float64(x / y_m) * 0.5) * x);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= -100000000000.0)
		tmp = -0.5 * ((z * z) / y_m);
	elseif (t_0 <= 5e+144)
		tmp = 0.5 * y_m;
	else
		tmp = ((x / y_m) * 0.5) * x;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -100000000000.0], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+144], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -1e11

   1. Initial program 94.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

   if -1e11 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.9999999999999999e144

   1. Initial program 93.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   if 4.9999999999999999e144 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 47.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in x around inf

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

      1. lift-/.f64 49.6

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

   8. Applied rewrites 49.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 54.5

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

   10. Applied rewrites 54.5%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 54.5

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

   13. Applied rewrites 54.5%

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

Alternative 5: 64.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -100000000000:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y\_m}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+144}:\\ \;\;\;\;0.5 \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_0 -100000000000.0)
      (* -0.5 (/ (* z z) y_m))
      (if (<= t_0 5e+144) (* 0.5 y_m) (/ (* x x) (+ y_m y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 5e+144) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x * x) / (y_m + y_m);
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    if (t_0 <= (-100000000000.0d0)) then
        tmp = (-0.5d0) * ((z * z) / y_m)
    else if (t_0 <= 5d+144) then
        tmp = 0.5d0 * y_m
    else
        tmp = (x * x) / (y_m + y_m)
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_0 <= -100000000000.0) {
		tmp = -0.5 * ((z * z) / y_m);
	} else if (t_0 <= 5e+144) {
		tmp = 0.5 * y_m;
	} else {
		tmp = (x * x) / (y_m + y_m);
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	tmp = 0
	if t_0 <= -100000000000.0:
		tmp = -0.5 * ((z * z) / y_m)
	elif t_0 <= 5e+144:
		tmp = 0.5 * y_m
	else:
		tmp = (x * x) / (y_m + y_m)
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_0 <= -100000000000.0)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y_m));
	elseif (t_0 <= 5e+144)
		tmp = Float64(0.5 * y_m);
	else
		tmp = Float64(Float64(x * x) / Float64(y_m + y_m));
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	t_0 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	tmp = 0.0;
	if (t_0 <= -100000000000.0)
		tmp = -0.5 * ((z * z) / y_m);
	elseif (t_0 <= 5e+144)
		tmp = 0.5 * y_m;
	else
		tmp = (x * x) / (y_m + y_m);
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$0, -100000000000.0], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+144], N[(0.5 * y$95$m), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -1e11

   1. Initial program 94.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 94.1

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

   5. Applied rewrites 94.1%

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

   if -1e11 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.9999999999999999e144

   1. Initial program 93.2%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 67.2

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

   5. Applied rewrites 67.2%

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

   if 4.9999999999999999e144 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 47.2%

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

   3. Taylor expanded in x around 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 49.6

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

   5. Applied rewrites 49.6%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   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 49.6

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

   7. Applied rewrites 49.6%

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

Alternative 6: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x - z}{y\_m}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.75 \cdot 10^{-88}:\\ \;\;\;\;\left(\left(z + x\right) \cdot t\_0\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 5.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t\_0, \frac{-0.5}{y\_m}, -0.5\right) \cdot \left(-y\_m\right)\\ \end{array} \end{array} \end{array} \]

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 (/ (- x z) y_m)))
   (*
    y_s
    (if (<= y_m 1.75e-88)
      (* (* (+ z x) t_0) 0.5)
      (if (<= y_m 5.2e+132)
        (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
        (* (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 tmp;
	if (y_m <= 1.75e-88) {
		tmp = ((z + x) * t_0) * 0.5;
	} else if (y_m <= 5.2e+132) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} 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)
	tmp = 0.0
	if (y_m <= 1.75e-88)
		tmp = Float64(Float64(Float64(z + x) * t_0) * 0.5);
	elseif (y_m <= 5.2e+132)
		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 * 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]}, N[(y$95$s * If[LessEqual[y$95$m, 1.75e-88], N[(N[(N[(z + x), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$95$m, 5.2e+132], 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 * t$95$0), $MachinePrecision] * N[(-0.5 / y$95$m), $MachinePrecision] + -0.5), $MachinePrecision] * (-y$95$m)), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if y < 1.7500000000000001e-88

   1. Initial program 89.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 83.5%

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

   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 95.1

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

   8. Applied rewrites 95.1%

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

   if 1.7500000000000001e-88 < y < 5.2e132

   1. Initial program 88.5%

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

   if 5.2e132 < y

   1. Initial program 17.7%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} - \frac{1}{2}\right)\right)} \]
   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. Taylor expanded in x around 0

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

   10. Applied rewrites 87.0%

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

Alternative 7: 90.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 1.75 \cdot 10^{-88}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \mathbf{elif}\;y\_m \leq 2.25 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m - z \cdot \frac{z}{y\_m}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= y_m 1.75e-88)
    (* (* (+ z x) (/ (- x z) y_m)) 0.5)
    (if (<= y_m 2.25e+154)
      (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))
      (* (- y_m (* z (/ z y_m))) 0.5)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.75e-88) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else if (y_m <= 2.25e+154) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = (y_m - (z * (z / y_m))) * 0.5;
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1.75d-88) then
        tmp = ((z + x) * ((x - z) / y_m)) * 0.5d0
    else if (y_m <= 2.25d+154) then
        tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)
    else
        tmp = (y_m - (z * (z / y_m))) * 0.5d0
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (y_m <= 1.75e-88) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else if (y_m <= 2.25e+154) {
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	} else {
		tmp = (y_m - (z * (z / y_m))) * 0.5;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if y_m <= 1.75e-88:
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5
	elif y_m <= 2.25e+154:
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)
	else:
		tmp = (y_m - (z * (z / y_m))) * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (y_m <= 1.75e-88)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	elseif (y_m <= 2.25e+154)
		tmp = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0));
	else
		tmp = Float64(Float64(y_m - Float64(z * Float64(z / y_m))) * 0.5);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if (y_m <= 1.75e-88)
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	elseif (y_m <= 2.25e+154)
		tmp = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	else
		tmp = (y_m - (z * (z / y_m))) * 0.5;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[y$95$m, 1.75e-88], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$95$m, 2.25e+154], 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[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < 1.7500000000000001e-88

   1. Initial program 89.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 83.5%

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

   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 95.1

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

   8. Applied rewrites 95.1%

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

   if 1.7500000000000001e-88 < y < 2.25000000000000005e154

   1. Initial program 87.4%

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

   if 2.25000000000000005e154 < y

   1. Initial program 9.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 35.2%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 71.8

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

   8. Applied rewrites 71.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 87.9

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

   10. Applied rewrites 87.9%

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

Alternative 8: 80.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{+196}:\\ \;\;\;\;\left(y\_m - z \cdot \frac{z}{y\_m}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (* x x) 5e+196)
    (* (- y_m (* z (/ z y_m))) 0.5)
    (* (* (/ x y_m) 0.5) x))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((x * x) <= 5e+196) {
		tmp = (y_m - (z * (z / y_m))) * 0.5;
	} else {
		tmp = ((x / y_m) * 0.5) * x;
	}
	return y_s * tmp;
}

Fortran

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

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

real(8) function code(y_s, x, y_m, z)
use fmin_fmax_functions
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 5d+196) then
        tmp = (y_m - (z * (z / y_m))) * 0.5d0
    else
        tmp = ((x / y_m) * 0.5d0) * x
    end if
    code = y_s * tmp
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
public static double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if ((x * x) <= 5e+196) {
		tmp = (y_m - (z * (z / y_m))) * 0.5;
	} else {
		tmp = ((x / y_m) * 0.5) * x;
	}
	return y_s * tmp;
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
def code(y_s, x, y_m, z):
	tmp = 0
	if (x * x) <= 5e+196:
		tmp = (y_m - (z * (z / y_m))) * 0.5
	else:
		tmp = ((x / y_m) * 0.5) * x
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(x * x) <= 5e+196)
		tmp = Float64(Float64(y_m - Float64(z * Float64(z / y_m))) * 0.5);
	else
		tmp = Float64(Float64(Float64(x / y_m) * 0.5) * x);
	end
	return Float64(y_s * tmp)
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x, y_m, z)
	tmp = 0.0;
	if ((x * x) <= 5e+196)
		tmp = (y_m - (z * (z / y_m))) * 0.5;
	else
		tmp = ((x / y_m) * 0.5) * x;
	end
	tmp_2 = y_s * tmp;
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(x * x), $MachinePrecision], 5e+196], N[(N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.9999999999999998e196

   1. Initial program 72.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 78.1

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

   8. Applied rewrites 78.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 85.0

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

   10. Applied rewrites 85.0%

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

   if 4.9999999999999998e196 < (*.f64 x x)

   1. Initial program 61.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in x around inf

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

      1. lift-/.f64 63.2

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

   8. Applied rewrites 63.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 70.9

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

   10. Applied rewrites 70.9%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 70.9

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

   13. Applied rewrites 70.9%

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

Alternative 9: 75.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 580000000000:\\ \;\;\;\;\left(y\_m - z \cdot \frac{z}{y\_m}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= x 580000000000.0)
    (* (- y_m (* z (/ z y_m))) 0.5)
    (* (* (+ z x) (/ (- x z) y_m)) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (x <= 580000000000.0) {
		tmp = (y_m - (z * (z / y_m))) * 0.5;
	} else {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	}
	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 (x <= 580000000000.0d0) then
        tmp = (y_m - (z * (z / y_m))) * 0.5d0
    else
        tmp = ((z + x) * ((x - z) / y_m)) * 0.5d0
    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 (x <= 580000000000.0) {
		tmp = (y_m - (z * (z / y_m))) * 0.5;
	} else {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	}
	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 x <= 580000000000.0:
		tmp = (y_m - (z * (z / y_m))) * 0.5
	else:
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5
	return y_s * tmp

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (x <= 580000000000.0)
		tmp = Float64(Float64(y_m - Float64(z * Float64(z / y_m))) * 0.5);
	else
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	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 (x <= 580000000000.0)
		tmp = (y_m - (z * (z / y_m))) * 0.5;
	else
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	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[x, 580000000000.0], N[(N[(y$95$m - N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.8e11

   1. Initial program 70.1%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift--.f64 68.2

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

   8. Applied rewrites 68.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 74.4

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

   10. Applied rewrites 74.4%

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

   if 5.8e11 < x

   1. Initial program 63.9%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(\frac{-1}{2} \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}} - \frac{1}{2}\right)\right)} \]
   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 81.3%

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

   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 80.0

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

   8. Applied rewrites 80.0%

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

Alternative 10: 52.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 540000000000:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 540000000000.0) (/ (* 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 <= 540000000000.0) {
		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 <= 540000000000.0d0) 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 <= 540000000000.0) {
		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 <= 540000000000.0:
		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 <= 540000000000.0)
		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 <= 540000000000.0)
		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, 540000000000.0], 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 < 5.4e11

   1. Initial program 91.2%

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

   3. Taylor expanded in x around 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 45.9

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

   5. Applied rewrites 45.9%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   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 45.9

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

   7. Applied rewrites 45.9%

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

   if 5.4e11 < y

   1. Initial program 44.4%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 59.6

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

   5. Applied rewrites 59.6%

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

Alternative 11: 34.6% 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 68.6%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 34.6

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

5. Applied rewrites 34.6%

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

Developer Target 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025089 
(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)))