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

Percentage Accurate: 69.1% → 98.8%

Time: 8.7s

Alternatives: 8

Speedup: 0.6×

• 42.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \left(z + x\right) \cdot \frac{x - z}{y\_m}\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-58}:\\ \;\;\;\;t\_0 \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \frac{-0.5}{y\_m}, -0.5\right) \cdot \left(-y\_m\right)\\ \end{array} \end{array} \end{array} \]

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)))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -2e-58)
      (* t_0 0.5)
      (if (<= t_1 INFINITY)
        (* (fma (/ x y_m) x y_m) 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 t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -2e-58) {
		tmp = t_0 * 0.5;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x / y_m), x, y_m) * 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))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= -2e-58)
		tmp = Float64(t_0 * 0.5);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 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]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, -2e-58], N[(t$95$0 * 0.5), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 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 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2.0000000000000001e-58

   1. Initial program 65.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 87.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 58.3

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

   8. Applied rewrites 58.3%

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

   if -2.0000000000000001e-58 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 73.7

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

   8. Applied rewrites 73.7%

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

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

   1. Initial program 0.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. 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 61.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

Alternative 2: 97.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{x - z}{y\_m}\\ t_1 := \frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2}\\ y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-58}:\\ \;\;\;\;\left(\left(z + x\right) \cdot t\_0\right) \cdot 0.5\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot t\_0, \frac{-0.5}{y\_m}, -0.5\right) \cdot \left(-y\_m\right)\\ \end{array} \end{array} \end{array} \]

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))
        (t_1 (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0))))
   (*
    y_s
    (if (<= t_1 -2e-58)
      (* (* (+ z x) t_0) 0.5)
      (if (<= t_1 INFINITY)
        (* (fma (/ x y_m) x y_m) 0.5)
        (* (fma (* z t_0) (/ -0.5 y_m) -0.5) (- y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double t_0 = (x - z) / y_m;
	double t_1 = (((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0);
	double tmp;
	if (t_1 <= -2e-58) {
		tmp = ((z + x) * t_0) * 0.5;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	} else {
		tmp = fma((z * t_0), (-0.5 / y_m), -0.5) * -y_m;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	t_0 = Float64(Float64(x - z) / y_m)
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0))
	tmp = 0.0
	if (t_1 <= -2e-58)
		tmp = Float64(Float64(Float64(z + x) * t_0) * 0.5);
	elseif (t_1 <= Inf)
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	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]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * If[LessEqual[t$95$1, -2e-58], N[(N[(N[(z + x), $MachinePrecision] * t$95$0), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $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 (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2.0000000000000001e-58

   1. Initial program 65.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 87.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 58.3

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

   8. Applied rewrites 58.3%

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

   if -2.0000000000000001e-58 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 73.7

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

   8. Applied rewrites 73.7%

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

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

   1. Initial program 0.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. 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 61.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 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 82.4%

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

Alternative 3: 93.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -2 \cdot 10^{-58}:\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y\_m}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -2e-58)
    (* (* (+ z x) (/ (- x z) y_m)) 0.5)
    (* (fma (/ x y_m) x y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e-58) {
		tmp = ((z + x) * ((x - z) / y_m)) * 0.5;
	} else {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -2e-58)
		tmp = Float64(Float64(Float64(z + x) * Float64(Float64(x - z) / y_m)) * 0.5);
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -2e-58], N[(N[(N[(z + x), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2.0000000000000001e-58

   1. Initial program 65.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 87.6%

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

   6. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. sub-div N/A

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

      8. lower-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. sub-div N/A

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

      12. lower-/.f64 N/A

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

      13. lift--.f64 58.3

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

   8. Applied rewrites 58.3%

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

   if -2.0000000000000001e-58 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 70.5

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

   8. Applied rewrites 70.5%

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

Alternative 4: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -2 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(y\_m + z\right) \cdot \left(y\_m - z\right)}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -2e-58)
    (/ (* (+ y_m z) (- y_m z)) (+ y_m y_m))
    (* (fma (/ x y_m) x y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e-58) {
		tmp = ((y_m + z) * (y_m - z)) / (y_m + y_m);
	} else {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -2e-58)
		tmp = Float64(Float64(Float64(y_m + z) * Float64(y_m - z)) / Float64(y_m + y_m));
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -2e-58], N[(N[(N[(y$95$m + z), $MachinePrecision] * N[(y$95$m - z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m + y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2.0000000000000001e-58

   1. Initial program 65.0%

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

   3. Taylor expanded in x around 0

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. difference-of-squares N/A

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

      4. lower-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower--.f64 38.2

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

   5. Applied rewrites 38.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 38.2

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

   7. Applied rewrites 38.2%

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

   if -2.0000000000000001e-58 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 70.5

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

   8. Applied rewrites 70.5%

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

Alternative 5: 92.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -2 \cdot 10^{-58}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y\_m}, x, y\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
(FPCore (y_s x y_m z)
 :precision binary64
 (*
  y_s
  (if (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -2e-58)
    (* -0.5 (/ (* z z) y_m))
    (* (fma (/ x y_m) x y_m) 0.5))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
double code(double y_s, double x, double y_m, double z) {
	double tmp;
	if (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e-58) {
		tmp = -0.5 * ((z * z) / y_m);
	} else {
		tmp = fma((x / y_m), x, y_m) * 0.5;
	}
	return y_s * tmp;
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
function code(y_s, x, y_m, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -2e-58)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y_m));
	else
		tmp = Float64(fma(Float64(x / y_m), x, y_m) * 0.5);
	end
	return Float64(y_s * tmp)
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x_, y$95$m_, z_] := N[(y$95$s * If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -2e-58], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] * x + y$95$m), $MachinePrecision] * 0.5), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2.0000000000000001e-58

   1. Initial program 65.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 25.5

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

   5. Applied rewrites 25.5%

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

   if -2.0000000000000001e-58 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 45.1%

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

   6. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 70.5

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

   8. Applied rewrites 70.5%

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

Alternative 6: 61.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y\_m \cdot y\_m\right) - z \cdot z}{y\_m \cdot 2} \leq -2 \cdot 10^{-58}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 (<= (/ (- (+ (* x x) (* y_m y_m)) (* z z)) (* y_m 2.0)) -2e-58)
    (* -0.5 (/ (* z z) 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 (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e-58) {
		tmp = -0.5 * ((z * z) / 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 (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0d0)) <= (-2d-58)) then
        tmp = (-0.5d0) * ((z * z) / 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 (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e-58) {
		tmp = -0.5 * ((z * z) / 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 ((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e-58:
		tmp = -0.5 * ((z * z) / 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 (Float64(Float64(Float64(Float64(x * x) + Float64(y_m * y_m)) - Float64(z * z)) / Float64(y_m * 2.0)) <= -2e-58)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / 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 (((((x * x) + (y_m * y_m)) - (z * z)) / (y_m * 2.0)) <= -2e-58)
		tmp = -0.5 * ((z * z) / 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[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision], -2e-58], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < -2.0000000000000001e-58

   1. Initial program 65.0%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 25.5

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

   5. Applied rewrites 25.5%

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

   if -2.0000000000000001e-58 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 39.7

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

   5. Applied rewrites 39.7%

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

Alternative 7: 52.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 4.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{x \cdot x}{y\_m + y\_m}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\_m\\ \end{array} \end{array} \]

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 4.8e+64) (/ (* 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 <= 4.8e+64) {
		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 <= 4.8d+64) 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 <= 4.8e+64) {
		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 <= 4.8e+64:
		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 <= 4.8e+64)
		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 <= 4.8e+64)
		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, 4.8e+64], 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 < 4.79999999999999999e64

   1. Initial program 71.5%

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

   3. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 35.6

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

   5. Applied rewrites 35.6%

      \[\leadsto \frac{\color{blue}{x \cdot x}}{y \cdot 2} \]
   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 35.6

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

   7. Applied rewrites 35.6%

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

   if 4.79999999999999999e64 < y

   1. Initial program 35.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

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

Alternative 8: 34.9% 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 63.6%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
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 41.3

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

5. Applied rewrites 41.3%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
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 2025043 
(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)))