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

Percentage Accurate: 69.1% → 99.9%

Time: 11.5s

Alternatives: 9

Speedup: 1.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.1% 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: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ 0.5 \cdot \mathsf{fma}\left(x - z\_m, \frac{z\_m + x}{y}, y\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (* 0.5 (fma (- x z_m) (/ (+ z_m x) y) y)))

C

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

Julia

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(0.5 * N[(N[(x - z$95$m), $MachinePrecision] * N[(N[(z$95$m + x), $MachinePrecision] / y), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.1%

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

3. Applied rewrites 91.6%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 99.9%

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

Alternative 2: 54.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \left(\left(-z\_m\right) \cdot \frac{z\_m}{y}\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y}, y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \mathsf{fma}\left(x - z\_m, \frac{z\_m}{y}, y\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -4e-80)
     (* 0.5 (* (- z_m) (/ z_m y)))
     (if (<= t_0 INFINITY)
       (* (fma x (/ x y) y) 0.5)
       (* 0.5 (fma (- x z_m) (/ z_m y) y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -4e-80) {
		tmp = 0.5 * (-z_m * (z_m / y));
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma(x, (x / y), y) * 0.5;
	} else {
		tmp = 0.5 * fma((x - z_m), (z_m / y), y);
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -4e-80)
		tmp = Float64(0.5 * Float64(Float64(-z_m) * Float64(z_m / y)));
	elseif (t_0 <= Inf)
		tmp = Float64(fma(x, Float64(x / y), y) * 0.5);
	else
		tmp = Float64(0.5 * fma(Float64(x - z_m), Float64(z_m / y), y));
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-80], N[(0.5 * N[((-z$95$m) * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[(x - z$95$m), $MachinePrecision] * N[(z$95$m / y), $MachinePrecision] + y), $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))) < -3.99999999999999985e-80

   1. Initial program 78.4%

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-neg-out N/A

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

      4. *-rgt-identity N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 32.8

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

   9. Applied rewrites 32.8%

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

   if -3.99999999999999985e-80 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 76.8%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 67.7%

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

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

   1. Initial program 0.0%

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

   3. Applied rewrites 61.4%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

      1. lower-/.f64 82.9

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

   9. Applied rewrites 82.9%

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

Alternative 3: 51.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2} \leq -4 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \left(\left(-z\_m\right) \cdot \frac{z\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y}, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0)) -4e-80)
   (* 0.5 (* (- z_m) (/ z_m y)))
   (* (fma x (/ x y) y) 0.5)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (((((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)) <= -4e-80) {
		tmp = 0.5 * (-z_m * (z_m / y));
	} else {
		tmp = fma(x, (x / y), y) * 0.5;
	}
	return tmp;
}

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0)) <= -4e-80)
		tmp = Float64(0.5 * Float64(Float64(-z_m) * Float64(z_m / y)));
	else
		tmp = Float64(fma(x, Float64(x / y), y) * 0.5);
	end
	return tmp
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -4e-80], N[(0.5 * N[((-z$95$m) * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(x / y), $MachinePrecision] + y), $MachinePrecision] * 0.5), $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))) < -3.99999999999999985e-80

   1. Initial program 78.4%

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-neg-out N/A

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

      4. *-rgt-identity N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 32.8

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

   9. Applied rewrites 32.8%

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

   if -3.99999999999999985e-80 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 62.3%

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

   2. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 64.9%

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

Alternative 4: 42.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \left(\left(-z\_m\right) \cdot \frac{z\_m}{y}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+152}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -4e-80)
     (* 0.5 (* (- z_m) (/ z_m y)))
     (if (<= t_0 4e+152) (* y 0.5) (* (* (/ x y) x) 0.5)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -4e-80) {
		tmp = 0.5 * (-z_m * (z_m / y));
	} else if (t_0 <= 4e+152) {
		tmp = y * 0.5;
	} else {
		tmp = ((x / y) * x) * 0.5;
	}
	return tmp;
}

Fortran

z_m =     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(x, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)
    if (t_0 <= (-4d-80)) then
        tmp = 0.5d0 * (-z_m * (z_m / y))
    else if (t_0 <= 4d+152) then
        tmp = y * 0.5d0
    else
        tmp = ((x / y) * x) * 0.5d0
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -4e-80) {
		tmp = 0.5 * (-z_m * (z_m / y));
	} else if (t_0 <= 4e+152) {
		tmp = y * 0.5;
	} else {
		tmp = ((x / y) * x) * 0.5;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_0 <= -4e-80:
		tmp = 0.5 * (-z_m * (z_m / y))
	elif t_0 <= 4e+152:
		tmp = y * 0.5
	else:
		tmp = ((x / y) * x) * 0.5
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -4e-80)
		tmp = Float64(0.5 * Float64(Float64(-z_m) * Float64(z_m / y)));
	elseif (t_0 <= 4e+152)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(Float64(Float64(x / y) * x) * 0.5);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -4e-80)
		tmp = 0.5 * (-z_m * (z_m / y));
	elseif (t_0 <= 4e+152)
		tmp = y * 0.5;
	else
		tmp = ((x / y) * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-80], N[(0.5 * N[((-z$95$m) * N[(z$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+152], N[(y * 0.5), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * 0.5), $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))) < -3.99999999999999985e-80

   1. Initial program 78.4%

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-rgt-neg-out N/A

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

      4. *-rgt-identity N/A

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

      5. pow2 N/A

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

      6. associate-/l* N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 32.8

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

   9. Applied rewrites 32.8%

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

   if -3.99999999999999985e-80 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.0000000000000002e152

   1. Initial program 89.1%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-rgt-identity 56.1

          \[\leadsto y \cdot 0.5 \]

   4. Applied rewrites 56.1%

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

   if 4.0000000000000002e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 53.8%

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 36.3%

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

Alternative 5: 37.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-80}:\\ \;\;\;\;\frac{-0.5}{y} \cdot \left(z\_m \cdot z\_m\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+152}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -4e-80)
     (* (/ -0.5 y) (* z_m z_m))
     (if (<= t_0 4e+152) (* y 0.5) (* (* (/ x y) x) 0.5)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -4e-80) {
		tmp = (-0.5 / y) * (z_m * z_m);
	} else if (t_0 <= 4e+152) {
		tmp = y * 0.5;
	} else {
		tmp = ((x / y) * x) * 0.5;
	}
	return tmp;
}

Fortran

z_m =     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(x, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)
    if (t_0 <= (-4d-80)) then
        tmp = ((-0.5d0) / y) * (z_m * z_m)
    else if (t_0 <= 4d+152) then
        tmp = y * 0.5d0
    else
        tmp = ((x / y) * x) * 0.5d0
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -4e-80) {
		tmp = (-0.5 / y) * (z_m * z_m);
	} else if (t_0 <= 4e+152) {
		tmp = y * 0.5;
	} else {
		tmp = ((x / y) * x) * 0.5;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_0 <= -4e-80:
		tmp = (-0.5 / y) * (z_m * z_m)
	elif t_0 <= 4e+152:
		tmp = y * 0.5
	else:
		tmp = ((x / y) * x) * 0.5
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -4e-80)
		tmp = Float64(Float64(-0.5 / y) * Float64(z_m * z_m));
	elseif (t_0 <= 4e+152)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(Float64(Float64(x / y) * x) * 0.5);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -4e-80)
		tmp = (-0.5 / y) * (z_m * z_m);
	elseif (t_0 <= 4e+152)
		tmp = y * 0.5;
	else
		tmp = ((x / y) * x) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-80], N[(N[(-0.5 / y), $MachinePrecision] * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+152], N[(y * 0.5), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * 0.5), $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))) < -3.99999999999999985e-80

   1. Initial program 78.4%

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 81.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 31.5%

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

   if -3.99999999999999985e-80 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.0000000000000002e152

   1. Initial program 89.1%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-rgt-identity 56.1

          \[\leadsto y \cdot 0.5 \]

   4. Applied rewrites 56.1%

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

   if 4.0000000000000002e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 53.8%

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 36.3%

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

Alternative 6: 37.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-80}:\\ \;\;\;\;\frac{-0.5}{y} \cdot \left(z\_m \cdot z\_m\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+152}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{0.5}{y} \cdot x\right) \cdot x\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -4e-80)
     (* (/ -0.5 y) (* z_m z_m))
     (if (<= t_0 4e+152) (* y 0.5) (* (* (/ 0.5 y) x) x)))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -4e-80) {
		tmp = (-0.5 / y) * (z_m * z_m);
	} else if (t_0 <= 4e+152) {
		tmp = y * 0.5;
	} else {
		tmp = ((0.5 / y) * x) * x;
	}
	return tmp;
}

Fortran

z_m =     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(x, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)
    if (t_0 <= (-4d-80)) then
        tmp = ((-0.5d0) / y) * (z_m * z_m)
    else if (t_0 <= 4d+152) then
        tmp = y * 0.5d0
    else
        tmp = ((0.5d0 / y) * x) * x
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -4e-80) {
		tmp = (-0.5 / y) * (z_m * z_m);
	} else if (t_0 <= 4e+152) {
		tmp = y * 0.5;
	} else {
		tmp = ((0.5 / y) * x) * x;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_0 <= -4e-80:
		tmp = (-0.5 / y) * (z_m * z_m)
	elif t_0 <= 4e+152:
		tmp = y * 0.5
	else:
		tmp = ((0.5 / y) * x) * x
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -4e-80)
		tmp = Float64(Float64(-0.5 / y) * Float64(z_m * z_m));
	elseif (t_0 <= 4e+152)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(Float64(Float64(0.5 / y) * x) * x);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -4e-80)
		tmp = (-0.5 / y) * (z_m * z_m);
	elseif (t_0 <= 4e+152)
		tmp = y * 0.5;
	else
		tmp = ((0.5 / y) * x) * x;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-80], N[(N[(-0.5 / y), $MachinePrecision] * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+152], N[(y * 0.5), $MachinePrecision], N[(N[(N[(0.5 / y), $MachinePrecision] * x), $MachinePrecision] * x), $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))) < -3.99999999999999985e-80

   1. Initial program 78.4%

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 81.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 31.5%

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

   if -3.99999999999999985e-80 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.0000000000000002e152

   1. Initial program 89.1%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-rgt-identity 56.1

          \[\leadsto y \cdot 0.5 \]

   4. Applied rewrites 56.1%

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

   if 4.0000000000000002e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 53.8%

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 36.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. pow2 N/A

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

      6. *-commutative N/A

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

      7. associate-*r/ N/A

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

      8. associate-*l/ N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. pow2 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. associate-*r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 36.3

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

   8. Applied rewrites 36.3%

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

Alternative 7: 37.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z\_m \cdot z\_m}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-80}:\\ \;\;\;\;\frac{-0.5}{y} \cdot \left(z\_m \cdot z\_m\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+152}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z_m z_m)) (* y 2.0))))
   (if (<= t_0 -4e-80)
     (* (/ -0.5 y) (* z_m z_m))
     (if (<= t_0 4e+152) (* y 0.5) (/ (* x x) (+ y y))))))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -4e-80) {
		tmp = (-0.5 / y) * (z_m * z_m);
	} else if (t_0 <= 4e+152) {
		tmp = y * 0.5;
	} else {
		tmp = (x * x) / (y + y);
	}
	return tmp;
}

Fortran

z_m =     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(x, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0d0)
    if (t_0 <= (-4d-80)) then
        tmp = ((-0.5d0) / y) * (z_m * z_m)
    else if (t_0 <= 4d+152) then
        tmp = y * 0.5d0
    else
        tmp = (x * x) / (y + y)
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	double tmp;
	if (t_0 <= -4e-80) {
		tmp = (-0.5 / y) * (z_m * z_m);
	} else if (t_0 <= 4e+152) {
		tmp = y * 0.5;
	} else {
		tmp = (x * x) / (y + y);
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0)
	tmp = 0
	if t_0 <= -4e-80:
		tmp = (-0.5 / y) * (z_m * z_m)
	elif t_0 <= 4e+152:
		tmp = y * 0.5
	else:
		tmp = (x * x) / (y + y)
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	t_0 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z_m * z_m)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -4e-80)
		tmp = Float64(Float64(-0.5 / y) * Float64(z_m * z_m));
	elseif (t_0 <= 4e+152)
		tmp = Float64(y * 0.5);
	else
		tmp = Float64(Float64(x * x) / Float64(y + y));
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	t_0 = (((x * x) + (y * y)) - (z_m * z_m)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -4e-80)
		tmp = (-0.5 / y) * (z_m * z_m);
	elseif (t_0 <= 4e+152)
		tmp = y * 0.5;
	else
		tmp = (x * x) / (y + y);
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e-80], N[(N[(-0.5 / y), $MachinePrecision] * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+152], N[(y * 0.5), $MachinePrecision], N[(N[(x * x), $MachinePrecision] / N[(y + y), $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))) < -3.99999999999999985e-80

   1. Initial program 78.4%

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

   3. Applied rewrites 94.9%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.9%

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

   8. Applied rewrites 81.4%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 31.5%

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

   if -3.99999999999999985e-80 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 4.0000000000000002e152

   1. Initial program 89.1%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-rgt-identity 56.1

          \[\leadsto y \cdot 0.5 \]

   4. Applied rewrites 56.1%

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

   if 4.0000000000000002e152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 53.8%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 33.5

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

   4. Applied rewrites 33.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 33.5

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

   6. Applied rewrites 33.5%

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

Alternative 8: 35.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+34}:\\ \;\;\;\;\frac{x \cdot x}{y + y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m)
 :precision binary64
 (if (<= y 2.7e+34) (/ (* x x) (+ y y)) (* y 0.5)))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	double tmp;
	if (y <= 2.7e+34) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Fortran

z_m =     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(x, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 2.7d+34) then
        tmp = (x * x) / (y + y)
    else
        tmp = y * 0.5d0
    end if
    code = tmp
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	double tmp;
	if (y <= 2.7e+34) {
		tmp = (x * x) / (y + y);
	} else {
		tmp = y * 0.5;
	}
	return tmp;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	tmp = 0
	if y <= 2.7e+34:
		tmp = (x * x) / (y + y)
	else:
		tmp = y * 0.5
	return tmp

Julia

z_m = abs(z)
function code(x, y, z_m)
	tmp = 0.0
	if (y <= 2.7e+34)
		tmp = Float64(Float64(x * x) / Float64(y + y));
	else
		tmp = Float64(y * 0.5);
	end
	return tmp
end

MATLAB

z_m = abs(z);
function tmp_2 = code(x, y, z_m)
	tmp = 0.0;
	if (y <= 2.7e+34)
		tmp = (x * x) / (y + y);
	else
		tmp = y * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := If[LessEqual[y, 2.7e+34], N[(N[(x * x), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision], N[(y * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7e34

   1. Initial program 76.7%

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

   2. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 36.8

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

   4. Applied rewrites 36.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 36.8

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

   6. Applied rewrites 36.8%

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

   if 2.7e34 < y

   1. Initial program 42.1%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. metadata-eval N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

      10. distribute-rgt-neg-in N/A

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

      11. metadata-eval N/A

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

      12. *-rgt-identity 63.6

          \[\leadsto y \cdot 0.5 \]

   4. Applied rewrites 63.6%

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

Alternative 9: 33.9% accurate, 5.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ y \cdot 0.5 \end{array} \]

FPCore

z_m = (fabs.f64 z)
(FPCore (x y z_m) :precision binary64 (* y 0.5))

C

z_m = fabs(z);
double code(double x, double y, double z_m) {
	return y * 0.5;
}

Fortran

z_m =     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(x, y, z_m)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = y * 0.5d0
end function

Java

z_m = Math.abs(z);
public static double code(double x, double y, double z_m) {
	return y * 0.5;
}

Python

z_m = math.fabs(z)
def code(x, y, z_m):
	return y * 0.5

Julia

z_m = abs(z)
function code(x, y, z_m)
	return Float64(y * 0.5)
end

MATLAB

z_m = abs(z);
function tmp = code(x, y, z_m)
	tmp = y * 0.5;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
code[x_, y_, z$95$m_] := N[(y * 0.5), $MachinePrecision]

Derivation

1. Initial program 69.1%

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

2. Taylor expanded in y around inf

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

   1. *-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. metadata-eval N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-commutative N/A

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

   6. mul-1-neg N/A

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

   7. lower-*.f64 N/A

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

   8. mul-1-neg N/A

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

   9. *-commutative N/A

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

   10. distribute-rgt-neg-in N/A

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

   11. metadata-eval N/A

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

   12. *-rgt-identity 33.9

       \[\leadsto y \cdot 0.5 \]

4. Applied rewrites 33.9%

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

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64
  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))