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

Percentage Accurate: 69.7% → 99.9%

Time: 5.7s

Alternatives: 8

Speedup: 1.3×

• 41.9% 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: 69.7% 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, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.9%

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

3. Taylor expanded in x around 0

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

5. Applied rewrites 99.9%

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

Alternative 2: 67.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-104} \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(z + x\_m\right) \cdot \frac{x\_m - z}{y}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{y}, x\_m, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y y)) (* z z)) (* y 2.0))))
   (if (or (<= t_0 -5e-104) (not (<= t_0 INFINITY)))
     (* (* (+ z x_m) (/ (- x_m z) y)) 0.5)
     (* (fma (/ x_m y) x_m y) 0.5))))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if ((t_0 <= -5e-104) || !(t_0 <= ((double) INFINITY))) {
		tmp = ((z + x_m) * ((x_m - z) / y)) * 0.5;
	} else {
		tmp = fma((x_m / y), x_m, y) * 0.5;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if ((t_0 <= -5e-104) || !(t_0 <= Inf))
		tmp = Float64(Float64(Float64(z + x_m) * Float64(Float64(x_m - z) / y)) * 0.5);
	else
		tmp = Float64(fma(Float64(x_m / y), x_m, y) * 0.5);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e-104], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(N[(z + x$95$m), $MachinePrecision] * N[(N[(x$95$m - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x$95$m / y), $MachinePrecision] * x$95$m + 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))) < -4.99999999999999979e-104 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 67.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. difference-of-squares N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower--.f64 72.4

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

   5. Applied rewrites 72.4%

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

   if -4.99999999999999979e-104 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 72.2%

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

   3. Taylor expanded in z around 0

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

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 66.6%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-104} \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty\right):\\ \;\;\;\;\left(\left(z + x\right) \cdot \frac{x - z}{y}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 37.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\left(-0.5 \cdot z\right) \cdot \frac{z}{y}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x\_m}{y} \cdot x\_m\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -5e-104)
     (* (* -0.5 z) (/ z y))
     (if (<= t_0 4e+146) (* 0.5 y) (* (* (/ x_m y) x_m) 0.5)))))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-104) {
		tmp = (-0.5 * z) * (z / y);
	} else if (t_0 <= 4e+146) {
		tmp = 0.5 * y;
	} else {
		tmp = ((x_m / y) * x_m) * 0.5;
	}
	return tmp;
}

Fortran

x_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_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0d0)
    if (t_0 <= (-5d-104)) then
        tmp = ((-0.5d0) * z) * (z / y)
    else if (t_0 <= 4d+146) then
        tmp = 0.5d0 * y
    else
        tmp = ((x_m / y) * x_m) * 0.5d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-104) {
		tmp = (-0.5 * z) * (z / y);
	} else if (t_0 <= 4e+146) {
		tmp = 0.5 * y;
	} else {
		tmp = ((x_m / y) * x_m) * 0.5;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= -5e-104:
		tmp = (-0.5 * z) * (z / y)
	elif t_0 <= 4e+146:
		tmp = 0.5 * y
	else:
		tmp = ((x_m / y) * x_m) * 0.5
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-104)
		tmp = Float64(Float64(-0.5 * z) * Float64(z / y));
	elseif (t_0 <= 4e+146)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(Float64(x_m / y) * x_m) * 0.5);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -5e-104)
		tmp = (-0.5 * z) * (z / y);
	elseif (t_0 <= 4e+146)
		tmp = 0.5 * y;
	else
		tmp = ((x_m / y) * x_m) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-104], N[(N[(-0.5 * z), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+146], N[(0.5 * y), $MachinePrecision], N[(N[(N[(x$95$m / y), $MachinePrecision] * x$95$m), $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))) < -4.99999999999999979e-104

   1. Initial program 82.8%

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

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 35.2

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

   5. Applied rewrites 35.2%

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

   7. Applied rewrites 35.1%

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

   if -4.99999999999999979e-104 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 3.99999999999999973e146

   1. Initial program 87.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

   if 3.99999999999999973e146 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 49.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 32.3

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

   5. Applied rewrites 32.3%

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

   7. Applied rewrites 35.4%

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

Alternative 4: 35.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\left(-0.5 \cdot z\right) \cdot \frac{z}{y}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y + y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -5e-104)
     (* (* -0.5 z) (/ z y))
     (if (<= t_0 4e+146) (* 0.5 y) (/ (* x_m x_m) (+ y y))))))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-104) {
		tmp = (-0.5 * z) * (z / y);
	} else if (t_0 <= 4e+146) {
		tmp = 0.5 * y;
	} else {
		tmp = (x_m * x_m) / (y + y);
	}
	return tmp;
}

Fortran

x_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_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0d0)
    if (t_0 <= (-5d-104)) then
        tmp = ((-0.5d0) * z) * (z / y)
    else if (t_0 <= 4d+146) then
        tmp = 0.5d0 * y
    else
        tmp = (x_m * x_m) / (y + y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-104) {
		tmp = (-0.5 * z) * (z / y);
	} else if (t_0 <= 4e+146) {
		tmp = 0.5 * y;
	} else {
		tmp = (x_m * x_m) / (y + y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= -5e-104:
		tmp = (-0.5 * z) * (z / y)
	elif t_0 <= 4e+146:
		tmp = 0.5 * y
	else:
		tmp = (x_m * x_m) / (y + y)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-104)
		tmp = Float64(Float64(-0.5 * z) * Float64(z / y));
	elseif (t_0 <= 4e+146)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x_m * x_m) / Float64(y + y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -5e-104)
		tmp = (-0.5 * z) * (z / y);
	elseif (t_0 <= 4e+146)
		tmp = 0.5 * y;
	else
		tmp = (x_m * x_m) / (y + y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-104], N[(N[(-0.5 * z), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+146], N[(0.5 * y), $MachinePrecision], N[(N[(x$95$m * x$95$m), $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))) < -4.99999999999999979e-104

   1. Initial program 82.8%

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

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 35.2

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

   5. Applied rewrites 35.2%

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

   7. Applied rewrites 35.1%

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

   if -4.99999999999999979e-104 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 3.99999999999999973e146

   1. Initial program 87.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

   if 3.99999999999999973e146 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 49.4%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. difference-of-squares N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 64.1

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

   5. Applied rewrites 64.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 64.1

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 32.3

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

   10. Applied rewrites 32.3%

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

Alternative 5: 35.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-104}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+146}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y + y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x_m x_m) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 -5e-104)
     (* -0.5 (/ (* z z) y))
     (if (<= t_0 4e+146) (* 0.5 y) (/ (* x_m x_m) (+ y y))))))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-104) {
		tmp = -0.5 * ((z * z) / y);
	} else if (t_0 <= 4e+146) {
		tmp = 0.5 * y;
	} else {
		tmp = (x_m * x_m) / (y + y);
	}
	return tmp;
}

Fortran

x_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_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0d0)
    if (t_0 <= (-5d-104)) then
        tmp = (-0.5d0) * ((z * z) / y)
    else if (t_0 <= 4d+146) then
        tmp = 0.5d0 * y
    else
        tmp = (x_m * x_m) / (y + y)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= -5e-104) {
		tmp = -0.5 * ((z * z) / y);
	} else if (t_0 <= 4e+146) {
		tmp = 0.5 * y;
	} else {
		tmp = (x_m * x_m) / (y + y);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_0 <= -5e-104:
		tmp = -0.5 * ((z * z) / y)
	elif t_0 <= 4e+146:
		tmp = 0.5 * y
	else:
		tmp = (x_m * x_m) / (y + y)
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	t_0 = Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_0 <= -5e-104)
		tmp = Float64(-0.5 * Float64(Float64(z * z) / y));
	elseif (t_0 <= 4e+146)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x_m * x_m) / Float64(y + y));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	t_0 = (((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_0 <= -5e-104)
		tmp = -0.5 * ((z * z) / y);
	elseif (t_0 <= 4e+146)
		tmp = 0.5 * y;
	else
		tmp = (x_m * x_m) / (y + y);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-104], N[(-0.5 * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+146], N[(0.5 * y), $MachinePrecision], N[(N[(x$95$m * x$95$m), $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))) < -4.99999999999999979e-104

   1. Initial program 82.8%

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

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 35.2

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

   5. Applied rewrites 35.2%

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

   if -4.99999999999999979e-104 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 3.99999999999999973e146

   1. Initial program 87.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 50.9

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

   5. Applied rewrites 50.9%

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

   if 3.99999999999999973e146 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 49.4%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. difference-of-squares N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 64.1

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

   5. Applied rewrites 64.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 64.1

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 32.3

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

   10. Applied rewrites 32.3%

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

Alternative 6: 50.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x\_m \cdot x\_m + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -5 \cdot 10^{-104}:\\ \;\;\;\;\left(-0.5 \cdot z\right) \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{y}, x\_m, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= (/ (- (+ (* x_m x_m) (* y y)) (* z z)) (* y 2.0)) -5e-104)
   (* (* -0.5 z) (/ z y))
   (* (fma (/ x_m y) x_m y) 0.5)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (((((x_m * x_m) + (y * y)) - (z * z)) / (y * 2.0)) <= -5e-104) {
		tmp = (-0.5 * z) * (z / y);
	} else {
		tmp = fma((x_m / y), x_m, y) * 0.5;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x_m * x_m) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0)) <= -5e-104)
		tmp = Float64(Float64(-0.5 * z) * Float64(z / y));
	else
		tmp = Float64(fma(Float64(x_m / y), x_m, y) * 0.5);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -5e-104], N[(N[(-0.5 * z), $MachinePrecision] * N[(z / y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / y), $MachinePrecision] * x$95$m + 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))) < -4.99999999999999979e-104

   1. Initial program 82.8%

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

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

      2. lower-/.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 35.2

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

   5. Applied rewrites 35.2%

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

   7. Applied rewrites 35.1%

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

   if -4.99999999999999979e-104 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 59.5%

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

   3. Taylor expanded in z around 0

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

      1. div-add N/A

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

      2. distribute-lft-in N/A

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

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

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. distribute-lft-in N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 64.2%

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

Alternative 7: 40.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+117}:\\ \;\;\;\;\frac{x\_m \cdot x\_m}{y + y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= y 5e+117) (/ (* x_m x_m) (+ y y)) (* 0.5 y)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 5e+117) {
		tmp = (x_m * x_m) / (y + y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_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_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 5d+117) then
        tmp = (x_m * x_m) / (y + y)
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 5e+117) {
		tmp = (x_m * x_m) / (y + y);
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if y <= 5e+117:
		tmp = (x_m * x_m) / (y + y)
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (y <= 5e+117)
		tmp = Float64(Float64(x_m * x_m) / Float64(y + y));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if (y <= 5e+117)
		tmp = (x_m * x_m) / (y + y);
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[y, 5e+117], N[(N[(x$95$m * x$95$m), $MachinePrecision] / N[(y + y), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999983e117

   1. Initial program 79.3%

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

   3. Taylor expanded in y around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. difference-of-squares N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 72.3

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

   5. Applied rewrites 72.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 72.3

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

   7. Applied rewrites 72.3%

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

   8. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 35.6

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

   10. Applied rewrites 35.6%

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

   if 4.99999999999999983e117 < y

   1. Initial program 15.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 78.1

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

   5. Applied rewrites 78.1%

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

Alternative 8: 35.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

x_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_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 69.9%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 32.6

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

5. Applied rewrites 32.6%

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

Developer Target 1: 99.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024352 
(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)))