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

Percentage Accurate: 67.7% → 99.9%

Time: 4.4s

Alternatives: 7

Speedup: 1.3×

• 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: 67.7% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

3. Taylor expanded in x around 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: 40.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ t_1 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+142}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\left(\frac{x}{y} \cdot x\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -0.5 (* z (/ z y))))
        (t_1 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_1 0.0)
     t_0
     (if (<= t_1 5e+142)
       (* 0.5 y)
       (if (<= t_1 INFINITY) (* (* (/ x y) x) 0.5) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+142) {
		tmp = 0.5 * y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = ((x / y) * x) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = -0.5 * (z * (z / y));
	double t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = t_0;
	} else if (t_1 <= 5e+142) {
		tmp = 0.5 * y;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = ((x / y) * x) * 0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -0.5 * (z * (z / y))
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if t_1 <= 0.0:
		tmp = t_0
	elif t_1 <= 5e+142:
		tmp = 0.5 * y
	elif t_1 <= math.inf:
		tmp = ((x / y) * x) * 0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-0.5 * Float64(z * Float64(z / y)))
	t_1 = Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+142)
		tmp = Float64(0.5 * y);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(Float64(x / y) * x) * 0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -0.5 * (z * (z / y));
	t_1 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = t_0;
	elseif (t_1 <= 5e+142)
		tmp = 0.5 * y;
	elseif (t_1 <= Inf)
		tmp = ((x / y) * x) * 0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], t$95$0, If[LessEqual[t$95$1, 5e+142], N[(0.5 * y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 36.7

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

   5. Applied rewrites 36.7%

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

   7. Applied rewrites 39.6%

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

   if 0.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < 5.0000000000000001e142

   1. Initial program 99.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 57.1

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

   5. Applied rewrites 57.1%

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

   if 5.0000000000000001e142 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64))) < +inf.0

   1. Initial program 70.9%

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

   3. Taylor expanded in x around 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 41.2

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

   5. Applied rewrites 41.2%

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

   7. Applied rewrites 44.4%

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

Alternative 3: 68.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;\left(\left(x - z\right) \cdot \frac{z + x}{y}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (or (<= t_0 0.0) (not (<= t_0 INFINITY)))
     (* (* (- x z) (/ (+ z x) y)) 0.5)
     (* (fma (/ x y) x y) 0.5))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= ((double) INFINITY))) {
		tmp = ((x - z) * ((z + x) / y)) * 0.5;
	} else {
		tmp = fma((x / y), x, y) * 0.5;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(N[(N[(x - z), $MachinePrecision] * N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + 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))) < 0.0 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 64.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Taylor expanded in y around 0

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

   8. Applied rewrites 71.1%

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

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

   1. Initial program 76.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 73.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 0 \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty\right):\\ \;\;\;\;\left(\left(x - z\right) \cdot \frac{z + x}{y}\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\left(\left(x - z\right) \cdot \frac{z + x}{y}\right) \cdot 0.5\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z + x, \frac{-z}{y}, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (<= t_0 0.0)
     (* (* (- x z) (/ (+ z x) y)) 0.5)
     (if (<= t_0 INFINITY)
       (* (fma (/ x y) x y) 0.5)
       (* (fma (+ z x) (/ (- z) y) y) 0.5)))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((x - z) * ((z + x) / y)) * 0.5;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = fma((x / y), x, y) * 0.5;
	} else {
		tmp = fma((z + x), (-z / y), y) * 0.5;
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(N[(x - z), $MachinePrecision] * N[(N[(z + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[t$95$0, Infinity], N[(N[(N[(x / y), $MachinePrecision] * x + y), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[(N[(z + x), $MachinePrecision] * N[((-z) / y), $MachinePrecision] + y), $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))) < 0.0

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 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. Taylor expanded in y around 0

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

   8. Applied rewrites 70.4%

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

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

   1. Initial program 76.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 73.0%

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

   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%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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. Taylor expanded in x around 0

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

   8. Applied rewrites 84.7%

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

Alternative 5: 36.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\\ \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq \infty\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0))))
   (if (or (<= t_0 0.0) (not (<= t_0 INFINITY)))
     (* -0.5 (* z (/ z y)))
     (* 0.5 y))))

C

double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= ((double) INFINITY))) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	double tmp;
	if ((t_0 <= 0.0) || !(t_0 <= Double.POSITIVE_INFINITY)) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0)
	tmp = 0
	if (t_0 <= 0.0) or not (t_0 <= math.inf):
		tmp = -0.5 * (z * (z / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
	tmp = 0.0;
	if ((t_0 <= 0.0) || ~((t_0 <= Inf)))
		tmp = -0.5 * (z * (z / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, Infinity]], $MachinePrecision]], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $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))) < 0.0 or +inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 64.8%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   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 36.7

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

   5. Applied rewrites 36.7%

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

   7. Applied rewrites 39.6%

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

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

   1. Initial program 76.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 34.7

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

   5. Applied rewrites 34.7%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq 0 \lor \neg \left(\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq \infty\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \leq -1 \cdot 10^{-152}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, x, y\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)) -1e-152)
   (* -0.5 (* z (/ z y)))
   (* (fma (/ x y) x y) 0.5)))

C

double code(double x, double y, double z) {
	double tmp;
	if (((((x * x) + (y * y)) - (z * z)) / (y * 2.0)) <= -1e-152) {
		tmp = -0.5 * (z * (z / y));
	} else {
		tmp = fma((x / y), x, y) * 0.5;
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0)) <= -1e-152)
		tmp = Float64(-0.5 * Float64(z * Float64(z / y)));
	else
		tmp = Float64(fma(Float64(x / y), x, y) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], -1e-152], N[(-0.5 * N[(z * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * x + 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))) < -1.00000000000000007e-152

   1. Initial program 83.2%

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

   3. Taylor expanded in z around 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 35.8

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

   5. Applied rewrites 35.8%

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

   7. Applied rewrites 36.2%

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

   if -1.00000000000000007e-152 < (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (*.f64 y #s(literal 2 binary64)))

   1. Initial program 57.6%

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

   3. Taylor expanded in z around 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-rgt-in N/A

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. +-commutative N/A

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

      15. distribute-lft-in N/A

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

      16. *-commutative N/A

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

      17. 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. Add Preprocessing

Alternative 7: 35.1% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* 0.5 y))

C

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

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 = 0.5d0 * y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 69.5%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 32.0

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

5. Applied rewrites 32.0%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
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 2025017 
(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)))