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

Percentage Accurate: 69.2% → 99.9%

Time: 9.8s

Alternatives: 8

Speedup: 0.9×

• 41.0% 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

real(8) function code(x, y, z)
    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: 69.2% 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

real(8) function code(x, y, z)
    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, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(0.5 * N[(N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] + z), $MachinePrecision] * N[(N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.9%

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

   1. remove-double-neg 71.9%

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

   2. distribute-lft-neg-out 71.9%

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

   3. distribute-frac-neg2 71.9%

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

   4. distribute-frac-neg 71.9%

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

   5. neg-mul-1 71.9%

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

   6. distribute-lft-neg-out 71.9%

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

   7. *-commutative 71.9%

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

   8. distribute-lft-neg-in 71.9%

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

   9. times-frac 71.9%

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

   10. metadata-eval 71.9%

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

   11. metadata-eval 71.9%

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

   12. associate--l+ 71.9%

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

   13. fma-define 72.7%

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

3. Simplified 72.7%

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

   1. fma-undefine 71.9%

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

   2. associate--l+ 71.9%

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

   3. add-sqr-sqrt 71.9%

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

   4. difference-of-squares 75.5%

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

   5. hypot-define 76.7%

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

   6. hypot-define 77.5%

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

6. Applied egg-rr 77.5%

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

   1. associate-/l* 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

   \[\leadsto 0.5 \cdot \left(\left(\mathsf{hypot}\left(x, y\right) + z\right) \cdot \frac{\mathsf{hypot}\left(x, y\right) - z}{y}\right) \]
10. Add Preprocessing

Alternative 2: 81.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3e+132)
   (* 0.5 (/ (fma x x (- (* y y) (* z z))) y))
   (* 0.5 (* (+ y z) (/ (- y z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e+132) {
		tmp = 0.5 * (fma(x, x, ((y * y) - (z * z))) / y);
	} else {
		tmp = 0.5 * ((y + z) * ((y - z) / y));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3e+132)
		tmp = Float64(0.5 * Float64(fma(x, x, Float64(Float64(y * y) - Float64(z * z))) / y));
	else
		tmp = Float64(0.5 * Float64(Float64(y + z) * Float64(Float64(y - z) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3e+132], N[(0.5 * N[(N[(x * x + N[(N[(y * y), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y + z), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9999999999999998e132

   1. Initial program 82.4%

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

      1. remove-double-neg 82.4%

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

      2. distribute-lft-neg-out 82.4%

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

      3. distribute-frac-neg2 82.4%

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

      4. distribute-frac-neg 82.4%

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

      5. neg-mul-1 82.4%

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

      6. distribute-lft-neg-out 82.4%

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

      7. *-commutative 82.4%

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

      8. distribute-lft-neg-in 82.4%

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

      9. times-frac 82.4%

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

      10. metadata-eval 82.4%

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

      11. metadata-eval 82.4%

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

      12. associate--l+ 82.4%

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

      13. fma-define 83.4%

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

   3. Simplified 83.4%

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

   if 2.9999999999999998e132 < y

   1. Initial program 11.4%

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

      1. remove-double-neg 11.4%

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

      2. distribute-lft-neg-out 11.4%

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

      3. distribute-frac-neg2 11.4%

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

      4. distribute-frac-neg 11.4%

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

      5. neg-mul-1 11.4%

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

      6. distribute-lft-neg-out 11.4%

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

      7. *-commutative 11.4%

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

      8. distribute-lft-neg-in 11.4%

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

      9. times-frac 11.4%

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

      10. metadata-eval 11.4%

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

      11. metadata-eval 11.4%

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

      12. associate--l+ 11.4%

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

      13. fma-define 11.4%

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

   3. Simplified 11.4%

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

      1. fma-undefine 11.4%

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

      2. associate--l+ 11.4%

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

      3. add-sqr-sqrt 11.4%

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

      4. difference-of-squares 13.3%

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

      5. hypot-define 18.6%

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

      6. hypot-define 21.6%

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

   6. Applied egg-rr 21.6%

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

   7. Taylor expanded in x around 0 21.6%

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

      1. associate-/l* 91.5%

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

   9. Simplified 91.5%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{+132}:\\ \;\;\;\;0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(\left(y + z\right) \cdot \frac{z}{-y}\right)\\ \mathbf{if}\;y \leq 9 \cdot 10^{-266}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-240}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 0.46:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{y \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 0.5 (* (+ y z) (/ z (- y))))))
   (if (<= y 9e-266)
     t_0
     (if (<= y 1.8e-240)
       (* (/ x y) (/ x 2.0))
       (if (<= y 0.46)
         t_0
         (if (<= y 5.6e+76) (* x (* x (/ 1.0 (* y 2.0)))) (* 0.5 y)))))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * ((y + z) * (z / -y));
	double tmp;
	if (y <= 9e-266) {
		tmp = t_0;
	} else if (y <= 1.8e-240) {
		tmp = (x / y) * (x / 2.0);
	} else if (y <= 0.46) {
		tmp = t_0;
	} else if (y <= 5.6e+76) {
		tmp = x * (x * (1.0 / (y * 2.0)));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * ((y + z) * (z / -y))
    if (y <= 9d-266) then
        tmp = t_0
    else if (y <= 1.8d-240) then
        tmp = (x / y) * (x / 2.0d0)
    else if (y <= 0.46d0) then
        tmp = t_0
    else if (y <= 5.6d+76) then
        tmp = x * (x * (1.0d0 / (y * 2.0d0)))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.5 * ((y + z) * (z / -y));
	double tmp;
	if (y <= 9e-266) {
		tmp = t_0;
	} else if (y <= 1.8e-240) {
		tmp = (x / y) * (x / 2.0);
	} else if (y <= 0.46) {
		tmp = t_0;
	} else if (y <= 5.6e+76) {
		tmp = x * (x * (1.0 / (y * 2.0)));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * ((y + z) * (z / -y))
	tmp = 0
	if y <= 9e-266:
		tmp = t_0
	elif y <= 1.8e-240:
		tmp = (x / y) * (x / 2.0)
	elif y <= 0.46:
		tmp = t_0
	elif y <= 5.6e+76:
		tmp = x * (x * (1.0 / (y * 2.0)))
	else:
		tmp = 0.5 * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(Float64(y + z) * Float64(z / Float64(-y))))
	tmp = 0.0
	if (y <= 9e-266)
		tmp = t_0;
	elseif (y <= 1.8e-240)
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	elseif (y <= 0.46)
		tmp = t_0;
	elseif (y <= 5.6e+76)
		tmp = Float64(x * Float64(x * Float64(1.0 / Float64(y * 2.0))));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 * ((y + z) * (z / -y));
	tmp = 0.0;
	if (y <= 9e-266)
		tmp = t_0;
	elseif (y <= 1.8e-240)
		tmp = (x / y) * (x / 2.0);
	elseif (y <= 0.46)
		tmp = t_0;
	elseif (y <= 5.6e+76)
		tmp = x * (x * (1.0 / (y * 2.0)));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(N[(y + z), $MachinePrecision] * N[(z / (-y)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 9e-266], t$95$0, If[LessEqual[y, 1.8e-240], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.46], t$95$0, If[LessEqual[y, 5.6e+76], N[(x * N[(x * N[(1.0 / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 9.0000000000000006e-266 or 1.7999999999999999e-240 < y < 0.46000000000000002

   1. Initial program 82.3%

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

      1. remove-double-neg 82.3%

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

      2. distribute-lft-neg-out 82.3%

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

      3. distribute-frac-neg2 82.3%

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

      4. distribute-frac-neg 82.3%

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

      5. neg-mul-1 82.3%

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

      6. distribute-lft-neg-out 82.3%

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

      7. *-commutative 82.3%

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

      8. distribute-lft-neg-in 82.3%

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

      9. times-frac 82.3%

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

      10. metadata-eval 82.3%

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

      11. metadata-eval 82.3%

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

      12. associate--l+ 82.3%

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

      13. fma-define 83.3%

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

   3. Simplified 83.3%

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

      1. fma-undefine 82.3%

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

      2. associate--l+ 82.3%

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

      3. add-sqr-sqrt 82.3%

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

      4. difference-of-squares 85.7%

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

      5. hypot-define 86.2%

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

      6. hypot-define 86.8%

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

   6. Applied egg-rr 86.8%

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

   7. Taylor expanded in x around 0 58.2%

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

      1. associate-/l* 68.3%

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

   9. Simplified 68.3%

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

   10. Taylor expanded in y around 0 45.0%

       \[\leadsto 0.5 \cdot \left(\left(y + z\right) \cdot \color{blue}{\left(-1 \cdot \frac{z}{y}\right)}\right) \]
   11. Step-by-step derivation

       1. neg-mul-1 45.0%

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

       2. distribute-neg-frac2 45.0%

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

   12. Simplified 45.0%

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

   if 9.0000000000000006e-266 < y < 1.7999999999999999e-240

   1. Initial program 99.6%

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

      1. clear-num 99.6%

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

      2. inv-pow 99.6%

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

      3. associate-/l* 99.2%

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

      4. add-sqr-sqrt 99.2%

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

      5. pow2 99.2%

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

      6. hypot-define 99.2%

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

      7. pow2 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

      2. associate-*r/ 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 75.1%

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

      1. clear-num 75.1%

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

      2. unpow2 75.1%

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

      3. times-frac 74.8%

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

   9. Applied egg-rr 74.8%

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

   if 0.46000000000000002 < y < 5.5999999999999997e76

   1. Initial program 78.8%

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

      1. clear-num 78.7%

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

      2. inv-pow 78.7%

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

      3. associate-/l* 78.6%

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

      4. add-sqr-sqrt 78.6%

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

      5. pow2 78.6%

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

      6. hypot-define 78.6%

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

      7. pow2 78.6%

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

   4. Applied egg-rr 78.6%

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

      1. unpow-1 78.6%

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

      2. associate-*r/ 78.7%

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

   6. Simplified 78.7%

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

   7. Taylor expanded in x around inf 46.7%

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

      1. associate-/r/ 46.7%

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

      2. unpow2 46.7%

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

      3. associate-*r* 56.8%

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

   9. Applied egg-rr 56.8%

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

   if 5.5999999999999997e76 < y

   1. Initial program 23.1%

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

   3. Taylor expanded in y around inf 70.6%

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

      1. *-commutative 70.6%

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

   5. Simplified 70.6%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-266}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{z}{-y}\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-240}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \mathbf{elif}\;y \leq 0.46:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{z}{-y}\right)\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+76}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{y \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3e-86)
   (* 0.5 (* (+ x z) (/ (- x z) y)))
   (if (<= y 2.65e+132)
     (/ (- (+ (* y y) (* x x)) (* z z)) (* y 2.0))
     (* 0.5 (* (+ y z) (/ (- y z) y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e-86) {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	} else if (y <= 2.65e+132) {
		tmp = (((y * y) + (x * x)) - (z * z)) / (y * 2.0);
	} else {
		tmp = 0.5 * ((y + z) * ((y - z) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3d-86) then
        tmp = 0.5d0 * ((x + z) * ((x - z) / y))
    else if (y <= 2.65d+132) then
        tmp = (((y * y) + (x * x)) - (z * z)) / (y * 2.0d0)
    else
        tmp = 0.5d0 * ((y + z) * ((y - z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3e-86) {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	} else if (y <= 2.65e+132) {
		tmp = (((y * y) + (x * x)) - (z * z)) / (y * 2.0);
	} else {
		tmp = 0.5 * ((y + z) * ((y - z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3e-86:
		tmp = 0.5 * ((x + z) * ((x - z) / y))
	elif y <= 2.65e+132:
		tmp = (((y * y) + (x * x)) - (z * z)) / (y * 2.0)
	else:
		tmp = 0.5 * ((y + z) * ((y - z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3e-86)
		tmp = Float64(0.5 * Float64(Float64(x + z) * Float64(Float64(x - z) / y)));
	elseif (y <= 2.65e+132)
		tmp = Float64(Float64(Float64(Float64(y * y) + Float64(x * x)) - Float64(z * z)) / Float64(y * 2.0));
	else
		tmp = Float64(0.5 * Float64(Float64(y + z) * Float64(Float64(y - z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3e-86)
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	elseif (y <= 2.65e+132)
		tmp = (((y * y) + (x * x)) - (z * z)) / (y * 2.0);
	else
		tmp = 0.5 * ((y + z) * ((y - z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3e-86], N[(0.5 * N[(N[(x + z), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.65e+132], N[(N[(N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y + z), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.0000000000000001e-86

   1. Initial program 81.0%

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

      1. remove-double-neg 81.0%

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

      2. distribute-lft-neg-out 81.0%

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

      3. distribute-frac-neg2 81.0%

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

      4. distribute-frac-neg 81.0%

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

      5. neg-mul-1 81.0%

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

      6. distribute-lft-neg-out 81.0%

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

      7. *-commutative 81.0%

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

      8. distribute-lft-neg-in 81.0%

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

      9. times-frac 81.0%

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

      10. metadata-eval 81.0%

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

      11. metadata-eval 81.0%

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

      12. associate--l+ 81.0%

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

      13. fma-define 82.1%

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

   3. Simplified 82.1%

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

      1. fma-undefine 81.0%

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

      2. associate--l+ 81.0%

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

      3. add-sqr-sqrt 81.0%

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

      4. difference-of-squares 84.7%

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

      5. hypot-define 85.2%

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

      6. hypot-define 85.8%

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

   6. Applied egg-rr 85.8%

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

   7. Taylor expanded in y around 0 74.0%

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

      1. associate-/l* 76.6%

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

      2. +-commutative 76.6%

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

   9. Simplified 76.6%

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

   if 3.0000000000000001e-86 < y < 2.65e132

   1. Initial program 88.5%

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

   if 2.65e132 < y

   1. Initial program 11.4%

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

      1. remove-double-neg 11.4%

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

      2. distribute-lft-neg-out 11.4%

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

      3. distribute-frac-neg2 11.4%

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

      4. distribute-frac-neg 11.4%

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

      5. neg-mul-1 11.4%

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

      6. distribute-lft-neg-out 11.4%

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

      7. *-commutative 11.4%

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

      8. distribute-lft-neg-in 11.4%

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

      9. times-frac 11.4%

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

      10. metadata-eval 11.4%

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

      11. metadata-eval 11.4%

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

      12. associate--l+ 11.4%

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

      13. fma-define 11.4%

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

   3. Simplified 11.4%

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

      1. fma-undefine 11.4%

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

      2. associate--l+ 11.4%

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

      3. add-sqr-sqrt 11.4%

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

      4. difference-of-squares 13.3%

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

      5. hypot-define 18.6%

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

      6. hypot-define 21.6%

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

   6. Applied egg-rr 21.6%

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

   7. Taylor expanded in x around 0 21.6%

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

      1. associate-/l* 91.5%

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

   9. Simplified 91.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-86}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \mathbf{elif}\;y \leq 2.65 \cdot 10^{+132}:\\ \;\;\;\;\frac{\left(y \cdot y + x \cdot x\right) - z \cdot z}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+227}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+227)
   (* 0.5 (* (+ y z) (/ (- y z) y)))
   (* 0.5 (* (+ x z) (/ (- x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+227) {
		tmp = 0.5 * ((y + z) * ((y - z) / y));
	} else {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 2d+227) then
        tmp = 0.5d0 * ((y + z) * ((y - z) / y))
    else
        tmp = 0.5d0 * ((x + z) * ((x - z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+227) {
		tmp = 0.5 * ((y + z) * ((y - z) / y));
	} else {
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+227:
		tmp = 0.5 * ((y + z) * ((y - z) / y))
	else:
		tmp = 0.5 * ((x + z) * ((x - z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+227)
		tmp = Float64(0.5 * Float64(Float64(y + z) * Float64(Float64(y - z) / y)));
	else
		tmp = Float64(0.5 * Float64(Float64(x + z) * Float64(Float64(x - z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+227)
		tmp = 0.5 * ((y + z) * ((y - z) / y));
	else
		tmp = 0.5 * ((x + z) * ((x - z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+227], N[(0.5 * N[(N[(y + z), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x + z), $MachinePrecision] * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000002e227

   1. Initial program 75.9%

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

      1. remove-double-neg 75.9%

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

      2. distribute-lft-neg-out 75.9%

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

      3. distribute-frac-neg2 75.9%

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

      4. distribute-frac-neg 75.9%

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

      5. neg-mul-1 75.9%

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

      6. distribute-lft-neg-out 75.9%

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

      7. *-commutative 75.9%

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

      8. distribute-lft-neg-in 75.9%

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

      9. times-frac 75.9%

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

      10. metadata-eval 75.9%

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

      11. metadata-eval 75.9%

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

      12. associate--l+ 75.9%

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

      13. fma-define 75.9%

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

   3. Simplified 75.9%

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

      1. fma-undefine 75.9%

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

      2. associate--l+ 75.9%

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

      3. add-sqr-sqrt 75.9%

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

      4. difference-of-squares 76.4%

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

      5. hypot-define 77.4%

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

      6. hypot-define 77.5%

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

   6. Applied egg-rr 77.5%

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

   7. Taylor expanded in x around 0 66.2%

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

      1. associate-/l* 88.1%

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

   9. Simplified 88.1%

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

   if 2.0000000000000002e227 < (*.f64 x x)

   1. Initial program 62.1%

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

      1. remove-double-neg 62.1%

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

      2. distribute-lft-neg-out 62.1%

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

      3. distribute-frac-neg2 62.1%

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

      4. distribute-frac-neg 62.1%

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

      5. neg-mul-1 62.1%

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

      6. distribute-lft-neg-out 62.1%

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

      7. *-commutative 62.1%

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

      8. distribute-lft-neg-in 62.1%

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

      9. times-frac 62.1%

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

      10. metadata-eval 62.1%

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

      11. metadata-eval 62.1%

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

      12. associate--l+ 62.1%

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

      13. fma-define 64.8%

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

   3. Simplified 64.8%

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

      1. fma-undefine 62.1%

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

      2. associate--l+ 62.1%

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

      3. add-sqr-sqrt 62.1%

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

      4. difference-of-squares 73.4%

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

      5. hypot-define 74.8%

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

      6. hypot-define 77.6%

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

   6. Applied egg-rr 77.6%

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

   7. Taylor expanded in y around 0 77.3%

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

      1. associate-/l* 88.3%

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

      2. +-commutative 88.3%

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

   9. Simplified 88.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+227}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\left(x + z\right) \cdot \frac{x - z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 4.5e+113) (* 0.5 (* (+ y z) (/ (- y z) y))) (* (/ x y) (/ x 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.5e+113) {
		tmp = 0.5 * ((y + z) * ((y - z) / y));
	} else {
		tmp = (x / y) * (x / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 4.5d+113) then
        tmp = 0.5d0 * ((y + z) * ((y - z) / y))
    else
        tmp = (x / y) * (x / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 4.5e+113) {
		tmp = 0.5 * ((y + z) * ((y - z) / y));
	} else {
		tmp = (x / y) * (x / 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 4.5e+113:
		tmp = 0.5 * ((y + z) * ((y - z) / y))
	else:
		tmp = (x / y) * (x / 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 4.5e+113)
		tmp = Float64(0.5 * Float64(Float64(y + z) * Float64(Float64(y - z) / y)));
	else
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 4.5e+113)
		tmp = 0.5 * ((y + z) * ((y - z) / y));
	else
		tmp = (x / y) * (x / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 4.5e+113], N[(0.5 * N[(N[(y + z), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.5000000000000001e113

   1. Initial program 72.6%

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

      1. remove-double-neg 72.6%

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

      2. distribute-lft-neg-out 72.6%

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

      3. distribute-frac-neg2 72.6%

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

      4. distribute-frac-neg 72.6%

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

      5. neg-mul-1 72.6%

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

      6. distribute-lft-neg-out 72.6%

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

      7. *-commutative 72.6%

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

      8. distribute-lft-neg-in 72.6%

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

      9. times-frac 72.6%

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

      10. metadata-eval 72.6%

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

      11. metadata-eval 72.6%

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

      12. associate--l+ 72.6%

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

      13. fma-define 73.1%

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

   3. Simplified 73.1%

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

      1. fma-undefine 72.6%

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

      2. associate--l+ 72.6%

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

      3. add-sqr-sqrt 72.6%

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

      4. difference-of-squares 75.0%

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

      5. hypot-define 76.4%

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

      6. hypot-define 76.4%

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

   6. Applied egg-rr 76.4%

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

   7. Taylor expanded in x around 0 56.5%

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

      1. associate-/l* 78.1%

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

   9. Simplified 78.1%

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

   if 4.5000000000000001e113 < x

   1. Initial program 67.1%

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

      1. clear-num 67.1%

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

      2. inv-pow 67.1%

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

      3. associate-/l* 67.1%

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

      4. add-sqr-sqrt 67.1%

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

      5. pow2 67.1%

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

      6. hypot-define 67.1%

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

      7. pow2 67.1%

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

   4. Applied egg-rr 67.1%

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

      1. unpow-1 67.1%

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

      2. associate-*r/ 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in x around inf 70.0%

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

      1. clear-num 70.0%

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

      2. unpow2 70.0%

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

      3. times-frac 85.5%

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

   9. Applied egg-rr 85.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+113}:\\ \;\;\;\;0.5 \cdot \left(\left(y + z\right) \cdot \frac{y - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+227}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= (* x x) 2e+227) (* 0.5 y) (* (/ x y) (/ x 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+227) {
		tmp = 0.5 * y;
	} else {
		tmp = (x / y) * (x / 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x * x) <= 2d+227) then
        tmp = 0.5d0 * y
    else
        tmp = (x / y) * (x / 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x * x) <= 2e+227) {
		tmp = 0.5 * y;
	} else {
		tmp = (x / y) * (x / 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x * x) <= 2e+227:
		tmp = 0.5 * y
	else:
		tmp = (x / y) * (x / 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (Float64(x * x) <= 2e+227)
		tmp = Float64(0.5 * y);
	else
		tmp = Float64(Float64(x / y) * Float64(x / 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x * x) <= 2e+227)
		tmp = 0.5 * y;
	else
		tmp = (x / y) * (x / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+227], N[(0.5 * y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(x / 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.0000000000000002e227

   1. Initial program 75.9%

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

   3. Taylor expanded in y around inf 39.7%

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

      1. *-commutative 39.7%

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

   5. Simplified 39.7%

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

   if 2.0000000000000002e227 < (*.f64 x x)

   1. Initial program 62.1%

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

      1. clear-num 62.1%

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

      2. inv-pow 62.1%

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

      3. associate-/l* 62.1%

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

      4. add-sqr-sqrt 62.1%

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

      5. pow2 62.1%

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

      6. hypot-define 62.1%

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

      7. pow2 62.1%

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

   4. Applied egg-rr 62.1%

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

      1. unpow-1 62.1%

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

      2. associate-*r/ 62.1%

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

   6. Simplified 62.1%

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

   7. Taylor expanded in x around inf 67.1%

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

      1. clear-num 67.1%

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

      2. unpow2 67.1%

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

      3. times-frac 76.8%

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

   9. Applied egg-rr 76.8%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+227}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.7% accurate, 5.0× 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

real(8) function code(x, y, z)
    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 71.9%

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

3. Taylor expanded in y around inf 32.3%

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

   1. *-commutative 32.3%

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

5. Simplified 32.3%

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

6. Final simplification 32.3%

   \[\leadsto 0.5 \cdot y \]
7. Add Preprocessing

Developer target: 99.9% accurate, 1.0× 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

real(8) function code(x, y, z)
    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 2024077 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))