Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5

Percentage Accurate: 70.2% → 97.4%

Time: 4.2s

Alternatives: 14

Speedup: 0.4×

• unsoundness detected in the egraph

Specification

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))

C

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

Fortran

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 = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 * Math.sqrt((((x * y) + (x * z)) + (y * z)));
}

Python

def code(x, y, z):
	return 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(2) * (sqrt((((x * y) + (x * z)) + (y * z))))
END code

Initial Program: 70.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))

C

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

Fortran

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 = 2.0d0 * sqrt((((x * y) + (x * z)) + (y * z)))
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 * Math.sqrt((((x * y) + (x * z)) + (y * z)));
}

Python

def code(x, y, z):
	return 2.0 * math.sqrt((((x * y) + (x * z)) + (y * z)))

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(2) * (sqrt((((x * y) + (x * z)) + (y * z))))
END code

Alternative 1: 97.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -9.55106670595963 \cdot 10^{+18}:\\ \;\;\;\;-2 \cdot \frac{t\_2}{\sqrt{\frac{t\_2}{t\_0}}}\\ \mathbf{elif}\;t\_2 \leq 3.162244367281735 \cdot 10^{-235}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(t\_3 + t\_0, t\_2, t\_3 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\sqrt{\left|t\_0 + t\_2\right|}}{\sqrt{\left|\frac{1}{t\_3}\right|}}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -9.55106670595963e+18)
    (* -2.0 (/ t_2 (sqrt (/ t_2 t_0))))
    (if (<= t_2 3.162244367281735e-235)
      (* 2.0 (sqrt (fma (+ t_3 t_0) t_2 (* t_3 t_0))))
      (*
       2.0
       (/ (sqrt (fabs (+ t_0 t_2))) (sqrt (fabs (/ 1.0 t_3)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -9.55106670595963e+18) {
		tmp = -2.0 * (t_2 / sqrt((t_2 / t_0)));
	} else if (t_2 <= 3.162244367281735e-235) {
		tmp = 2.0 * sqrt(fma((t_3 + t_0), t_2, (t_3 * t_0)));
	} else {
		tmp = 2.0 * (sqrt(fabs((t_0 + t_2))) / sqrt(fabs((1.0 / t_3))));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -9.55106670595963e+18)
		tmp = Float64(-2.0 * Float64(t_2 / sqrt(Float64(t_2 / t_0))));
	elseif (t_2 <= 3.162244367281735e-235)
		tmp = Float64(2.0 * sqrt(fma(Float64(t_3 + t_0), t_2, Float64(t_3 * t_0))));
	else
		tmp = Float64(2.0 * Float64(sqrt(abs(Float64(t_0 + t_2))) / sqrt(abs(Float64(1.0 / t_3)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -9.55106670595963e+18], N[(-2.0 * N[(t$95$2 / N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 3.162244367281735e-235], N[(2.0 * N[Sqrt[N[(N[(t$95$3 + t$95$0), $MachinePrecision] * t$95$2 + N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Sqrt[N[Abs[N[(t$95$0 + t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[N[(1.0 / t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 < z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 > z) THEN tmp_7 ELSE z ENDIF IN
		LET t_1 = tmp_5 IN
			LET tmp_10 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_11 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_9 = IF (tmp_10 < t_1) THEN tmp_11 ELSE t_1 ENDIF IN
			LET t_2 = tmp_9 IN
				LET tmp_14 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_15 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_13 = IF (tmp_14 > t_1) THEN tmp_15 ELSE t_1 ENDIF IN
				LET t_3 = tmp_13 IN
					LET tmp_17 = IF (t_2 <= (316224436728173517826490973030509114229484794791842938981949366113033743905093575595070068931708986606031530590568878421439620112075383086559308277572143517002028425189710606188571420147253332190817165135574622012695516986016156887496460553570965363747898426956714723321241293488917411653104042808627000800146047233409691273946971141723821674580190015263436783195002367241747046585248673115329038756374660839475041316073620469550192881615364386555569876592995114409979428009784180005454796405399109467891387927330205984337789666693321185111793640178028488331440915004577618674375116825103759765625e-831)) THEN ((2) * (sqrt((((t_3 + t_0) * t_2) + (t_3 * t_0))))) ELSE ((2) * ((sqrt((abs((t_0 + t_2))))) / (sqrt((abs(((1) / t_3))))))) ENDIF IN
					LET tmp_16 = IF (t_2 <= (-9551066705959630848)) THEN ((-2) * (t_2 / (sqrt((t_2 / t_0))))) ELSE tmp_17 ENDIF IN
	tmp_16
END code

Derivation

1. Split input into 3 regimes

2. if y < -9551066705959630800

   1. Initial program 70.2%

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

   2. Taylor expanded in y around -inf

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

   4. Applied rewrites 29.2%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 15.4%

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

   9. Applied rewrites 15.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto -2 \cdot \frac{y}{\sqrt{\frac{y}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.4%

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

   if -9551066705959630800 < y < 3.1622443672817352e-235

   1. Initial program 70.2%

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

   3. Applied rewrites 70.2%

      \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z + x, y, z \cdot x\right)} \]

   if 3.1622443672817352e-235 < y

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 30.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 15.6%

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

   9. Applied rewrites 16.5%

      \[\leadsto 2 \cdot \frac{z \cdot \sqrt{\left|y\right|}}{\sqrt{\left|z\right|}} \]

   10. Taylor expanded in z around inf

       \[\leadsto 2 \cdot \frac{\sqrt{\left|x + y\right|}}{\sqrt{\left|\frac{1}{z}\right|}} \]
   11. Step-by-step derivation

   12. Applied rewrites 67.9%

       \[\leadsto 2 \cdot \frac{\sqrt{\left|x + y\right|}}{\sqrt{\left|\frac{1}{z}\right|}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 96.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -9.55106670595963 \cdot 10^{+18}:\\ \;\;\;\;-2 \cdot \frac{t\_2}{\sqrt{\frac{t\_2}{t\_0}}}\\ \mathbf{elif}\;t\_2 \leq 233162798460332.5:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{fma}\left(t\_3 + t\_0, t\_2, t\_3 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \sqrt{\frac{t\_0 + t\_2}{t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -9.55106670595963e+18)
    (* -2.0 (/ t_2 (sqrt (/ t_2 t_0))))
    (if (<= t_2 233162798460332.5)
      (* 2.0 (sqrt (fma (+ t_3 t_0) t_2 (* t_3 t_0))))
      (* 2.0 (* t_3 (sqrt (/ (+ t_0 t_2) t_3))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -9.55106670595963e+18) {
		tmp = -2.0 * (t_2 / sqrt((t_2 / t_0)));
	} else if (t_2 <= 233162798460332.5) {
		tmp = 2.0 * sqrt(fma((t_3 + t_0), t_2, (t_3 * t_0)));
	} else {
		tmp = 2.0 * (t_3 * sqrt(((t_0 + t_2) / t_3)));
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -9.55106670595963e+18)
		tmp = Float64(-2.0 * Float64(t_2 / sqrt(Float64(t_2 / t_0))));
	elseif (t_2 <= 233162798460332.5)
		tmp = Float64(2.0 * sqrt(fma(Float64(t_3 + t_0), t_2, Float64(t_3 * t_0))));
	else
		tmp = Float64(2.0 * Float64(t_3 * sqrt(Float64(Float64(t_0 + t_2) / t_3))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -9.55106670595963e+18], N[(-2.0 * N[(t$95$2 / N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 233162798460332.5], N[(2.0 * N[Sqrt[N[(N[(t$95$3 + t$95$0), $MachinePrecision] * t$95$2 + N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[Sqrt[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 < z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 > z) THEN tmp_7 ELSE z ENDIF IN
		LET t_1 = tmp_5 IN
			LET tmp_10 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_11 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_9 = IF (tmp_10 < t_1) THEN tmp_11 ELSE t_1 ENDIF IN
			LET t_2 = tmp_9 IN
				LET tmp_14 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_15 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_13 = IF (tmp_14 > t_1) THEN tmp_15 ELSE t_1 ENDIF IN
				LET t_3 = tmp_13 IN
					LET tmp_17 = IF (t_2 <= (2331627984603325e-1)) THEN ((2) * (sqrt((((t_3 + t_0) * t_2) + (t_3 * t_0))))) ELSE ((2) * (t_3 * (sqrt(((t_0 + t_2) / t_3))))) ENDIF IN
					LET tmp_16 = IF (t_2 <= (-9551066705959630848)) THEN ((-2) * (t_2 / (sqrt((t_2 / t_0))))) ELSE tmp_17 ENDIF IN
	tmp_16
END code

Derivation

1. Split input into 3 regimes

2. if y < -9551066705959630800

   1. Initial program 70.2%

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

   2. Taylor expanded in y around -inf

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

   4. Applied rewrites 29.2%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 15.4%

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

   9. Applied rewrites 15.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto -2 \cdot \frac{y}{\sqrt{\frac{y}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.4%

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

   if -9551066705959630800 < y < 233162798460332.5

   1. Initial program 70.2%

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

   3. Applied rewrites 70.2%

      \[\leadsto 2 \cdot \sqrt{\mathsf{fma}\left(z + x, y, z \cdot x\right)} \]

   if 233162798460332.5 < y

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 30.6%

      \[\leadsto 2 \cdot \left(z \cdot \sqrt{\frac{x + y}{z}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 96.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -4.239319354108506 \cdot 10^{+22}:\\ \;\;\;\;-2 \cdot \frac{t\_2}{\sqrt{\frac{t\_2}{t\_0}}}\\ \mathbf{elif}\;t\_2 \leq -2.0821119553426003 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \sqrt{t\_0 \cdot \left(t\_2 + t\_3\right)}\\ \mathbf{elif}\;t\_2 \leq 3.502817809010857 \cdot 10^{+40}:\\ \;\;\;\;2 \cdot \sqrt{t\_3 \cdot \left(t\_0 + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t\_3}{\sqrt{\left|\frac{t\_3}{t\_2}\right|}}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -4.239319354108506e+22)
    (* -2.0 (/ t_2 (sqrt (/ t_2 t_0))))
    (if (<= t_2 -2.0821119553426003e-307)
      (* 2.0 (sqrt (* t_0 (+ t_2 t_3))))
      (if (<= t_2 3.502817809010857e+40)
        (* 2.0 (sqrt (* t_3 (+ t_0 t_2))))
        (* 2.0 (/ t_3 (sqrt (fabs (/ t_3 t_2))))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -4.239319354108506e+22) {
		tmp = -2.0 * (t_2 / sqrt((t_2 / t_0)));
	} else if (t_2 <= -2.0821119553426003e-307) {
		tmp = 2.0 * sqrt((t_0 * (t_2 + t_3)));
	} else if (t_2 <= 3.502817809010857e+40) {
		tmp = 2.0 * sqrt((t_3 * (t_0 + t_2)));
	} else {
		tmp = 2.0 * (t_3 / sqrt(fabs((t_3 / t_2))));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(fmin(x, y), z)
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmax(x, y), t_1)
    t_3 = fmax(fmax(x, y), t_1)
    if (t_2 <= (-4.239319354108506d+22)) then
        tmp = (-2.0d0) * (t_2 / sqrt((t_2 / t_0)))
    else if (t_2 <= (-2.0821119553426003d-307)) then
        tmp = 2.0d0 * sqrt((t_0 * (t_2 + t_3)))
    else if (t_2 <= 3.502817809010857d+40) then
        tmp = 2.0d0 * sqrt((t_3 * (t_0 + t_2)))
    else
        tmp = 2.0d0 * (t_3 / sqrt(abs((t_3 / t_2))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -4.239319354108506e+22) {
		tmp = -2.0 * (t_2 / Math.sqrt((t_2 / t_0)));
	} else if (t_2 <= -2.0821119553426003e-307) {
		tmp = 2.0 * Math.sqrt((t_0 * (t_2 + t_3)));
	} else if (t_2 <= 3.502817809010857e+40) {
		tmp = 2.0 * Math.sqrt((t_3 * (t_0 + t_2)));
	} else {
		tmp = 2.0 * (t_3 / Math.sqrt(Math.abs((t_3 / t_2))));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0
	if t_2 <= -4.239319354108506e+22:
		tmp = -2.0 * (t_2 / math.sqrt((t_2 / t_0)))
	elif t_2 <= -2.0821119553426003e-307:
		tmp = 2.0 * math.sqrt((t_0 * (t_2 + t_3)))
	elif t_2 <= 3.502817809010857e+40:
		tmp = 2.0 * math.sqrt((t_3 * (t_0 + t_2)))
	else:
		tmp = 2.0 * (t_3 / math.sqrt(math.fabs((t_3 / t_2))))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -4.239319354108506e+22)
		tmp = Float64(-2.0 * Float64(t_2 / sqrt(Float64(t_2 / t_0))));
	elseif (t_2 <= -2.0821119553426003e-307)
		tmp = Float64(2.0 * sqrt(Float64(t_0 * Float64(t_2 + t_3))));
	elseif (t_2 <= 3.502817809010857e+40)
		tmp = Float64(2.0 * sqrt(Float64(t_3 * Float64(t_0 + t_2))));
	else
		tmp = Float64(2.0 * Float64(t_3 / sqrt(abs(Float64(t_3 / t_2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(min(x, y), z);
	t_1 = max(min(x, y), z);
	t_2 = min(max(x, y), t_1);
	t_3 = max(max(x, y), t_1);
	tmp = 0.0;
	if (t_2 <= -4.239319354108506e+22)
		tmp = -2.0 * (t_2 / sqrt((t_2 / t_0)));
	elseif (t_2 <= -2.0821119553426003e-307)
		tmp = 2.0 * sqrt((t_0 * (t_2 + t_3)));
	elseif (t_2 <= 3.502817809010857e+40)
		tmp = 2.0 * sqrt((t_3 * (t_0 + t_2)));
	else
		tmp = 2.0 * (t_3 / sqrt(abs((t_3 / t_2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -4.239319354108506e+22], N[(-2.0 * N[(t$95$2 / N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2.0821119553426003e-307], N[(2.0 * N[Sqrt[N[(t$95$0 * N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 3.502817809010857e+40], N[(2.0 * N[Sqrt[N[(t$95$3 * N[(t$95$0 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 / N[Sqrt[N[Abs[N[(t$95$3 / t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 < z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 > z) THEN tmp_7 ELSE z ENDIF IN
		LET t_1 = tmp_5 IN
			LET tmp_10 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_11 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_9 = IF (tmp_10 < t_1) THEN tmp_11 ELSE t_1 ENDIF IN
			LET t_2 = tmp_9 IN
				LET tmp_14 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_15 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_13 = IF (tmp_14 > t_1) THEN tmp_15 ELSE t_1 ENDIF IN
				LET t_3 = tmp_13 IN
					LET tmp_18 = IF (t_2 <= (35028178090108567626637793564331708776448)) THEN ((2) * (sqrt((t_3 * (t_0 + t_2))))) ELSE ((2) * (t_3 / (sqrt((abs((t_3 / t_2))))))) ENDIF IN
					LET tmp_17 = IF (t_2 <= (-20821119553426002996347196025835561405697388316458785341201450968945481337899787060796348824120331353716831384653559042802444302870388592096072855535603447317557718459616853259036799654135211449890320111830447389091546692768151236822209289170550596893643694413771911601967028135087778619056808672744571706375809831044721854351011064522583608860826505495103951456489907897832350584149531818016471179631461411340733316020548517331064350679128700240284622363133223411595889728810283440094047708382451353374854177713818705066625558097641844158456743352244625319670852416002805273539681595064610839178578219790827007709304063359587392669722010972310253657209587854495895223801308770056296081482026360902481756205365894494768524113081920035028815618716180324554443359375e-1070)) THEN ((2) * (sqrt((t_0 * (t_2 + t_3))))) ELSE tmp_18 ENDIF IN
					LET tmp_16 = IF (t_2 <= (-42393193541085056466944)) THEN ((-2) * (t_2 / (sqrt((t_2 / t_0))))) ELSE tmp_17 ENDIF IN
	tmp_16
END code

Derivation

1. Split input into 4 regimes

2. if y < -4.2393193541085056e22

   1. Initial program 70.2%

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

   2. Taylor expanded in y around -inf

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

   4. Applied rewrites 29.2%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 15.4%

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

   9. Applied rewrites 15.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto -2 \cdot \frac{y}{\sqrt{\frac{y}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.4%

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

   if -4.2393193541085056e22 < y < -2.0821119553426003e-307

   1. Initial program 70.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 24.5%

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 47.7%

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

   if -2.0821119553426003e-307 < y < 3.5028178090108568e40

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.5%

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

   if 3.5028178090108568e40 < y

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 30.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 15.6%

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

   9. Applied rewrites 16.5%

      \[\leadsto 2 \cdot \frac{z \cdot \sqrt{\left|y\right|}}{\sqrt{\left|z\right|}} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.7%

       \[\leadsto 2 \cdot \frac{z}{\sqrt{\left|\frac{z}{y}\right|}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -4.239319354108506 \cdot 10^{+22}:\\ \;\;\;\;-2 \cdot \frac{t\_2}{\sqrt{\frac{t\_2}{t\_0}}}\\ \mathbf{elif}\;t\_2 \leq -2.0821119553426003 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \sqrt{t\_0 \cdot \left(t\_2 + t\_3\right)}\\ \mathbf{elif}\;t\_2 \leq 1.048916818449786 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \sqrt{t\_3 \cdot \left(t\_0 + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \sqrt{\frac{t\_2}{t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -4.239319354108506e+22)
    (* -2.0 (/ t_2 (sqrt (/ t_2 t_0))))
    (if (<= t_2 -2.0821119553426003e-307)
      (* 2.0 (sqrt (* t_0 (+ t_2 t_3))))
      (if (<= t_2 1.048916818449786e+22)
        (* 2.0 (sqrt (* t_3 (+ t_0 t_2))))
        (* 2.0 (* t_3 (sqrt (/ t_2 t_3)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -4.239319354108506e+22) {
		tmp = -2.0 * (t_2 / sqrt((t_2 / t_0)));
	} else if (t_2 <= -2.0821119553426003e-307) {
		tmp = 2.0 * sqrt((t_0 * (t_2 + t_3)));
	} else if (t_2 <= 1.048916818449786e+22) {
		tmp = 2.0 * sqrt((t_3 * (t_0 + t_2)));
	} else {
		tmp = 2.0 * (t_3 * sqrt((t_2 / t_3)));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(fmin(x, y), z)
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmax(x, y), t_1)
    t_3 = fmax(fmax(x, y), t_1)
    if (t_2 <= (-4.239319354108506d+22)) then
        tmp = (-2.0d0) * (t_2 / sqrt((t_2 / t_0)))
    else if (t_2 <= (-2.0821119553426003d-307)) then
        tmp = 2.0d0 * sqrt((t_0 * (t_2 + t_3)))
    else if (t_2 <= 1.048916818449786d+22) then
        tmp = 2.0d0 * sqrt((t_3 * (t_0 + t_2)))
    else
        tmp = 2.0d0 * (t_3 * sqrt((t_2 / t_3)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -4.239319354108506e+22) {
		tmp = -2.0 * (t_2 / Math.sqrt((t_2 / t_0)));
	} else if (t_2 <= -2.0821119553426003e-307) {
		tmp = 2.0 * Math.sqrt((t_0 * (t_2 + t_3)));
	} else if (t_2 <= 1.048916818449786e+22) {
		tmp = 2.0 * Math.sqrt((t_3 * (t_0 + t_2)));
	} else {
		tmp = 2.0 * (t_3 * Math.sqrt((t_2 / t_3)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0
	if t_2 <= -4.239319354108506e+22:
		tmp = -2.0 * (t_2 / math.sqrt((t_2 / t_0)))
	elif t_2 <= -2.0821119553426003e-307:
		tmp = 2.0 * math.sqrt((t_0 * (t_2 + t_3)))
	elif t_2 <= 1.048916818449786e+22:
		tmp = 2.0 * math.sqrt((t_3 * (t_0 + t_2)))
	else:
		tmp = 2.0 * (t_3 * math.sqrt((t_2 / t_3)))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -4.239319354108506e+22)
		tmp = Float64(-2.0 * Float64(t_2 / sqrt(Float64(t_2 / t_0))));
	elseif (t_2 <= -2.0821119553426003e-307)
		tmp = Float64(2.0 * sqrt(Float64(t_0 * Float64(t_2 + t_3))));
	elseif (t_2 <= 1.048916818449786e+22)
		tmp = Float64(2.0 * sqrt(Float64(t_3 * Float64(t_0 + t_2))));
	else
		tmp = Float64(2.0 * Float64(t_3 * sqrt(Float64(t_2 / t_3))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(min(x, y), z);
	t_1 = max(min(x, y), z);
	t_2 = min(max(x, y), t_1);
	t_3 = max(max(x, y), t_1);
	tmp = 0.0;
	if (t_2 <= -4.239319354108506e+22)
		tmp = -2.0 * (t_2 / sqrt((t_2 / t_0)));
	elseif (t_2 <= -2.0821119553426003e-307)
		tmp = 2.0 * sqrt((t_0 * (t_2 + t_3)));
	elseif (t_2 <= 1.048916818449786e+22)
		tmp = 2.0 * sqrt((t_3 * (t_0 + t_2)));
	else
		tmp = 2.0 * (t_3 * sqrt((t_2 / t_3)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -4.239319354108506e+22], N[(-2.0 * N[(t$95$2 / N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2.0821119553426003e-307], N[(2.0 * N[Sqrt[N[(t$95$0 * N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.048916818449786e+22], N[(2.0 * N[Sqrt[N[(t$95$3 * N[(t$95$0 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[Sqrt[N[(t$95$2 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 < z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 > z) THEN tmp_7 ELSE z ENDIF IN
		LET t_1 = tmp_5 IN
			LET tmp_10 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_11 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_9 = IF (tmp_10 < t_1) THEN tmp_11 ELSE t_1 ENDIF IN
			LET t_2 = tmp_9 IN
				LET tmp_14 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_15 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_13 = IF (tmp_14 > t_1) THEN tmp_15 ELSE t_1 ENDIF IN
				LET t_3 = tmp_13 IN
					LET tmp_18 = IF (t_2 <= (10489168184497861033984)) THEN ((2) * (sqrt((t_3 * (t_0 + t_2))))) ELSE ((2) * (t_3 * (sqrt((t_2 / t_3))))) ENDIF IN
					LET tmp_17 = IF (t_2 <= (-20821119553426002996347196025835561405697388316458785341201450968945481337899787060796348824120331353716831384653559042802444302870388592096072855535603447317557718459616853259036799654135211449890320111830447389091546692768151236822209289170550596893643694413771911601967028135087778619056808672744571706375809831044721854351011064522583608860826505495103951456489907897832350584149531818016471179631461411340733316020548517331064350679128700240284622363133223411595889728810283440094047708382451353374854177713818705066625558097641844158456743352244625319670852416002805273539681595064610839178578219790827007709304063359587392669722010972310253657209587854495895223801308770056296081482026360902481756205365894494768524113081920035028815618716180324554443359375e-1070)) THEN ((2) * (sqrt((t_0 * (t_2 + t_3))))) ELSE tmp_18 ENDIF IN
					LET tmp_16 = IF (t_2 <= (-42393193541085056466944)) THEN ((-2) * (t_2 / (sqrt((t_2 / t_0))))) ELSE tmp_17 ENDIF IN
	tmp_16
END code

Derivation

1. Split input into 4 regimes

2. if y < -4.2393193541085056e22

   1. Initial program 70.2%

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

   2. Taylor expanded in y around -inf

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

   4. Applied rewrites 29.2%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 15.4%

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

   9. Applied rewrites 15.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto -2 \cdot \frac{y}{\sqrt{\frac{y}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.4%

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

   if -4.2393193541085056e22 < y < -2.0821119553426003e-307

   1. Initial program 70.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 24.5%

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 47.7%

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

   if -2.0821119553426003e-307 < y < 1.0489168184497861e22

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.5%

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

   if 1.0489168184497861e22 < y

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 30.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 15.6%

      \[\leadsto 2 \cdot \left(z \cdot \sqrt{\frac{y}{z}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -4.239319354108506 \cdot 10^{+22}:\\ \;\;\;\;-2 \cdot \left(t\_0 \cdot \sqrt{\frac{t\_2}{t\_0}}\right)\\ \mathbf{elif}\;t\_2 \leq -2.0821119553426003 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \sqrt{t\_0 \cdot \left(t\_2 + t\_3\right)}\\ \mathbf{elif}\;t\_2 \leq 1.048916818449786 \cdot 10^{+22}:\\ \;\;\;\;2 \cdot \sqrt{t\_3 \cdot \left(t\_0 + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \sqrt{\frac{t\_2}{t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -4.239319354108506e+22)
    (* -2.0 (* t_0 (sqrt (/ t_2 t_0))))
    (if (<= t_2 -2.0821119553426003e-307)
      (* 2.0 (sqrt (* t_0 (+ t_2 t_3))))
      (if (<= t_2 1.048916818449786e+22)
        (* 2.0 (sqrt (* t_3 (+ t_0 t_2))))
        (* 2.0 (* t_3 (sqrt (/ t_2 t_3)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -4.239319354108506e+22) {
		tmp = -2.0 * (t_0 * sqrt((t_2 / t_0)));
	} else if (t_2 <= -2.0821119553426003e-307) {
		tmp = 2.0 * sqrt((t_0 * (t_2 + t_3)));
	} else if (t_2 <= 1.048916818449786e+22) {
		tmp = 2.0 * sqrt((t_3 * (t_0 + t_2)));
	} else {
		tmp = 2.0 * (t_3 * sqrt((t_2 / t_3)));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(fmin(x, y), z)
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmax(x, y), t_1)
    t_3 = fmax(fmax(x, y), t_1)
    if (t_2 <= (-4.239319354108506d+22)) then
        tmp = (-2.0d0) * (t_0 * sqrt((t_2 / t_0)))
    else if (t_2 <= (-2.0821119553426003d-307)) then
        tmp = 2.0d0 * sqrt((t_0 * (t_2 + t_3)))
    else if (t_2 <= 1.048916818449786d+22) then
        tmp = 2.0d0 * sqrt((t_3 * (t_0 + t_2)))
    else
        tmp = 2.0d0 * (t_3 * sqrt((t_2 / t_3)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -4.239319354108506e+22) {
		tmp = -2.0 * (t_0 * Math.sqrt((t_2 / t_0)));
	} else if (t_2 <= -2.0821119553426003e-307) {
		tmp = 2.0 * Math.sqrt((t_0 * (t_2 + t_3)));
	} else if (t_2 <= 1.048916818449786e+22) {
		tmp = 2.0 * Math.sqrt((t_3 * (t_0 + t_2)));
	} else {
		tmp = 2.0 * (t_3 * Math.sqrt((t_2 / t_3)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0
	if t_2 <= -4.239319354108506e+22:
		tmp = -2.0 * (t_0 * math.sqrt((t_2 / t_0)))
	elif t_2 <= -2.0821119553426003e-307:
		tmp = 2.0 * math.sqrt((t_0 * (t_2 + t_3)))
	elif t_2 <= 1.048916818449786e+22:
		tmp = 2.0 * math.sqrt((t_3 * (t_0 + t_2)))
	else:
		tmp = 2.0 * (t_3 * math.sqrt((t_2 / t_3)))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -4.239319354108506e+22)
		tmp = Float64(-2.0 * Float64(t_0 * sqrt(Float64(t_2 / t_0))));
	elseif (t_2 <= -2.0821119553426003e-307)
		tmp = Float64(2.0 * sqrt(Float64(t_0 * Float64(t_2 + t_3))));
	elseif (t_2 <= 1.048916818449786e+22)
		tmp = Float64(2.0 * sqrt(Float64(t_3 * Float64(t_0 + t_2))));
	else
		tmp = Float64(2.0 * Float64(t_3 * sqrt(Float64(t_2 / t_3))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(min(x, y), z);
	t_1 = max(min(x, y), z);
	t_2 = min(max(x, y), t_1);
	t_3 = max(max(x, y), t_1);
	tmp = 0.0;
	if (t_2 <= -4.239319354108506e+22)
		tmp = -2.0 * (t_0 * sqrt((t_2 / t_0)));
	elseif (t_2 <= -2.0821119553426003e-307)
		tmp = 2.0 * sqrt((t_0 * (t_2 + t_3)));
	elseif (t_2 <= 1.048916818449786e+22)
		tmp = 2.0 * sqrt((t_3 * (t_0 + t_2)));
	else
		tmp = 2.0 * (t_3 * sqrt((t_2 / t_3)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -4.239319354108506e+22], N[(-2.0 * N[(t$95$0 * N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2.0821119553426003e-307], N[(2.0 * N[Sqrt[N[(t$95$0 * N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1.048916818449786e+22], N[(2.0 * N[Sqrt[N[(t$95$3 * N[(t$95$0 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[Sqrt[N[(t$95$2 / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 < z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 > z) THEN tmp_7 ELSE z ENDIF IN
		LET t_1 = tmp_5 IN
			LET tmp_10 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_11 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_9 = IF (tmp_10 < t_1) THEN tmp_11 ELSE t_1 ENDIF IN
			LET t_2 = tmp_9 IN
				LET tmp_14 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_15 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_13 = IF (tmp_14 > t_1) THEN tmp_15 ELSE t_1 ENDIF IN
				LET t_3 = tmp_13 IN
					LET tmp_18 = IF (t_2 <= (10489168184497861033984)) THEN ((2) * (sqrt((t_3 * (t_0 + t_2))))) ELSE ((2) * (t_3 * (sqrt((t_2 / t_3))))) ENDIF IN
					LET tmp_17 = IF (t_2 <= (-20821119553426002996347196025835561405697388316458785341201450968945481337899787060796348824120331353716831384653559042802444302870388592096072855535603447317557718459616853259036799654135211449890320111830447389091546692768151236822209289170550596893643694413771911601967028135087778619056808672744571706375809831044721854351011064522583608860826505495103951456489907897832350584149531818016471179631461411340733316020548517331064350679128700240284622363133223411595889728810283440094047708382451353374854177713818705066625558097641844158456743352244625319670852416002805273539681595064610839178578219790827007709304063359587392669722010972310253657209587854495895223801308770056296081482026360902481756205365894494768524113081920035028815618716180324554443359375e-1070)) THEN ((2) * (sqrt((t_0 * (t_2 + t_3))))) ELSE tmp_18 ENDIF IN
					LET tmp_16 = IF (t_2 <= (-42393193541085056466944)) THEN ((-2) * (t_0 * (sqrt((t_2 / t_0))))) ELSE tmp_17 ENDIF IN
	tmp_16
END code

Derivation

1. Split input into 4 regimes

2. if y < -4.2393193541085056e22

   1. Initial program 70.2%

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

   2. Taylor expanded in x around -inf

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

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 15.4%

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

   if -4.2393193541085056e22 < y < -2.0821119553426003e-307

   1. Initial program 70.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 24.5%

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 47.7%

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

   if -2.0821119553426003e-307 < y < 1.0489168184497861e22

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.5%

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

   if 1.0489168184497861e22 < y

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 30.6%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 15.6%

      \[\leadsto 2 \cdot \left(z \cdot \sqrt{\frac{y}{z}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 96.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -4.239319354108506 \cdot 10^{+22}:\\ \;\;\;\;-2 \cdot \left(t\_0 \cdot \sqrt{\frac{t\_2}{t\_0}}\right)\\ \mathbf{elif}\;t\_2 \leq -2.0821119553426003 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \sqrt{t\_0 \cdot \left(t\_2 + t\_3\right)}\\ \mathbf{elif}\;t\_2 \leq 8.50636575992331 \cdot 10^{+21}:\\ \;\;\;\;2 \cdot \sqrt{t\_3 \cdot \left(t\_0 + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_2 \cdot \sqrt{\frac{t\_3}{t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -4.239319354108506e+22)
    (* -2.0 (* t_0 (sqrt (/ t_2 t_0))))
    (if (<= t_2 -2.0821119553426003e-307)
      (* 2.0 (sqrt (* t_0 (+ t_2 t_3))))
      (if (<= t_2 8.50636575992331e+21)
        (* 2.0 (sqrt (* t_3 (+ t_0 t_2))))
        (* 2.0 (* t_2 (sqrt (/ t_3 t_2)))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -4.239319354108506e+22) {
		tmp = -2.0 * (t_0 * sqrt((t_2 / t_0)));
	} else if (t_2 <= -2.0821119553426003e-307) {
		tmp = 2.0 * sqrt((t_0 * (t_2 + t_3)));
	} else if (t_2 <= 8.50636575992331e+21) {
		tmp = 2.0 * sqrt((t_3 * (t_0 + t_2)));
	} else {
		tmp = 2.0 * (t_2 * sqrt((t_3 / t_2)));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(fmin(x, y), z)
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmax(x, y), t_1)
    t_3 = fmax(fmax(x, y), t_1)
    if (t_2 <= (-4.239319354108506d+22)) then
        tmp = (-2.0d0) * (t_0 * sqrt((t_2 / t_0)))
    else if (t_2 <= (-2.0821119553426003d-307)) then
        tmp = 2.0d0 * sqrt((t_0 * (t_2 + t_3)))
    else if (t_2 <= 8.50636575992331d+21) then
        tmp = 2.0d0 * sqrt((t_3 * (t_0 + t_2)))
    else
        tmp = 2.0d0 * (t_2 * sqrt((t_3 / t_2)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -4.239319354108506e+22) {
		tmp = -2.0 * (t_0 * Math.sqrt((t_2 / t_0)));
	} else if (t_2 <= -2.0821119553426003e-307) {
		tmp = 2.0 * Math.sqrt((t_0 * (t_2 + t_3)));
	} else if (t_2 <= 8.50636575992331e+21) {
		tmp = 2.0 * Math.sqrt((t_3 * (t_0 + t_2)));
	} else {
		tmp = 2.0 * (t_2 * Math.sqrt((t_3 / t_2)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0
	if t_2 <= -4.239319354108506e+22:
		tmp = -2.0 * (t_0 * math.sqrt((t_2 / t_0)))
	elif t_2 <= -2.0821119553426003e-307:
		tmp = 2.0 * math.sqrt((t_0 * (t_2 + t_3)))
	elif t_2 <= 8.50636575992331e+21:
		tmp = 2.0 * math.sqrt((t_3 * (t_0 + t_2)))
	else:
		tmp = 2.0 * (t_2 * math.sqrt((t_3 / t_2)))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -4.239319354108506e+22)
		tmp = Float64(-2.0 * Float64(t_0 * sqrt(Float64(t_2 / t_0))));
	elseif (t_2 <= -2.0821119553426003e-307)
		tmp = Float64(2.0 * sqrt(Float64(t_0 * Float64(t_2 + t_3))));
	elseif (t_2 <= 8.50636575992331e+21)
		tmp = Float64(2.0 * sqrt(Float64(t_3 * Float64(t_0 + t_2))));
	else
		tmp = Float64(2.0 * Float64(t_2 * sqrt(Float64(t_3 / t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(min(x, y), z);
	t_1 = max(min(x, y), z);
	t_2 = min(max(x, y), t_1);
	t_3 = max(max(x, y), t_1);
	tmp = 0.0;
	if (t_2 <= -4.239319354108506e+22)
		tmp = -2.0 * (t_0 * sqrt((t_2 / t_0)));
	elseif (t_2 <= -2.0821119553426003e-307)
		tmp = 2.0 * sqrt((t_0 * (t_2 + t_3)));
	elseif (t_2 <= 8.50636575992331e+21)
		tmp = 2.0 * sqrt((t_3 * (t_0 + t_2)));
	else
		tmp = 2.0 * (t_2 * sqrt((t_3 / t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -4.239319354108506e+22], N[(-2.0 * N[(t$95$0 * N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2.0821119553426003e-307], N[(2.0 * N[Sqrt[N[(t$95$0 * N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 8.50636575992331e+21], N[(2.0 * N[Sqrt[N[(t$95$3 * N[(t$95$0 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[Sqrt[N[(t$95$3 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 < z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 > z) THEN tmp_7 ELSE z ENDIF IN
		LET t_1 = tmp_5 IN
			LET tmp_10 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_11 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_9 = IF (tmp_10 < t_1) THEN tmp_11 ELSE t_1 ENDIF IN
			LET t_2 = tmp_9 IN
				LET tmp_14 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_15 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_13 = IF (tmp_14 > t_1) THEN tmp_15 ELSE t_1 ENDIF IN
				LET t_3 = tmp_13 IN
					LET tmp_18 = IF (t_2 <= (8506365759923310034944)) THEN ((2) * (sqrt((t_3 * (t_0 + t_2))))) ELSE ((2) * (t_2 * (sqrt((t_3 / t_2))))) ENDIF IN
					LET tmp_17 = IF (t_2 <= (-20821119553426002996347196025835561405697388316458785341201450968945481337899787060796348824120331353716831384653559042802444302870388592096072855535603447317557718459616853259036799654135211449890320111830447389091546692768151236822209289170550596893643694413771911601967028135087778619056808672744571706375809831044721854351011064522583608860826505495103951456489907897832350584149531818016471179631461411340733316020548517331064350679128700240284622363133223411595889728810283440094047708382451353374854177713818705066625558097641844158456743352244625319670852416002805273539681595064610839178578219790827007709304063359587392669722010972310253657209587854495895223801308770056296081482026360902481756205365894494768524113081920035028815618716180324554443359375e-1070)) THEN ((2) * (sqrt((t_0 * (t_2 + t_3))))) ELSE tmp_18 ENDIF IN
					LET tmp_16 = IF (t_2 <= (-42393193541085056466944)) THEN ((-2) * (t_0 * (sqrt((t_2 / t_0))))) ELSE tmp_17 ENDIF IN
	tmp_16
END code

Derivation

1. Split input into 4 regimes

2. if y < -4.2393193541085056e22

   1. Initial program 70.2%

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

   2. Taylor expanded in x around -inf

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

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 15.4%

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

   if -4.2393193541085056e22 < y < -2.0821119553426003e-307

   1. Initial program 70.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 24.5%

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 47.7%

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

   if -2.0821119553426003e-307 < y < 8.50636575992331e21

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.5%

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

   if 8.50636575992331e21 < y

   1. Initial program 70.2%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 30.1%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 15.6%

      \[\leadsto 2 \cdot \left(y \cdot \sqrt{\frac{z}{y}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -209266119953830.34:\\ \;\;\;\;-2 \cdot \frac{t\_2}{\sqrt{\frac{t\_2}{t\_0}}}\\ \mathbf{elif}\;t\_2 \leq 4.70325674994519 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot \sqrt{t\_2 \cdot \left(t\_0 + t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_3 \cdot \sqrt{\frac{t\_0 + t\_2}{t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -209266119953830.34)
    (* -2.0 (/ t_2 (sqrt (/ t_2 t_0))))
    (if (<= t_2 4.70325674994519e-45)
      (* 2.0 (sqrt (* t_2 (+ t_0 t_3))))
      (* 2.0 (* t_3 (sqrt (/ (+ t_0 t_2) t_3))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -209266119953830.34) {
		tmp = -2.0 * (t_2 / sqrt((t_2 / t_0)));
	} else if (t_2 <= 4.70325674994519e-45) {
		tmp = 2.0 * sqrt((t_2 * (t_0 + t_3)));
	} else {
		tmp = 2.0 * (t_3 * sqrt(((t_0 + t_2) / t_3)));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(fmin(x, y), z)
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmax(x, y), t_1)
    t_3 = fmax(fmax(x, y), t_1)
    if (t_2 <= (-209266119953830.34d0)) then
        tmp = (-2.0d0) * (t_2 / sqrt((t_2 / t_0)))
    else if (t_2 <= 4.70325674994519d-45) then
        tmp = 2.0d0 * sqrt((t_2 * (t_0 + t_3)))
    else
        tmp = 2.0d0 * (t_3 * sqrt(((t_0 + t_2) / t_3)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -209266119953830.34) {
		tmp = -2.0 * (t_2 / Math.sqrt((t_2 / t_0)));
	} else if (t_2 <= 4.70325674994519e-45) {
		tmp = 2.0 * Math.sqrt((t_2 * (t_0 + t_3)));
	} else {
		tmp = 2.0 * (t_3 * Math.sqrt(((t_0 + t_2) / t_3)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0
	if t_2 <= -209266119953830.34:
		tmp = -2.0 * (t_2 / math.sqrt((t_2 / t_0)))
	elif t_2 <= 4.70325674994519e-45:
		tmp = 2.0 * math.sqrt((t_2 * (t_0 + t_3)))
	else:
		tmp = 2.0 * (t_3 * math.sqrt(((t_0 + t_2) / t_3)))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -209266119953830.34)
		tmp = Float64(-2.0 * Float64(t_2 / sqrt(Float64(t_2 / t_0))));
	elseif (t_2 <= 4.70325674994519e-45)
		tmp = Float64(2.0 * sqrt(Float64(t_2 * Float64(t_0 + t_3))));
	else
		tmp = Float64(2.0 * Float64(t_3 * sqrt(Float64(Float64(t_0 + t_2) / t_3))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(min(x, y), z);
	t_1 = max(min(x, y), z);
	t_2 = min(max(x, y), t_1);
	t_3 = max(max(x, y), t_1);
	tmp = 0.0;
	if (t_2 <= -209266119953830.34)
		tmp = -2.0 * (t_2 / sqrt((t_2 / t_0)));
	elseif (t_2 <= 4.70325674994519e-45)
		tmp = 2.0 * sqrt((t_2 * (t_0 + t_3)));
	else
		tmp = 2.0 * (t_3 * sqrt(((t_0 + t_2) / t_3)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -209266119953830.34], N[(-2.0 * N[(t$95$2 / N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4.70325674994519e-45], N[(2.0 * N[Sqrt[N[(t$95$2 * N[(t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$3 * N[Sqrt[N[(N[(t$95$0 + t$95$2), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 < z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 > z) THEN tmp_7 ELSE z ENDIF IN
		LET t_1 = tmp_5 IN
			LET tmp_10 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_11 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_9 = IF (tmp_10 < t_1) THEN tmp_11 ELSE t_1 ENDIF IN
			LET t_2 = tmp_9 IN
				LET tmp_14 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_15 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_13 = IF (tmp_14 > t_1) THEN tmp_15 ELSE t_1 ENDIF IN
				LET t_3 = tmp_13 IN
					LET tmp_17 = IF (t_2 <= (47032567499451896966770065570197330390239495806000378200405502854169807643298450069663922324136090924137243660389968991086817595714819617569446563720703125e-199)) THEN ((2) * (sqrt((t_2 * (t_0 + t_3))))) ELSE ((2) * (t_3 * (sqrt(((t_0 + t_2) / t_3))))) ENDIF IN
					LET tmp_16 = IF (t_2 <= (-20926611995383034375e-5)) THEN ((-2) * (t_2 / (sqrt((t_2 / t_0))))) ELSE tmp_17 ENDIF IN
	tmp_16
END code

Derivation

1. Split input into 3 regimes

2. if y < -209266119953830.34

   1. Initial program 70.2%

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

   2. Taylor expanded in y around -inf

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

   4. Applied rewrites 29.2%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 15.4%

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

   9. Applied rewrites 15.4%

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

   10. Taylor expanded in x around inf

       \[\leadsto -2 \cdot \frac{y}{\sqrt{\frac{y}{x}}} \]
   11. Step-by-step derivation

   12. Applied rewrites 15.4%

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

   if -209266119953830.34 < y < 4.7032567499451897e-45

   1. Initial program 70.2%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 46.8%

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

   if 4.7032567499451897e-45 < y

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 30.6%

      \[\leadsto 2 \cdot \left(z \cdot \sqrt{\frac{x + y}{z}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 83.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_2 \leq -4.239319354108506 \cdot 10^{+22}:\\ \;\;\;\;-2 \cdot \left(t\_0 \cdot \sqrt{\frac{t\_2}{t\_0}}\right)\\ \mathbf{elif}\;t\_2 \leq -2.0821119553426003 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \sqrt{t\_0 \cdot \left(t\_2 + t\_3\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{t\_3 \cdot \left(t\_0 + t\_2\right)}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmin (fmax x y) t_1))
       (t_3 (fmax (fmax x y) t_1)))
  (if (<= t_2 -4.239319354108506e+22)
    (* -2.0 (* t_0 (sqrt (/ t_2 t_0))))
    (if (<= t_2 -2.0821119553426003e-307)
      (* 2.0 (sqrt (* t_0 (+ t_2 t_3))))
      (* 2.0 (sqrt (* t_3 (+ t_0 t_2))))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -4.239319354108506e+22) {
		tmp = -2.0 * (t_0 * sqrt((t_2 / t_0)));
	} else if (t_2 <= -2.0821119553426003e-307) {
		tmp = 2.0 * sqrt((t_0 * (t_2 + t_3)));
	} else {
		tmp = 2.0 * sqrt((t_3 * (t_0 + t_2)));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(fmin(x, y), z)
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmin(fmax(x, y), t_1)
    t_3 = fmax(fmax(x, y), t_1)
    if (t_2 <= (-4.239319354108506d+22)) then
        tmp = (-2.0d0) * (t_0 * sqrt((t_2 / t_0)))
    else if (t_2 <= (-2.0821119553426003d-307)) then
        tmp = 2.0d0 * sqrt((t_0 * (t_2 + t_3)))
    else
        tmp = 2.0d0 * sqrt((t_3 * (t_0 + t_2)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmin(fmax(x, y), t_1);
	double t_3 = fmax(fmax(x, y), t_1);
	double tmp;
	if (t_2 <= -4.239319354108506e+22) {
		tmp = -2.0 * (t_0 * Math.sqrt((t_2 / t_0)));
	} else if (t_2 <= -2.0821119553426003e-307) {
		tmp = 2.0 * Math.sqrt((t_0 * (t_2 + t_3)));
	} else {
		tmp = 2.0 * Math.sqrt((t_3 * (t_0 + t_2)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0
	if t_2 <= -4.239319354108506e+22:
		tmp = -2.0 * (t_0 * math.sqrt((t_2 / t_0)))
	elif t_2 <= -2.0821119553426003e-307:
		tmp = 2.0 * math.sqrt((t_0 * (t_2 + t_3)))
	else:
		tmp = 2.0 * math.sqrt((t_3 * (t_0 + t_2)))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmin(fmax(x, y), t_1)
	t_3 = fmax(fmax(x, y), t_1)
	tmp = 0.0
	if (t_2 <= -4.239319354108506e+22)
		tmp = Float64(-2.0 * Float64(t_0 * sqrt(Float64(t_2 / t_0))));
	elseif (t_2 <= -2.0821119553426003e-307)
		tmp = Float64(2.0 * sqrt(Float64(t_0 * Float64(t_2 + t_3))));
	else
		tmp = Float64(2.0 * sqrt(Float64(t_3 * Float64(t_0 + t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(min(x, y), z);
	t_1 = max(min(x, y), z);
	t_2 = min(max(x, y), t_1);
	t_3 = max(max(x, y), t_1);
	tmp = 0.0;
	if (t_2 <= -4.239319354108506e+22)
		tmp = -2.0 * (t_0 * sqrt((t_2 / t_0)));
	elseif (t_2 <= -2.0821119553426003e-307)
		tmp = 2.0 * sqrt((t_0 * (t_2 + t_3)));
	else
		tmp = 2.0 * sqrt((t_3 * (t_0 + t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$2, -4.239319354108506e+22], N[(-2.0 * N[(t$95$0 * N[Sqrt[N[(t$95$2 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -2.0821119553426003e-307], N[(2.0 * N[Sqrt[N[(t$95$0 * N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(t$95$3 * N[(t$95$0 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 < z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 > z) THEN tmp_7 ELSE z ENDIF IN
		LET t_1 = tmp_5 IN
			LET tmp_10 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_11 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_9 = IF (tmp_10 < t_1) THEN tmp_11 ELSE t_1 ENDIF IN
			LET t_2 = tmp_9 IN
				LET tmp_14 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_15 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_13 = IF (tmp_14 > t_1) THEN tmp_15 ELSE t_1 ENDIF IN
				LET t_3 = tmp_13 IN
					LET tmp_17 = IF (t_2 <= (-20821119553426002996347196025835561405697388316458785341201450968945481337899787060796348824120331353716831384653559042802444302870388592096072855535603447317557718459616853259036799654135211449890320111830447389091546692768151236822209289170550596893643694413771911601967028135087778619056808672744571706375809831044721854351011064522583608860826505495103951456489907897832350584149531818016471179631461411340733316020548517331064350679128700240284622363133223411595889728810283440094047708382451353374854177713818705066625558097641844158456743352244625319670852416002805273539681595064610839178578219790827007709304063359587392669722010972310253657209587854495895223801308770056296081482026360902481756205365894494768524113081920035028815618716180324554443359375e-1070)) THEN ((2) * (sqrt((t_0 * (t_2 + t_3))))) ELSE ((2) * (sqrt((t_3 * (t_0 + t_2))))) ENDIF IN
					LET tmp_16 = IF (t_2 <= (-42393193541085056466944)) THEN ((-2) * (t_0 * (sqrt((t_2 / t_0))))) ELSE tmp_17 ENDIF IN
	tmp_16
END code

Derivation

1. Split input into 3 regimes

2. if y < -4.2393193541085056e22

   1. Initial program 70.2%

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

   2. Taylor expanded in x around -inf

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

   4. Applied rewrites 29.9%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 15.4%

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

   if -4.2393193541085056e22 < y < -2.0821119553426003e-307

   1. Initial program 70.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 24.5%

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 47.7%

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

   if -2.0821119553426003e-307 < y

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.5%

      \[\leadsto 2 \cdot \sqrt{z \cdot \left(x + y\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_2 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ t_3 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_1\right)\\ \mathbf{if}\;t\_3 \leq -2.0821119553426003 \cdot 10^{-307}:\\ \;\;\;\;2 \cdot \sqrt{t\_0 \cdot \left(t\_3 + t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{t\_2 \cdot \left(t\_0 + t\_3\right)}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmin (fmin x y) z))
       (t_1 (fmax (fmin x y) z))
       (t_2 (fmax (fmax x y) t_1))
       (t_3 (fmin (fmax x y) t_1)))
  (if (<= t_3 -2.0821119553426003e-307)
    (* 2.0 (sqrt (* t_0 (+ t_3 t_2))))
    (* 2.0 (sqrt (* t_2 (+ t_0 t_3)))))))

C

double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double tmp;
	if (t_3 <= -2.0821119553426003e-307) {
		tmp = 2.0 * sqrt((t_0 * (t_3 + t_2)));
	} else {
		tmp = 2.0 * sqrt((t_2 * (t_0 + t_3)));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = fmin(fmin(x, y), z)
    t_1 = fmax(fmin(x, y), z)
    t_2 = fmax(fmax(x, y), t_1)
    t_3 = fmin(fmax(x, y), t_1)
    if (t_3 <= (-2.0821119553426003d-307)) then
        tmp = 2.0d0 * sqrt((t_0 * (t_3 + t_2)))
    else
        tmp = 2.0d0 * sqrt((t_2 * (t_0 + t_3)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmin(fmin(x, y), z);
	double t_1 = fmax(fmin(x, y), z);
	double t_2 = fmax(fmax(x, y), t_1);
	double t_3 = fmin(fmax(x, y), t_1);
	double tmp;
	if (t_3 <= -2.0821119553426003e-307) {
		tmp = 2.0 * Math.sqrt((t_0 * (t_3 + t_2)));
	} else {
		tmp = 2.0 * Math.sqrt((t_2 * (t_0 + t_3)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	tmp = 0
	if t_3 <= -2.0821119553426003e-307:
		tmp = 2.0 * math.sqrt((t_0 * (t_3 + t_2)))
	else:
		tmp = 2.0 * math.sqrt((t_2 * (t_0 + t_3)))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmin(fmin(x, y), z)
	t_1 = fmax(fmin(x, y), z)
	t_2 = fmax(fmax(x, y), t_1)
	t_3 = fmin(fmax(x, y), t_1)
	tmp = 0.0
	if (t_3 <= -2.0821119553426003e-307)
		tmp = Float64(2.0 * sqrt(Float64(t_0 * Float64(t_3 + t_2))));
	else
		tmp = Float64(2.0 * sqrt(Float64(t_2 * Float64(t_0 + t_3))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = min(min(x, y), z);
	t_1 = max(min(x, y), z);
	t_2 = max(max(x, y), t_1);
	t_3 = min(max(x, y), t_1);
	tmp = 0.0;
	if (t_3 <= -2.0821119553426003e-307)
		tmp = 2.0 * sqrt((t_0 * (t_3 + t_2)));
	else
		tmp = 2.0 * sqrt((t_2 * (t_0 + t_3)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$2 = N[Max[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Min[N[Max[x, y], $MachinePrecision], t$95$1], $MachinePrecision]}, If[LessEqual[t$95$3, -2.0821119553426003e-307], N[(2.0 * N[Sqrt[N[(t$95$0 * N[(t$95$3 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(t$95$2 * N[(t$95$0 + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 < z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 > z) THEN tmp_7 ELSE z ENDIF IN
		LET t_1 = tmp_5 IN
			LET tmp_10 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_11 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_9 = IF (tmp_10 > t_1) THEN tmp_11 ELSE t_1 ENDIF IN
			LET t_2 = tmp_9 IN
				LET tmp_14 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_15 = IF (x > y) THEN x ELSE y ENDIF IN
				LET tmp_13 = IF (tmp_14 < t_1) THEN tmp_15 ELSE t_1 ENDIF IN
				LET t_3 = tmp_13 IN
					LET tmp_16 = IF (t_3 <= (-20821119553426002996347196025835561405697388316458785341201450968945481337899787060796348824120331353716831384653559042802444302870388592096072855535603447317557718459616853259036799654135211449890320111830447389091546692768151236822209289170550596893643694413771911601967028135087778619056808672744571706375809831044721854351011064522583608860826505495103951456489907897832350584149531818016471179631461411340733316020548517331064350679128700240284622363133223411595889728810283440094047708382451353374854177713818705066625558097641844158456743352244625319670852416002805273539681595064610839178578219790827007709304063359587392669722010972310253657209587854495895223801308770056296081482026360902481756205365894494768524113081920035028815618716180324554443359375e-1070)) THEN ((2) * (sqrt((t_0 * (t_3 + t_2))))) ELSE ((2) * (sqrt((t_2 * (t_0 + t_3))))) ENDIF IN
	tmp_16
END code

Derivation

1. Split input into 2 regimes

2. if y < -2.0821119553426003e-307

   1. Initial program 70.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 24.5%

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 47.7%

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

   if -2.0821119553426003e-307 < y

   1. Initial program 70.2%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 47.5%

      \[\leadsto 2 \cdot \sqrt{z \cdot \left(x + y\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_0\right)\\ t_2 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_0\right)\\ \mathbf{if}\;t\_2 \leq 1.4761403185952647 \cdot 10^{-282}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right) \cdot \left(t\_2 + t\_1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{t\_2 \cdot t\_1}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmax (fmin x y) z))
       (t_1 (fmax (fmax x y) t_0))
       (t_2 (fmin (fmax x y) t_0)))
  (if (<= t_2 1.4761403185952647e-282)
    (* 2.0 (sqrt (* (fmin (fmin x y) z) (+ t_2 t_1))))
    (* 2.0 (sqrt (* t_2 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	double t_1 = fmax(fmax(x, y), t_0);
	double t_2 = fmin(fmax(x, y), t_0);
	double tmp;
	if (t_2 <= 1.4761403185952647e-282) {
		tmp = 2.0 * sqrt((fmin(fmin(x, y), z) * (t_2 + t_1)));
	} else {
		tmp = 2.0 * sqrt((t_2 * t_1));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = fmax(fmin(x, y), z)
    t_1 = fmax(fmax(x, y), t_0)
    t_2 = fmin(fmax(x, y), t_0)
    if (t_2 <= 1.4761403185952647d-282) then
        tmp = 2.0d0 * sqrt((fmin(fmin(x, y), z) * (t_2 + t_1)))
    else
        tmp = 2.0d0 * sqrt((t_2 * t_1))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	double t_1 = fmax(fmax(x, y), t_0);
	double t_2 = fmin(fmax(x, y), t_0);
	double tmp;
	if (t_2 <= 1.4761403185952647e-282) {
		tmp = 2.0 * Math.sqrt((fmin(fmin(x, y), z) * (t_2 + t_1)));
	} else {
		tmp = 2.0 * Math.sqrt((t_2 * t_1));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmax(fmin(x, y), z)
	t_1 = fmax(fmax(x, y), t_0)
	t_2 = fmin(fmax(x, y), t_0)
	tmp = 0
	if t_2 <= 1.4761403185952647e-282:
		tmp = 2.0 * math.sqrt((fmin(fmin(x, y), z) * (t_2 + t_1)))
	else:
		tmp = 2.0 * math.sqrt((t_2 * t_1))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmax(fmin(x, y), z)
	t_1 = fmax(fmax(x, y), t_0)
	t_2 = fmin(fmax(x, y), t_0)
	tmp = 0.0
	if (t_2 <= 1.4761403185952647e-282)
		tmp = Float64(2.0 * sqrt(Float64(fmin(fmin(x, y), z) * Float64(t_2 + t_1))));
	else
		tmp = Float64(2.0 * sqrt(Float64(t_2 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = max(min(x, y), z);
	t_1 = max(max(x, y), t_0);
	t_2 = min(max(x, y), t_0);
	tmp = 0.0;
	if (t_2 <= 1.4761403185952647e-282)
		tmp = 2.0 * sqrt((min(min(x, y), z) * (t_2 + t_1)));
	else
		tmp = 2.0 * sqrt((t_2 * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Max[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Min[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]}, If[LessEqual[t$95$2, 1.4761403185952647e-282], N[(2.0 * N[Sqrt[N[(N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision] * N[(t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(t$95$2 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 > z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x > y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x > y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 > t_0) THEN tmp_7 ELSE t_0 ENDIF IN
		LET t_1 = tmp_5 IN
			LET tmp_10 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_11 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_9 = IF (tmp_10 < t_0) THEN tmp_11 ELSE t_0 ENDIF IN
			LET t_2 = tmp_9 IN
				LET tmp_19 = IF (x < y) THEN x ELSE y ENDIF IN
				LET tmp_20 = IF (x < y) THEN x ELSE y ENDIF IN
				LET tmp_18 = IF (tmp_19 < z) THEN tmp_20 ELSE z ENDIF IN
				LET tmp_16 = IF (t_2 <= (147614031859526474294695555120036569203493253587906631741363114182822367452555741948625603719696370079218651899270199883937175042294348162936777892152030379774690996112117432155452999013227131344824692676142297446714506578736307558845373440360629212674446536568204693280933006092880251257347589990388984225905761378770884773692028498248366866495649279766248633125262134831262319337700791916148064909606923302018440166843784311815662013150181057329693028143299828371665355137271564579070674694120398413588833117679488268135834993217022892670487471298687625506840315358419336060640977551746803857282090231563053167573586193576958242089781440015681144269727812857315369689248374385215356596745550632476806640625e-989)) THEN ((2) * (sqrt((tmp_18 * (t_2 + t_1))))) ELSE ((2) * (sqrt((t_2 * t_1)))) ENDIF IN
	tmp_16
END code

Derivation

1. Split input into 2 regimes

2. if y < 1.4761403185952647e-282

   1. Initial program 70.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 24.5%

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 47.7%

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

   if 1.4761403185952647e-282 < y

   1. Initial program 70.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 24.5%

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 68.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ 2 \cdot \sqrt{\mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_0\right) \cdot \left(\mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right) + \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_0\right)\right)} \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmax (fmin x y) z)))
  (*
   2.0
   (sqrt
    (*
     (fmin (fmax x y) t_0)
     (+ (fmin (fmin x y) z) (fmax (fmax x y) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	return 2.0 * sqrt((fmin(fmax(x, y), t_0) * (fmin(fmin(x, y), z) + fmax(fmax(x, y), t_0))));
}

Fortran

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
    real(8) :: t_0
    t_0 = fmax(fmin(x, y), z)
    code = 2.0d0 * sqrt((fmin(fmax(x, y), t_0) * (fmin(fmin(x, y), z) + fmax(fmax(x, y), t_0))))
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	return 2.0 * Math.sqrt((fmin(fmax(x, y), t_0) * (fmin(fmin(x, y), z) + fmax(fmax(x, y), t_0))));
}

Python

def code(x, y, z):
	t_0 = fmax(fmin(x, y), z)
	return 2.0 * math.sqrt((fmin(fmax(x, y), t_0) * (fmin(fmin(x, y), z) + fmax(fmax(x, y), t_0))))

Julia

function code(x, y, z)
	t_0 = fmax(fmin(x, y), z)
	return Float64(2.0 * sqrt(Float64(fmin(fmax(x, y), t_0) * Float64(fmin(fmin(x, y), z) + fmax(fmax(x, y), t_0)))))
end

MATLAB

function tmp = code(x, y, z)
	t_0 = max(min(x, y), z);
	tmp = 2.0 * sqrt((min(max(x, y), t_0) * (min(min(x, y), z) + max(max(x, y), t_0))));
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, N[(2.0 * N[Sqrt[N[(N[Min[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision] * N[(N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision] + N[Max[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 > z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x > y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x > y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 < t_0) THEN tmp_7 ELSE t_0 ENDIF IN
		LET tmp_10 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_11 = IF (x < y) THEN x ELSE y ENDIF IN
		LET tmp_9 = IF (tmp_10 < z) THEN tmp_11 ELSE z ENDIF IN
		LET tmp_14 = IF (x > y) THEN x ELSE y ENDIF IN
		LET tmp_15 = IF (x > y) THEN x ELSE y ENDIF IN
		LET tmp_13 = IF (tmp_14 > t_0) THEN tmp_15 ELSE t_0 ENDIF IN
	(2) * (sqrt((tmp_5 * (tmp_9 + tmp_13))))
END code

Derivation

1. Initial program 70.2%

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

2. Taylor expanded in y around inf

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

4. Applied rewrites 46.8%

   \[\leadsto 2 \cdot \sqrt{y \cdot \left(x + z\right)} \]
5. Add Preprocessing

Alternative 12: 68.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{max}\left(\mathsf{min}\left(x, y\right), z\right)\\ t_1 := \mathsf{min}\left(\mathsf{max}\left(x, y\right), t\_0\right)\\ \mathbf{if}\;t\_1 \leq -2.6897059964324 \cdot 10^{-311}:\\ \;\;\;\;2 \cdot \sqrt{\mathsf{min}\left(\mathsf{min}\left(x, y\right), z\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \sqrt{t\_1 \cdot \mathsf{max}\left(\mathsf{max}\left(x, y\right), t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (fmax (fmin x y) z)) (t_1 (fmin (fmax x y) t_0)))
  (if (<= t_1 -2.6897059964324e-311)
    (* 2.0 (sqrt (* (fmin (fmin x y) z) t_1)))
    (* 2.0 (sqrt (* t_1 (fmax (fmax x y) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	double t_1 = fmin(fmax(x, y), t_0);
	double tmp;
	if (t_1 <= -2.6897059964324e-311) {
		tmp = 2.0 * sqrt((fmin(fmin(x, y), z) * t_1));
	} else {
		tmp = 2.0 * sqrt((t_1 * fmax(fmax(x, y), t_0)));
	}
	return tmp;
}

Fortran

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = fmax(fmin(x, y), z)
    t_1 = fmin(fmax(x, y), t_0)
    if (t_1 <= (-2.6897059964324d-311)) then
        tmp = 2.0d0 * sqrt((fmin(fmin(x, y), z) * t_1))
    else
        tmp = 2.0d0 * sqrt((t_1 * fmax(fmax(x, y), t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = fmax(fmin(x, y), z);
	double t_1 = fmin(fmax(x, y), t_0);
	double tmp;
	if (t_1 <= -2.6897059964324e-311) {
		tmp = 2.0 * Math.sqrt((fmin(fmin(x, y), z) * t_1));
	} else {
		tmp = 2.0 * Math.sqrt((t_1 * fmax(fmax(x, y), t_0)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = fmax(fmin(x, y), z)
	t_1 = fmin(fmax(x, y), t_0)
	tmp = 0
	if t_1 <= -2.6897059964324e-311:
		tmp = 2.0 * math.sqrt((fmin(fmin(x, y), z) * t_1))
	else:
		tmp = 2.0 * math.sqrt((t_1 * fmax(fmax(x, y), t_0)))
	return tmp

Julia

function code(x, y, z)
	t_0 = fmax(fmin(x, y), z)
	t_1 = fmin(fmax(x, y), t_0)
	tmp = 0.0
	if (t_1 <= -2.6897059964324e-311)
		tmp = Float64(2.0 * sqrt(Float64(fmin(fmin(x, y), z) * t_1)));
	else
		tmp = Float64(2.0 * sqrt(Float64(t_1 * fmax(fmax(x, y), t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = max(min(x, y), z);
	t_1 = min(max(x, y), t_0);
	tmp = 0.0;
	if (t_1 <= -2.6897059964324e-311)
		tmp = 2.0 * sqrt((min(min(x, y), z) * t_1));
	else
		tmp = 2.0 * sqrt((t_1 * max(max(x, y), t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[Max[N[Min[x, y], $MachinePrecision], z], $MachinePrecision]}, Block[{t$95$1 = N[Min[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]}, If[LessEqual[t$95$1, -2.6897059964324e-311], N[(2.0 * N[Sqrt[N[(N[Min[N[Min[x, y], $MachinePrecision], z], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[Sqrt[N[(t$95$1 * N[Max[N[Max[x, y], $MachinePrecision], t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp_2 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_3 = IF (x < y) THEN x ELSE y ENDIF IN
	LET tmp_1 = IF (tmp_2 > z) THEN tmp_3 ELSE z ENDIF IN
	LET t_0 = tmp_1 IN
		LET tmp_6 = IF (x > y) THEN x ELSE y ENDIF IN
		LET tmp_7 = IF (x > y) THEN x ELSE y ENDIF IN
		LET tmp_5 = IF (tmp_6 < t_0) THEN tmp_7 ELSE t_0 ENDIF IN
		LET t_1 = tmp_5 IN
			LET tmp_15 = IF (x < y) THEN x ELSE y ENDIF IN
			LET tmp_16 = IF (x < y) THEN x ELSE y ENDIF IN
			LET tmp_14 = IF (tmp_15 < z) THEN tmp_16 ELSE z ENDIF IN
			LET tmp_19 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_20 = IF (x > y) THEN x ELSE y ENDIF IN
			LET tmp_18 = IF (tmp_19 > t_0) THEN tmp_20 ELSE t_0 ENDIF IN
			LET tmp_12 = IF (t_1 <= (-26897059964323854984457557864932070009391760445761542339610042391898471264167558917467452153474676443451598028034104521207273697671388710456741579151503375942491805480867554058601375679031929379841899677138694754849947586707799076427600701760716577954404046117370353189601887577132813986855293964384192631637024090350202024909109057708678377585884243011476684260990300959142009593983450240393981367437356629969535955191842499707760111790392180536494543950897150335227668719357342881491634566025949032280083164549879316522819185289625374971018595660552551188504461198992108056734632740404091147400207050086966255877787736094318401396669937412149837413333323879339155402985652600721684971836514946507656532050807413655640399664792372647070806124247610569000244140625e-1074)) THEN ((2) * (sqrt((tmp_14 * t_1)))) ELSE ((2) * (sqrt((t_1 * tmp_18)))) ENDIF IN
	tmp_12
END code

Derivation

1. Split input into 2 regimes

2. if y < -2.6897059964323855e-311

   1. Initial program 70.2%

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

   2. Taylor expanded in z around 0

      \[\leadsto 2 \cdot \sqrt{x \cdot y} \]
   3. Step-by-step derivation

   4. Applied rewrites 24.6%

      \[\leadsto 2 \cdot \sqrt{x \cdot y} \]

   if -2.6897059964323855e-311 < y

   1. Initial program 70.2%

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

   2. Taylor expanded in x around 0

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 24.5%

      \[\leadsto 2 \cdot \sqrt{y \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 34.9% accurate, 1.1× speedup

Math

\[2 \cdot \sqrt{\mathsf{min}\left(x, z\right) \cdot \mathsf{min}\left(y, \mathsf{max}\left(x, z\right)\right)} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (* 2.0 (sqrt (* (fmin x z) (fmin y (fmax x z))))))

C

double code(double x, double y, double z) {
	return 2.0 * sqrt((fmin(x, z) * fmin(y, fmax(x, z))));
}

Fortran

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 = 2.0d0 * sqrt((fmin(x, z) * fmin(y, fmax(x, z))))
end function

Java

public static double code(double x, double y, double z) {
	return 2.0 * Math.sqrt((fmin(x, z) * fmin(y, fmax(x, z))));
}

Python

def code(x, y, z):
	return 2.0 * math.sqrt((fmin(x, z) * fmin(y, fmax(x, z))))

Julia

function code(x, y, z)
	return Float64(2.0 * sqrt(Float64(fmin(x, z) * fmin(y, fmax(x, z)))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0 * sqrt((min(x, z) * min(y, max(x, z))));
end

Wolfram

code[x_, y_, z_] := N[(2.0 * N[Sqrt[N[(N[Min[x, z], $MachinePrecision] * N[Min[y, N[Max[x, z], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET tmp = IF (x < z) THEN x ELSE z ENDIF IN
	LET tmp_2 = IF (x > z) THEN x ELSE z ENDIF IN
	LET tmp_3 = IF (x > z) THEN x ELSE z ENDIF IN
	LET tmp_1 = IF (y < tmp_2) THEN y ELSE tmp_3 ENDIF IN
	(2) * (sqrt((tmp * tmp_1)))
END code

Derivation

1. Initial program 70.2%

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

2. Taylor expanded in z around 0

   \[\leadsto 2 \cdot \sqrt{x \cdot y} \]
3. Step-by-step derivation

4. Applied rewrites 24.6%

   \[\leadsto 2 \cdot \sqrt{x \cdot y} \]
5. Add Preprocessing

Alternative 14: 0.0% accurate, 4.2× speedup

Math

\[\frac{0}{0} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (/ 0.0 0.0))

C

double code(double x, double y, double z) {
	return 0.0 / 0.0;
}

Fortran

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 = 0.0d0 / 0.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 0.0 / 0.0;
}

Python

def code(x, y, z):
	return 0.0 / 0.0

Julia

function code(x, y, z)
	return Float64(0.0 / 0.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = 0.0 / 0.0;
end

Wolfram

code[x_, y_, z_] := N[(0.0 / 0.0), $MachinePrecision]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(0) / (0)
END code

Derivation

1. Initial program 70.2%

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

3. Applied rewrites 0.0%

   \[\leadsto \frac{0}{0} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:descartes from diagrams-contrib-1.3.0.5"
  :precision binary64
  (* 2.0 (sqrt (+ (+ (* x y) (* x z)) (* y z)))))