Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B

Percentage Accurate: 99.8% → 99.8%

Time: 7.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return (1.0 / 2.0) * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / 2.0) * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))

C

double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * sqrt(z)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (1.0d0 / 2.0d0) * (x + (y * sqrt(z)))
end function

Java

public static double code(double x, double y, double z) {
	return (1.0 / 2.0) * (x + (y * Math.sqrt(z)));
}

Python

def code(x, y, z):
	return (1.0 / 2.0) * (x + (y * math.sqrt(z)))

Julia

function code(x, y, z)
	return Float64(Float64(1.0 / 2.0) * Float64(x + Float64(y * sqrt(z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (1.0 / 2.0) * (x + (y * sqrt(z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(1.0 / 2.0), $MachinePrecision] * N[(x + N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * (x + (y * sqrt(z)))
end function

Java

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

Python

def code(x, y, z):
	return 0.5 * (x + (y * math.sqrt(z)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
4. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto 0.5 \cdot \left(x + y \cdot \sqrt{z}\right) \]
6. Add Preprocessing

Alternative 2: 72.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+117} \lor \neg \left(t\_0 \leq -2000000000\right) \land \left(t\_0 \leq -1 \cdot 10^{-74} \lor \neg \left(t\_0 \leq 200\right)\right):\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -5e+117)
           (and (not (<= t_0 -2000000000.0))
                (or (<= t_0 -1e-74) (not (<= t_0 200.0)))))
     (* 0.5 t_0)
     (* 0.5 x))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if ((t_0 <= -5e+117) || (!(t_0 <= -2000000000.0) && ((t_0 <= -1e-74) || !(t_0 <= 200.0)))) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(z)
    if ((t_0 <= (-5d+117)) .or. (.not. (t_0 <= (-2000000000.0d0))) .and. (t_0 <= (-1d-74)) .or. (.not. (t_0 <= 200.0d0))) then
        tmp = 0.5d0 * t_0
    else
        tmp = 0.5d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * Math.sqrt(z);
	double tmp;
	if ((t_0 <= -5e+117) || (!(t_0 <= -2000000000.0) && ((t_0 <= -1e-74) || !(t_0 <= 200.0)))) {
		tmp = 0.5 * t_0;
	} else {
		tmp = 0.5 * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if (t_0 <= -5e+117) or (not (t_0 <= -2000000000.0) and ((t_0 <= -1e-74) or not (t_0 <= 200.0))):
		tmp = 0.5 * t_0
	else:
		tmp = 0.5 * x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if ((t_0 <= -5e+117) || (!(t_0 <= -2000000000.0) && ((t_0 <= -1e-74) || !(t_0 <= 200.0))))
		tmp = Float64(0.5 * t_0);
	else
		tmp = Float64(0.5 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if ((t_0 <= -5e+117) || (~((t_0 <= -2000000000.0)) && ((t_0 <= -1e-74) || ~((t_0 <= 200.0)))))
		tmp = 0.5 * t_0;
	else
		tmp = 0.5 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+117], And[N[Not[LessEqual[t$95$0, -2000000000.0]], $MachinePrecision], Or[LessEqual[t$95$0, -1e-74], N[Not[LessEqual[t$95$0, 200.0]], $MachinePrecision]]]], N[(0.5 * t$95$0), $MachinePrecision], N[(0.5 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (sqrt.f64 z)) < -4.99999999999999983e117 or -2e9 < (*.f64 y (sqrt.f64 z)) < -9.99999999999999958e-75 or 200 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.8%

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

      1. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. add-sqr-sqrt 99.4%

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

      4. associate-*l* 99.4%

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

      5. fma-define 99.4%

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

      6. pow1/2 99.4%

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

      7. sqrt-pow1 99.6%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left(\color{blue}{{z}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{z}} \cdot y, x\right) \]

      8. metadata-eval 99.6%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{\color{blue}{0.25}}, \sqrt{\sqrt{z}} \cdot y, x\right) \]

      9. pow1/2 99.6%

         \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \sqrt{\color{blue}{{z}^{0.5}}} \cdot y, x\right) \]

      10. sqrt-pow1 99.5%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, \color{blue}{{z}^{\left(\frac{0.5}{2}\right)}} \cdot y, x\right) \]

      11. metadata-eval 99.5%

          \[\leadsto 0.5 \cdot \mathsf{fma}\left({z}^{0.25}, {z}^{\color{blue}{0.25}} \cdot y, x\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left({z}^{0.25}, {z}^{0.25} \cdot y, x\right)} \]

   7. Taylor expanded in z around inf 82.6%

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

   if -4.99999999999999983e117 < (*.f64 y (sqrt.f64 z)) < -2e9 or -9.99999999999999958e-75 < (*.f64 y (sqrt.f64 z)) < 200

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 79.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -5 \cdot 10^{+117} \lor \neg \left(y \cdot \sqrt{z} \leq -2000000000\right) \land \left(y \cdot \sqrt{z} \leq -1 \cdot 10^{-74} \lor \neg \left(y \cdot \sqrt{z} \leq 200\right)\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{1}{z} \cdot \frac{z}{\frac{1}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.32e+165) (* 0.5 x) (* 0.5 (* (/ 1.0 z) (/ z (/ 1.0 x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.32e+165) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * ((1.0 / z) * (z / (1.0 / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 1.32d+165) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * ((1.0d0 / z) * (z / (1.0d0 / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.32e+165) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * ((1.0 / z) * (z / (1.0 / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.32e+165:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * ((1.0 / z) * (z / (1.0 / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.32e+165)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(Float64(1.0 / z) * Float64(z / Float64(1.0 / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.32e+165)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * ((1.0 / z) * (z / (1.0 / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.32e+165], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(N[(1.0 / z), $MachinePrecision] * N[(z / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.31999999999999998e165

   1. Initial program 99.9%

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

      1. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 54.2%

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

   if 1.31999999999999998e165 < y

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 67.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot \left(y \cdot \sqrt{\frac{1}{z}} + \frac{x}{z}\right)\right)} \]

   6. Taylor expanded in y around 0 2.8%

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

      1. clear-num 2.8%

         \[\leadsto 0.5 \cdot \left(z \cdot \color{blue}{\frac{1}{\frac{z}{x}}}\right) \]

      2. un-div-inv 2.8%

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

   8. Applied egg-rr 2.8%

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

      1. *-un-lft-identity 2.8%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{1 \cdot z}}{\frac{z}{x}} \]

      2. div-inv 2.8%

         \[\leadsto 0.5 \cdot \frac{1 \cdot z}{\color{blue}{z \cdot \frac{1}{x}}} \]

      3. times-frac 26.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{1}{z} \cdot \frac{z}{\frac{1}{x}}\right)} \]

   10. Applied egg-rr 26.7%

       \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{1}{z} \cdot \frac{z}{\frac{1}{x}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.32 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{1}{z} \cdot \frac{z}{\frac{1}{x}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{z}{x \cdot z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.5e+166) (* 0.5 x) (* 0.5 (/ 1.0 (/ z (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e+166) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (1.0 / (z / (x * z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.5d+166) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * (1.0d0 / (z / (x * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e+166) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (1.0 / (z / (x * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.5e+166:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * (1.0 / (z / (x * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.5e+166)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(1.0 / Float64(z / Float64(x * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.5e+166)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * (1.0 / (z / (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.5e+166], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(1.0 / N[(z / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4999999999999999e166

   1. Initial program 99.9%

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

      1. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 54.2%

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

   if 3.4999999999999999e166 < y

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 67.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot \left(y \cdot \sqrt{\frac{1}{z}} + \frac{x}{z}\right)\right)} \]

   6. Taylor expanded in y around 0 2.8%

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

      1. associate-*r/ 26.7%

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

      2. clear-num 26.7%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{z}{z \cdot x}}} \]

   8. Applied egg-rr 26.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{1}{\frac{z}{z \cdot x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{+166}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{1}{\frac{z}{x \cdot z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot z}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 6.5e+165) (* 0.5 x) (* 0.5 (/ (* x z) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+165) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * ((x * z) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 6.5d+165) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * ((x * z) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 6.5e+165) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * ((x * z) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 6.5e+165:
		tmp = 0.5 * x
	else:
		tmp = 0.5 * ((x * z) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 6.5e+165)
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(Float64(x * z) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 6.5e+165)
		tmp = 0.5 * x;
	else
		tmp = 0.5 * ((x * z) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 6.5e+165], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(N[(x * z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.4999999999999999e165

   1. Initial program 99.9%

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

      1. metadata-eval 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 54.2%

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

   if 6.4999999999999999e165 < y

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 67.1%

      \[\leadsto 0.5 \cdot \color{blue}{\left(z \cdot \left(y \cdot \sqrt{\frac{1}{z}} + \frac{x}{z}\right)\right)} \]

   6. Taylor expanded in y around 0 2.8%

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

      1. associate-*r/ 26.7%

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

   8. Applied egg-rr 26.7%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.5 \cdot 10^{+165}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot z}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.9% accurate, 36.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{0.5 \cdot \left(x + y \cdot \sqrt{z}\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 48.9%

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

6. Final simplification 48.9%

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

Reproduce

herbie shell --seed 2024079 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quadForm from diagrams-solve-0.1, B"
  :precision binary64
  (* (/ 1.0 2.0) (+ x (* y (sqrt z)))))