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

Percentage Accurate: 99.8% → 99.8%

Time: 7.7s

Alternatives: 5

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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%

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

   2. +-commutative 99.8%

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

   3. fma-define 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 74.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{elif}\;t\_0 \leq -5 \cdot 10^{-33} \lor \neg \left(t\_0 \leq -5 \cdot 10^{-53}\right) \land t\_0 \leq 5 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{z}{\sqrt{z}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (<= t_0 -4e+93)
     (* 0.5 t_0)
     (if (or (<= t_0 -5e-33) (and (not (<= t_0 -5e-53)) (<= t_0 5e+36)))
       (* 0.5 x)
       (* 0.5 (* y (/ z (sqrt z))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * sqrt(z);
	double tmp;
	if (t_0 <= -4e+93) {
		tmp = 0.5 * t_0;
	} else if ((t_0 <= -5e-33) || (!(t_0 <= -5e-53) && (t_0 <= 5e+36))) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (y * (z / sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(z)
    if (t_0 <= (-4d+93)) then
        tmp = 0.5d0 * t_0
    else if ((t_0 <= (-5d-33)) .or. (.not. (t_0 <= (-5d-53))) .and. (t_0 <= 5d+36)) then
        tmp = 0.5d0 * x
    else
        tmp = 0.5d0 * (y * (z / sqrt(z)))
    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 <= -4e+93) {
		tmp = 0.5 * t_0;
	} else if ((t_0 <= -5e-33) || (!(t_0 <= -5e-53) && (t_0 <= 5e+36))) {
		tmp = 0.5 * x;
	} else {
		tmp = 0.5 * (y * (z / Math.sqrt(z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * math.sqrt(z)
	tmp = 0
	if t_0 <= -4e+93:
		tmp = 0.5 * t_0
	elif (t_0 <= -5e-33) or (not (t_0 <= -5e-53) and (t_0 <= 5e+36)):
		tmp = 0.5 * x
	else:
		tmp = 0.5 * (y * (z / math.sqrt(z)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * sqrt(z))
	tmp = 0.0
	if (t_0 <= -4e+93)
		tmp = Float64(0.5 * t_0);
	elseif ((t_0 <= -5e-33) || (!(t_0 <= -5e-53) && (t_0 <= 5e+36)))
		tmp = Float64(0.5 * x);
	else
		tmp = Float64(0.5 * Float64(y * Float64(z / sqrt(z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * sqrt(z);
	tmp = 0.0;
	if (t_0 <= -4e+93)
		tmp = 0.5 * t_0;
	elseif ((t_0 <= -5e-33) || (~((t_0 <= -5e-53)) && (t_0 <= 5e+36)))
		tmp = 0.5 * x;
	else
		tmp = 0.5 * (y * (z / sqrt(z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -4e+93], N[(0.5 * t$95$0), $MachinePrecision], If[Or[LessEqual[t$95$0, -5e-33], And[N[Not[LessEqual[t$95$0, -5e-53]], $MachinePrecision], LessEqual[t$95$0, 5e+36]]], N[(0.5 * x), $MachinePrecision], N[(0.5 * N[(y * N[(z / N[Sqrt[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (sqrt.f64 z)) < -4.00000000000000017e93

   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 99.5%

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

      4. associate-*l* 99.3%

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

      5. fma-define 99.3%

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

      6. pow1/2 99.3%

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

      7. sqrt-pow1 99.4%

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

      8. metadata-eval 99.4%

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

      9. pow1/2 99.4%

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

      10. sqrt-pow1 99.3%

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

      11. metadata-eval 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in z around inf 80.8%

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

   if -4.00000000000000017e93 < (*.f64 y (sqrt.f64 z)) < -5.00000000000000028e-33 or -5e-53 < (*.f64 y (sqrt.f64 z)) < 4.99999999999999977e36

   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 79.3%

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

   if -5.00000000000000028e-33 < (*.f64 y (sqrt.f64 z)) < -5e-53 or 4.99999999999999977e36 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.6%

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

      1. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in z around inf 80.1%

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

   6. Taylor expanded in y around inf 70.9%

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

      1. rem-exp-log 67.8%

         \[\leadsto 0.5 \cdot \left(z \cdot \left(y \cdot \sqrt{\frac{1}{\color{blue}{e^{\log z}}}}\right)\right) \]

      2. exp-neg 67.8%

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

      3. unpow1/2 67.8%

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

      4. exp-prod 67.8%

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

      5. distribute-lft-neg-out 67.8%

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

      6. distribute-rgt-neg-in 67.8%

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

      7. metadata-eval 67.8%

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

      8. exp-to-pow 70.9%

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

   8. Simplified 70.9%

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

      1. *-commutative 70.9%

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

      2. metadata-eval 70.9%

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

      3. pow-flip 70.9%

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

      4. pow1/2 70.9%

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

      5. associate-/r/ 70.4%

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

      6. un-div-inv 70.9%

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

   10. Applied egg-rr 70.9%

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

       1. associate-/r/ 85.0%

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

   12. Simplified 85.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -4 \cdot 10^{+93}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{elif}\;y \cdot \sqrt{z} \leq -5 \cdot 10^{-33} \lor \neg \left(y \cdot \sqrt{z} \leq -5 \cdot 10^{-53}\right) \land y \cdot \sqrt{z} \leq 5 \cdot 10^{+36}:\\ \;\;\;\;0.5 \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{z}{\sqrt{z}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{z}\\ \mathbf{if}\;t\_0 \leq -4 \cdot 10^{+93} \lor \neg \left(t\_0 \leq -5 \cdot 10^{-17} \lor \neg \left(t\_0 \leq -5 \cdot 10^{-53}\right) \land t\_0 \leq 5 \cdot 10^{+36}\right):\\ \;\;\;\;0.5 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (sqrt z))))
   (if (or (<= t_0 -4e+93)
           (not
            (or (<= t_0 -5e-17) (and (not (<= t_0 -5e-53)) (<= t_0 5e+36)))))
     (* 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 <= -4e+93) || !((t_0 <= -5e-17) || (!(t_0 <= -5e-53) && (t_0 <= 5e+36)))) {
		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 <= (-4d+93)) .or. (.not. (t_0 <= (-5d-17)) .or. (.not. (t_0 <= (-5d-53))) .and. (t_0 <= 5d+36))) 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 <= -4e+93) || !((t_0 <= -5e-17) || (!(t_0 <= -5e-53) && (t_0 <= 5e+36)))) {
		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 <= -4e+93) or not ((t_0 <= -5e-17) or (not (t_0 <= -5e-53) and (t_0 <= 5e+36))):
		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 <= -4e+93) || !((t_0 <= -5e-17) || (!(t_0 <= -5e-53) && (t_0 <= 5e+36))))
		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 <= -4e+93) || ~(((t_0 <= -5e-17) || (~((t_0 <= -5e-53)) && (t_0 <= 5e+36)))))
		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, -4e+93], N[Not[Or[LessEqual[t$95$0, -5e-17], And[N[Not[LessEqual[t$95$0, -5e-53]], $MachinePrecision], LessEqual[t$95$0, 5e+36]]]], $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)) < -4.00000000000000017e93 or -4.9999999999999999e-17 < (*.f64 y (sqrt.f64 z)) < -5e-53 or 4.99999999999999977e36 < (*.f64 y (sqrt.f64 z))

   1. Initial program 99.7%

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

      1. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 99.3%

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

      4. associate-*l* 99.3%

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

      5. fma-define 99.3%

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

      6. pow1/2 99.3%

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

      7. sqrt-pow1 99.4%

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

      8. metadata-eval 99.4%

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

      9. pow1/2 99.4%

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

      10. sqrt-pow1 99.3%

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

      11. metadata-eval 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in z around inf 82.3%

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

   if -4.00000000000000017e93 < (*.f64 y (sqrt.f64 z)) < -4.9999999999999999e-17 or -5e-53 < (*.f64 y (sqrt.f64 z)) < 4.99999999999999977e36

   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 80.1%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \sqrt{z} \leq -4 \cdot 10^{+93} \lor \neg \left(y \cdot \sqrt{z} \leq -5 \cdot 10^{-17} \lor \neg \left(y \cdot \sqrt{z} \leq -5 \cdot 10^{-53}\right) \land y \cdot \sqrt{z} \leq 5 \cdot 10^{+36}\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \sqrt{z}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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. Add Preprocessing

Alternative 5: 50.3% 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 99.8%

   \[\frac{1}{2} \cdot \left(x + y \cdot \sqrt{z}\right) \]
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. Taylor expanded in x around inf 50.2%

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

Reproduce

herbie shell --seed 2024100 
(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)))))