Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E

Percentage Accurate: 99.9% → 99.8%

Time: 8.3s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

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

Java

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

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

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

Java

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

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \left(1 - x\right) + {x}^{0.125} \cdot \left(y \cdot {x}^{0.375}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- 1.0 x) (* (pow x 0.125) (* y (pow x 0.375)))))

C

double code(double x, double y) {
	return (1.0 - x) + (pow(x, 0.125) * (y * pow(x, 0.375)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (1.0d0 - x) + ((x ** 0.125d0) * (y * (x ** 0.375d0)))
end function

Java

public static double code(double x, double y) {
	return (1.0 - x) + (Math.pow(x, 0.125) * (y * Math.pow(x, 0.375)));
}

Python

def code(x, y):
	return (1.0 - x) + (math.pow(x, 0.125) * (y * math.pow(x, 0.375)))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64((x ^ 0.125) * Float64(y * (x ^ 0.375))))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + ((x ^ 0.125) * (y * (x ^ 0.375)));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(N[Power[x, 0.125], $MachinePrecision] * N[(y * N[Power[x, 0.375], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. add-cube-cbrt 99.3%

      \[\leadsto \left(1 - x\right) + \color{blue}{\left(\sqrt[3]{y \cdot \sqrt{x}} \cdot \sqrt[3]{y \cdot \sqrt{x}}\right) \cdot \sqrt[3]{y \cdot \sqrt{x}}} \]

   2. pow3 99.3%

      \[\leadsto \left(1 - x\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{x}}\right)}^{3}} \]

4. Applied egg-rr 99.3%

   \[\leadsto \left(1 - x\right) + \color{blue}{{\left(\sqrt[3]{y \cdot \sqrt{x}}\right)}^{3}} \]
5. Step-by-step derivation

   1. rem-cube-cbrt 99.9%

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

   2. *-commutative 99.9%

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

   3. pow1/2 99.9%

      \[\leadsto \left(1 - x\right) + \color{blue}{{x}^{0.5}} \cdot y \]

   4. metadata-eval 99.9%

      \[\leadsto \left(1 - x\right) + {x}^{\color{blue}{\left(0.25 + 0.25\right)}} \cdot y \]

   5. pow-prod-up 99.8%

      \[\leadsto \left(1 - x\right) + \color{blue}{\left({x}^{0.25} \cdot {x}^{0.25}\right)} \cdot y \]

   6. associate-*r* 99.8%

      \[\leadsto \left(1 - x\right) + \color{blue}{{x}^{0.25} \cdot \left({x}^{0.25} \cdot y\right)} \]

   7. sqr-pow 99.7%

      \[\leadsto \left(1 - x\right) + \color{blue}{\left({x}^{\left(\frac{0.25}{2}\right)} \cdot {x}^{\left(\frac{0.25}{2}\right)}\right)} \cdot \left({x}^{0.25} \cdot y\right) \]

   8. associate-*l* 99.7%

      \[\leadsto \left(1 - x\right) + \color{blue}{{x}^{\left(\frac{0.25}{2}\right)} \cdot \left({x}^{\left(\frac{0.25}{2}\right)} \cdot \left({x}^{0.25} \cdot y\right)\right)} \]

   9. metadata-eval 99.7%

      \[\leadsto \left(1 - x\right) + {x}^{\color{blue}{0.125}} \cdot \left({x}^{\left(\frac{0.25}{2}\right)} \cdot \left({x}^{0.25} \cdot y\right)\right) \]

   10. metadata-eval 99.7%

       \[\leadsto \left(1 - x\right) + {x}^{0.125} \cdot \left({x}^{\color{blue}{0.125}} \cdot \left({x}^{0.25} \cdot y\right)\right) \]

6. Applied egg-rr 99.7%

   \[\leadsto \left(1 - x\right) + \color{blue}{{x}^{0.125} \cdot \left({x}^{0.125} \cdot \left({x}^{0.25} \cdot y\right)\right)} \]
7. Step-by-step derivation

   1. pow1 99.7%

      \[\leadsto \left(1 - x\right) + {x}^{0.125} \cdot \color{blue}{{\left({x}^{0.125} \cdot \left({x}^{0.25} \cdot y\right)\right)}^{1}} \]

   2. *-commutative 99.7%

      \[\leadsto \left(1 - x\right) + {x}^{0.125} \cdot {\color{blue}{\left(\left({x}^{0.25} \cdot y\right) \cdot {x}^{0.125}\right)}}^{1} \]

   3. *-commutative 99.7%

      \[\leadsto \left(1 - x\right) + {x}^{0.125} \cdot {\left(\color{blue}{\left(y \cdot {x}^{0.25}\right)} \cdot {x}^{0.125}\right)}^{1} \]

   4. associate-*l* 99.8%

      \[\leadsto \left(1 - x\right) + {x}^{0.125} \cdot {\color{blue}{\left(y \cdot \left({x}^{0.25} \cdot {x}^{0.125}\right)\right)}}^{1} \]

   5. pow-prod-up 99.9%

      \[\leadsto \left(1 - x\right) + {x}^{0.125} \cdot {\left(y \cdot \color{blue}{{x}^{\left(0.25 + 0.125\right)}}\right)}^{1} \]

   6. metadata-eval 99.9%

      \[\leadsto \left(1 - x\right) + {x}^{0.125} \cdot {\left(y \cdot {x}^{\color{blue}{0.375}}\right)}^{1} \]

8. Applied egg-rr 99.9%

   \[\leadsto \left(1 - x\right) + {x}^{0.125} \cdot \color{blue}{{\left(y \cdot {x}^{0.375}\right)}^{1}} \]
9. Step-by-step derivation

   1. unpow1 99.9%

      \[\leadsto \left(1 - x\right) + {x}^{0.125} \cdot \color{blue}{\left(y \cdot {x}^{0.375}\right)} \]

10. Simplified 99.9%

    \[\leadsto \left(1 - x\right) + {x}^{0.125} \cdot \color{blue}{\left(y \cdot {x}^{0.375}\right)} \]

11. Final simplification 99.9%

    \[\leadsto \left(1 - x\right) + {x}^{0.125} \cdot \left(y \cdot {x}^{0.375}\right) \]
12. Add Preprocessing

Alternative 2: 95.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+74} \lor \neg \left(y \leq 1.35 \cdot 10^{+72}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.7e+74) (not (<= y 1.35e+72)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.7e+74) || !(y <= 1.35e+72)) {
		tmp = 1.0 + (y * sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.7d+74)) .or. (.not. (y <= 1.35d+72))) then
        tmp = 1.0d0 + (y * sqrt(x))
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.7e+74) || !(y <= 1.35e+72)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.7e+74) or not (y <= 1.35e+72):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.7e+74) || !(y <= 1.35e+72))
		tmp = Float64(1.0 + Float64(y * sqrt(x)));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.7e+74) || ~((y <= 1.35e+72)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.7e+74], N[Not[LessEqual[y, 1.35e+72]], $MachinePrecision]], N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e74 or 1.35e72 < y

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{1 + \sqrt{x} \cdot y} \]

   if -1.7e74 < y < 1.35e72

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.2%

      \[\leadsto \color{blue}{1 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+74} \lor \neg \left(y \leq 1.35 \cdot 10^{+72}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+83} \lor \neg \left(y \leq 2.45 \cdot 10^{+81}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.7e+83) (not (<= y 2.45e+81))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.7e+83) || !(y <= 2.45e+81)) {
		tmp = y * sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.7d+83)) .or. (.not. (y <= 2.45d+81))) then
        tmp = y * sqrt(x)
    else
        tmp = 1.0d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.7e+83) || !(y <= 2.45e+81)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.7e+83) or not (y <= 2.45e+81):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.7e+83) || !(y <= 2.45e+81))
		tmp = Float64(y * sqrt(x));
	else
		tmp = Float64(1.0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.7e+83) || ~((y <= 2.45e+81)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.7e+83], N[Not[LessEqual[y, 2.45e+81]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.70000000000000007e83 or 2.45000000000000011e81 < y

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. *-commutative 99.6%

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

      3. add-sqr-sqrt 99.3%

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

      4. associate-*l* 99.2%

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

      5. fma-define 99.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{x}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right)} \]

      6. pow1/2 99.2%

         \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{{x}^{0.5}}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right) \]

      7. sqrt-pow1 99.4%

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right) \]

      8. metadata-eval 99.4%

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{0.25}}, \sqrt{\sqrt{x}} \cdot y, 1 - x\right) \]

      9. pow1/2 99.4%

         \[\leadsto \mathsf{fma}\left({x}^{0.25}, \sqrt{\color{blue}{{x}^{0.5}}} \cdot y, 1 - x\right) \]

      10. sqrt-pow1 99.3%

          \[\leadsto \mathsf{fma}\left({x}^{0.25}, \color{blue}{{x}^{\left(\frac{0.5}{2}\right)}} \cdot y, 1 - x\right) \]

      11. metadata-eval 99.3%

          \[\leadsto \mathsf{fma}\left({x}^{0.25}, {x}^{\color{blue}{0.25}} \cdot y, 1 - x\right) \]

   4. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{0.25}, {x}^{0.25} \cdot y, 1 - x\right)} \]

   5. Taylor expanded in y around inf 98.2%

      \[\leadsto \color{blue}{\sqrt{x} \cdot y} \]

   if -2.70000000000000007e83 < y < 2.45000000000000011e81

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 96.7%

      \[\leadsto \color{blue}{1 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+83} \lor \neg \left(y \leq 2.45 \cdot 10^{+81}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x)))) (if (<= x 1.0) (+ 1.0 t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt(x)
    if (x <= 1.0d0) then
        tmp = 1.0d0 + t_0
    else
        tmp = t_0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * Math.sqrt(x);
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * math.sqrt(x)
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 + t_0
	else:
		tmp = t_0 - x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * sqrt(x))
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 + t_0);
	else
		tmp = Float64(t_0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * sqrt(x);
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 + t_0;
	else
		tmp = t_0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.0], N[(1.0 + t$95$0), $MachinePrecision], N[(t$95$0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 97.7%

      \[\leadsto \color{blue}{1 + \sqrt{x} \cdot y} \]

   if 1 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 97.5%

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

   4. Taylor expanded in x around 0 97.6%

      \[\leadsto \color{blue}{-1 \cdot x + \sqrt{x} \cdot y} \]
   5. Step-by-step derivation

      1. neg-mul-1 97.6%

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

      2. +-commutative 97.6%

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

      3. *-commutative 97.6%

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

      4. sub-neg 97.6%

         \[\leadsto \color{blue}{y \cdot \sqrt{x} - x} \]

   6. Simplified 97.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

C

double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}

Fortran

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

Java

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

Python

def code(x, y):
	return (1.0 - x) + (y * math.sqrt(x))

Julia

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = (1.0 - x) + (y * sqrt(x));
end

Wolfram

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

   \[\leadsto \left(1 - x\right) + y \cdot \sqrt{x} \]
4. Add Preprocessing

Alternative 6: 62.5% accurate, 35.7× speedup

Math

\[\begin{array}{l} \\ 1 - x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 x))

C

double code(double x, double y) {
	return 1.0 - x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - x
end function

Java

public static double code(double x, double y) {
	return 1.0 - x;
}

Python

def code(x, y):
	return 1.0 - x

Julia

function code(x, y)
	return Float64(1.0 - x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - x;
end

Wolfram

code[x_, y_] := N[(1.0 - x), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0 65.3%

   \[\leadsto \color{blue}{1 - x} \]

4. Final simplification 65.3%

   \[\leadsto 1 - x \]
5. Add Preprocessing

Alternative 7: 31.6% accurate, 53.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x))

C

double code(double x, double y) {
	return -x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -x
end function

Java

public static double code(double x, double y) {
	return -x;
}

Python

def code(x, y):
	return -x

Julia

function code(x, y)
	return Float64(-x)
end

MATLAB

function tmp = code(x, y)
	tmp = -x;
end

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around inf 59.0%

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

4. Taylor expanded in y around 0 31.0%

   \[\leadsto \color{blue}{-1 \cdot x} \]
5. Step-by-step derivation

   1. neg-mul-1 31.0%

      \[\leadsto \color{blue}{-x} \]

6. Simplified 31.0%

   \[\leadsto \color{blue}{-x} \]

7. Final simplification 31.0%

   \[\leadsto -x \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))