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

Percentage Accurate: 99.9% → 99.9%

Time: 6.6s

Alternatives: 6

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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y, \sqrt{x}, 1 - x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. fma-define 99.9%

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

3. Simplified 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1 - x\right) \]
6. Add Preprocessing

Alternative 2: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+40} \lor \neg \left(y \leq 6 \cdot 10^{+55}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -8.2e+40) (not (<= y 6e+55))) (+ 1.0 (* y (sqrt x))) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -8.2e+40) || !(y <= 6e+55)) {
		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 <= (-8.2d+40)) .or. (.not. (y <= 6d+55))) 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 <= -8.2e+40) || !(y <= 6e+55)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -8.2e+40) or not (y <= 6e+55):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -8.2e+40) || !(y <= 6e+55))
		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 <= -8.2e+40) || ~((y <= 6e+55)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -8.2e+40], N[Not[LessEqual[y, 6e+55]], $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 < -8.2000000000000003e40 or 6.00000000000000033e55 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 88.5%

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

   if -8.2000000000000003e40 < y < 6.00000000000000033e55

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 97.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+40} \lor \neg \left(y \leq 6 \cdot 10^{+55}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{\sqrt{x}} + -1\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 99.7%

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

   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 97.9%

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

      1. *-commutative 97.9%

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

      2. sqrt-div 97.8%

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

      3. metadata-eval 97.8%

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

      4. un-div-inv 97.9%

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

   5. Applied egg-rr 97.9%

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

4. Final simplification 98.8%

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

Alternative 4: 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 5: 62.0% 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 66.6%

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

4. Final simplification 66.6%

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

Alternative 6: 31.0% 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 60.8%

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

4. Taylor expanded in y around 0 33.2%

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

   1. neg-mul-1 33.2%

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

6. Simplified 33.2%

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

7. Final simplification 33.2%

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

Reproduce

herbie shell --seed 2024059 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))