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

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 7

Speedup: 1.1×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

   4. *-inverses N/A

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

   5. associate-/l* N/A

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

   6. associate-*l/ N/A

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

   7. mul-1-neg N/A

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

   8. distribute-neg-frac N/A

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

   9. +-commutative N/A

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

   10. associate-+r+ N/A

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

   11. lft-mult-inverse N/A

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

   12. distribute-rgt-in N/A

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

   13. +-commutative N/A

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

   14. sub-neg N/A

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

   15. div-sub N/A

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

   16. *-commutative N/A

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

   17. distribute-lft-in N/A

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

   18. distribute-rgt-in N/A

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

   19. +-commutative N/A

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

   20. accelerator-lowering-fma.f64 N/A

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

   21. sqrt-lowering-sqrt.f64 N/A

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

   22. *-commutative N/A

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

   23. div-sub N/A

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

   24. sub-neg N/A

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

5. Simplified 99.9%

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

Alternative 2: 63.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq -400:\\ \;\;\;\;0 - x\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (- 1.0 x) (* (sqrt x) y)) -400.0) (- 0.0 x) (+ x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) + (sqrt(x) * y)) <= -400.0) {
		tmp = 0.0 - x;
	} else {
		tmp = 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 (((1.0d0 - x) + (sqrt(x) * y)) <= (-400.0d0)) then
        tmp = 0.0d0 - x
    else
        tmp = x + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - x) + (Math.sqrt(x) * y)) <= -400.0) {
		tmp = 0.0 - x;
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - x) + (math.sqrt(x) * y)) <= -400.0:
		tmp = 0.0 - x
	else:
		tmp = x + 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) + Float64(sqrt(x) * y)) <= -400.0)
		tmp = Float64(0.0 - x);
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - x) + (sqrt(x) * y)) <= -400.0)
		tmp = 0.0 - x;
	else
		tmp = x + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], -400.0], N[(0.0 - x), $MachinePrecision], N[(x + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x))) < -400

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 73.4

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

   5. Simplified 73.4%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - x} \]

      3. --lowering--.f64 71.2

         \[\leadsto \color{blue}{0 - x} \]

   8. Simplified 71.2%

      \[\leadsto \color{blue}{0 - x} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. neg-lowering-neg.f64 71.2

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

   10. Applied egg-rr 71.2%

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

   if -400 < (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x)))

   1. Initial program 99.9%

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

   4. Applied egg-rr 91.7%

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

   5. Taylor expanded in y around 0

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

      1. +-lowering-+.f64 63.8

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

   7. Simplified 63.8%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq -400:\\ \;\;\;\;0 - x\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq -400:\\ \;\;\;\;0 - x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (- 1.0 x) (* (sqrt x) y)) -400.0) (- 0.0 x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) + (sqrt(x) * y)) <= -400.0) {
		tmp = 0.0 - x;
	} else {
		tmp = 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 (((1.0d0 - x) + (sqrt(x) * y)) <= (-400.0d0)) then
        tmp = 0.0d0 - x
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((1.0 - x) + (Math.sqrt(x) * y)) <= -400.0) {
		tmp = 0.0 - x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - x) + (math.sqrt(x) * y)) <= -400.0:
		tmp = 0.0 - x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(1.0 - x) + Float64(sqrt(x) * y)) <= -400.0)
		tmp = Float64(0.0 - x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((1.0 - x) + (sqrt(x) * y)) <= -400.0)
		tmp = 0.0 - x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(1.0 - x), $MachinePrecision] + N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], -400.0], N[(0.0 - x), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x))) < -400

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 73.4

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

   5. Simplified 73.4%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - x} \]

      3. --lowering--.f64 71.2

         \[\leadsto \color{blue}{0 - x} \]

   8. Simplified 71.2%

      \[\leadsto \color{blue}{0 - x} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. neg-lowering-neg.f64 71.2

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

   10. Applied egg-rr 71.2%

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

   if -400 < (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 62.4

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

   5. Simplified 62.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 63.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq -400:\\ \;\;\;\;0 - x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \mathbf{if}\;y \leq -9 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+33}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt x) y 1.0)))
   (if (<= y -9e+88) t_0 (if (<= y 2.9e+33) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(sqrt(x), y, 1.0);
	double tmp;
	if (y <= -9e+88) {
		tmp = t_0;
	} else if (y <= 2.9e+33) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(sqrt(x), y, 1.0)
	tmp = 0.0
	if (y <= -9e+88)
		tmp = t_0;
	elseif (y <= 2.9e+33)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * y + 1.0), $MachinePrecision]}, If[LessEqual[y, -9e+88], t$95$0, If[LessEqual[y, 2.9e+33], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -9e88 or 2.90000000000000025e33 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 94.4

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

   5. Simplified 94.4%

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

   if -9e88 < y < 2.90000000000000025e33

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 95.7

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

   5. Simplified 95.7%

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

Alternative 5: 92.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot y\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+111}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{+82}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) y)))
   (if (<= y -2.4e+111) t_0 (if (<= y 5.8e+82) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (y <= -2.4e+111) {
		tmp = t_0;
	} else if (y <= 5.8e+82) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	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 = sqrt(x) * y
    if (y <= (-2.4d+111)) then
        tmp = t_0
    else if (y <= 5.8d+82) then
        tmp = 1.0d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * y;
	double tmp;
	if (y <= -2.4e+111) {
		tmp = t_0;
	} else if (y <= 5.8e+82) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * y
	tmp = 0
	if y <= -2.4e+111:
		tmp = t_0
	elif y <= 5.8e+82:
		tmp = 1.0 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * y)
	tmp = 0.0
	if (y <= -2.4e+111)
		tmp = t_0;
	elseif (y <= 5.8e+82)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * y;
	tmp = 0.0;
	if (y <= -2.4e+111)
		tmp = t_0;
	elseif (y <= 5.8e+82)
		tmp = 1.0 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.4e+111], t$95$0, If[LessEqual[y, 5.8e+82], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.40000000000000006e111 or 5.8000000000000003e82 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-frac N/A

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

      9. +-commutative N/A

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

      10. associate-+r+ N/A

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

      11. lft-mult-inverse N/A

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

      12. distribute-rgt-in N/A

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

      13. +-commutative N/A

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

      14. sub-neg N/A

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

      15. div-sub N/A

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

      16. *-commutative N/A

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

      17. distribute-lft-in N/A

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

      18. distribute-rgt-in N/A

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

      19. +-commutative N/A

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

      20. accelerator-lowering-fma.f64 N/A

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

      21. sqrt-lowering-sqrt.f64 N/A

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

      22. *-commutative N/A

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

      23. div-sub N/A

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

      24. sub-neg N/A

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 93.2

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

   8. Simplified 93.2%

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

   if -2.40000000000000006e111 < y < 5.8000000000000003e82

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 92.5

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

   5. Simplified 92.5%

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

Alternative 6: 63.5% accurate, 5.5× 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

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

   1. --lowering--.f64 67.6

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

5. Simplified 67.6%

   \[\leadsto \color{blue}{1 - x} \]
6. Add Preprocessing

Alternative 7: 32.7% accurate, 22.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0

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

   1. --lowering--.f64 67.6

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

5. Simplified 67.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 33.5%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))