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

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

Alternatives: 9

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. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-+l- N/A

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

   6. sub-neg N/A

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

   7. *-lft-identity N/A

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

   8. lft-mult-inverse N/A

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

   9. distribute-lft-neg-out N/A

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

   10. distribute-rgt-in N/A

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

   11. sub-neg N/A

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

   12. sub-neg N/A

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

   13. +-commutative N/A

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

   14. distribute-lft-in N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. rgt-mult-inverse N/A

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

   17. metadata-eval N/A

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

   18. *-rgt-identity N/A

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

   19. associate--r+ N/A

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

5. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, y, 1\right) - x} \]
6. Step-by-step derivation

   1. associate--l+ N/A

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

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

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

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

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

   4. --lowering--.f64 99.9

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

7. Applied egg-rr 99.9%

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

Alternative 2: 63.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq -20:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (- 1.0 x) (* (sqrt x) y)) -20.0) (- x) (+ x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) + (sqrt(x) * y)) <= -20.0) {
		tmp = -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)) <= (-20.0d0)) then
        tmp = -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)) <= -20.0) {
		tmp = -x;
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - x) + (math.sqrt(x) * y)) <= -20.0:
		tmp = -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)) <= -20.0)
		tmp = Float64(-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)) <= -20.0)
		tmp = -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], -20.0], (-x), 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))) < -20

   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 60.5

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

   5. Simplified 60.5%

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

   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-lowering-neg.f64 58.2

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

   8. Simplified 58.2%

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

   if -20 < (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x)))

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 54.7

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

   5. Simplified 54.7%

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

      1. flip3-- N/A

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

      2. metadata-eval N/A

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

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{1 + \left(\mathsf{neg}\left({x}^{3}\right)\right)}}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{{1}^{3}} + \left(\mathsf{neg}\left({x}^{3}\right)\right)}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)} \]

      5. cube-neg N/A

         \[\leadsto \frac{{1}^{3} + \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)} \]

      6. sqr-pow N/A

         \[\leadsto \frac{{1}^{3} + \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)} \]

      7. pow-prod-down N/A

         \[\leadsto \frac{{1}^{3} + \color{blue}{{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}^{\left(\frac{3}{2}\right)}}}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)} \]

      8. sqr-neg N/A

         \[\leadsto \frac{{1}^{3} + {\color{blue}{\left(x \cdot x\right)}}^{\left(\frac{3}{2}\right)}}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)} \]

      9. pow-prod-down N/A

         \[\leadsto \frac{{1}^{3} + \color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{1 \cdot 1 + \left(x \cdot x + 1 \cdot x\right)} \]

      10. sqr-pow N/A

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

      11. flip3-+ N/A

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

      12. *-lft-identity N/A

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

      13. sqr-pow N/A

          \[\leadsto \frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \frac{{\left(x \cdot x\right)}^{3} + \color{blue}{{x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(1 \cdot x\right) \cdot \left(1 \cdot x\right) - \left(x \cdot x\right) \cdot \left(1 \cdot x\right)\right)}} \]

      14. pow-prod-down N/A

          \[\leadsto \frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \frac{{\left(x \cdot x\right)}^{3} + \color{blue}{{\left(x \cdot x\right)}^{\left(\frac{3}{2}\right)}}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(1 \cdot x\right) \cdot \left(1 \cdot x\right) - \left(x \cdot x\right) \cdot \left(1 \cdot x\right)\right)}} \]

      15. sqr-neg N/A

          \[\leadsto \frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \frac{{\left(x \cdot x\right)}^{3} + {\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}^{\left(\frac{3}{2}\right)}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(1 \cdot x\right) \cdot \left(1 \cdot x\right) - \left(x \cdot x\right) \cdot \left(1 \cdot x\right)\right)}} \]

      16. pow-prod-down N/A

          \[\leadsto \frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \frac{{\left(x \cdot x\right)}^{3} + \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x\right)\right)}^{\left(\frac{3}{2}\right)}}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(1 \cdot x\right) \cdot \left(1 \cdot x\right) - \left(x \cdot x\right) \cdot \left(1 \cdot x\right)\right)}} \]

      17. sqr-pow N/A

          \[\leadsto \frac{{1}^{3} + {x}^{3}}{1 \cdot 1 + \frac{{\left(x \cdot x\right)}^{3} + \color{blue}{{\left(\mathsf{neg}\left(x\right)\right)}^{3}}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(1 \cdot x\right) \cdot \left(1 \cdot x\right) - \left(x \cdot x\right) \cdot \left(1 \cdot x\right)\right)}} \]

   7. Applied egg-rr 54.6%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq -20:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq -20:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (- 1.0 x) (* (sqrt x) y)) -20.0) (- x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((1.0 - x) + (sqrt(x) * y)) <= -20.0) {
		tmp = -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)) <= (-20.0d0)) then
        tmp = -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)) <= -20.0) {
		tmp = -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((1.0 - x) + (math.sqrt(x) * y)) <= -20.0:
		tmp = -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)) <= -20.0)
		tmp = Float64(-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)) <= -20.0)
		tmp = -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], -20.0], (-x), 1.0]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x))) < -20

   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 60.5

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

   5. Simplified 60.5%

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

   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-lowering-neg.f64 58.2

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

   8. Simplified 58.2%

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

   if -20 < (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x)))

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. --lowering--.f64 54.7

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

   5. Simplified 54.7%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 53.9%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - x\right) + \sqrt{x} \cdot y \leq -20:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+49}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt x) y 1.0)))
   (if (<= y -3.6e+44) t_0 (if (<= y 3.5e+49) (- 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 <= -3.6e+44) {
		tmp = t_0;
	} else if (y <= 3.5e+49) {
		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 <= -3.6e+44)
		tmp = t_0;
	elseif (y <= 3.5e+49)
		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, -3.6e+44], t$95$0, If[LessEqual[y, 3.5e+49], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.6e44 or 3.49999999999999975e49 < y

   1. Initial program 99.8%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   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 91.6

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

   5. Simplified 91.6%

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

   if -3.6e44 < y < 3.49999999999999975e49

   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 99.2

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

   5. Simplified 99.2%

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

Alternative 5: 93.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot y\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+54}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) y)))
   (if (<= y -3.8e+82) t_0 (if (<= y 1.25e+54) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (y <= -3.8e+82) {
		tmp = t_0;
	} else if (y <= 1.25e+54) {
		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 <= (-3.8d+82)) then
        tmp = t_0
    else if (y <= 1.25d+54) 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 <= -3.8e+82) {
		tmp = t_0;
	} else if (y <= 1.25e+54) {
		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 <= -3.8e+82:
		tmp = t_0
	elif y <= 1.25e+54:
		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 <= -3.8e+82)
		tmp = t_0;
	elseif (y <= 1.25e+54)
		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 <= -3.8e+82)
		tmp = t_0;
	elseif (y <= 1.25e+54)
		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, -3.8e+82], t$95$0, If[LessEqual[y, 1.25e+54], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.80000000000000033e82 or 1.25000000000000001e54 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 87.8

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

   5. Simplified 87.8%

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

   if -3.80000000000000033e82 < y < 1.25000000000000001e54

   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 96.9

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

   5. Simplified 96.9%

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

Alternative 6: 98.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

      \[\leadsto \color{blue}{1 + \sqrt{x} \cdot y} \]
   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 98.1

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

   5. Simplified 98.1%

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

   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

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt-in N/A

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

      5. cancel-sign-sub N/A

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

      6. distribute-rgt-neg-out N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. rem-square-sqrt N/A

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

      10. unpow2 N/A

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

      11. *-commutative N/A

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

      12. *-commutative N/A

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

      13. distribute-rgt-out-- N/A

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

      14. unsub-neg N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-in N/A

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

   5. Simplified 97.2%

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

      1. distribute-rgt-in N/A

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

      2. neg-mul-1 N/A

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

      3. unsub-neg N/A

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

      4. associate-*l* N/A

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

      5. pow1/2 N/A

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

      6. inv-pow N/A

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

      7. pow-pow N/A

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

      8. pow-plus N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

          \[\leadsto y \cdot {x}^{\color{blue}{\frac{1}{2}}} - x \]

      11. pow1/2 N/A

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

      12. --lowering--.f64 N/A

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

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

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

      14. sqrt-lowering-sqrt.f64 97.2

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

   7. Applied egg-rr 97.2%

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

4. Final simplification 97.6%

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

Alternative 7: 99.9% accurate, 1.1× speedup

Math

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

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 Float64(fma(sqrt(x), y, 1.0) - x)
end

Wolfram

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

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

   3. mul-1-neg N/A

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

   4. sub-neg N/A

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

   5. associate-+l- N/A

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

   6. sub-neg N/A

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

   7. *-lft-identity N/A

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

   8. lft-mult-inverse N/A

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

   9. distribute-lft-neg-out N/A

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

   10. distribute-rgt-in N/A

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

   11. sub-neg N/A

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

   12. sub-neg N/A

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

   13. +-commutative N/A

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

   14. distribute-lft-in N/A

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

   15. distribute-rgt-neg-out N/A

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

   16. rgt-mult-inverse N/A

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

   17. metadata-eval N/A

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

   18. *-rgt-identity N/A

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

   19. associate--r+ N/A

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

5. Simplified 99.9%

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

Alternative 8: 63.4% 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 57.5

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

5. Simplified 57.5%

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

Alternative 9: 31.8% 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 57.5

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

5. Simplified 57.5%

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

6. Taylor expanded in x around 0

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

8. Simplified 28.3%

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

Reproduce

herbie shell --seed 2024199 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))