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

Percentage Accurate: 99.9% → 99.9%

Time: 7.7s

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: 99.9% 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: 92.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+81}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{x}^{-0.5}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8e+46)
   (* y (sqrt x))
   (if (<= y 2.3e+81) (- 1.0 x) (/ 1.0 (/ (pow x -0.5) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8e+46) {
		tmp = y * sqrt(x);
	} else if (y <= 2.3e+81) {
		tmp = 1.0 - x;
	} else {
		tmp = 1.0 / (pow(x, -0.5) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.8d+46)) then
        tmp = y * sqrt(x)
    else if (y <= 2.3d+81) then
        tmp = 1.0d0 - x
    else
        tmp = 1.0d0 / ((x ** (-0.5d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.8e+46) {
		tmp = y * Math.sqrt(x);
	} else if (y <= 2.3e+81) {
		tmp = 1.0 - x;
	} else {
		tmp = 1.0 / (Math.pow(x, -0.5) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8e+46:
		tmp = y * math.sqrt(x)
	elif y <= 2.3e+81:
		tmp = 1.0 - x
	else:
		tmp = 1.0 / (math.pow(x, -0.5) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8e+46)
		tmp = Float64(y * sqrt(x));
	elseif (y <= 2.3e+81)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(1.0 / Float64((x ^ -0.5) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.8e+46)
		tmp = y * sqrt(x);
	elseif (y <= 2.3e+81)
		tmp = 1.0 - x;
	else
		tmp = 1.0 / ((x ^ -0.5) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8e+46], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e+81], N[(1.0 - x), $MachinePrecision], N[(1.0 / N[(N[Power[x, -0.5], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.7999999999999999e46

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. *-commutative 99.8%

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

      3. add-sqr-sqrt 99.2%

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

      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 99.2%

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

      11. metadata-eval 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in y around inf 89.6%

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

   if -1.7999999999999999e46 < y < 2.2999999999999999e81

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 95.9%

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

   if 2.2999999999999999e81 < y

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 99.2%

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

      4. associate-*l* 99.3%

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

      5. fma-define 99.3%

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

      6. pow1/2 99.3%

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

      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 99.4%

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

      11. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in y around inf 88.4%

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

      1. associate-*r/ 88.6%

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

      2. *-rgt-identity 88.6%

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

   9. Simplified 88.6%

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

   10. Taylor expanded in y around 0 88.4%

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

       1. associate-*r/ 88.6%

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

       2. *-rgt-identity 88.6%

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

       3. unpow1/2 88.6%

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

       4. rem-exp-log 84.3%

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

       5. rec-exp 84.3%

          \[\leadsto \frac{1}{\frac{{\color{blue}{\left(e^{-\log x}\right)}}^{0.5}}{y}} \]

       6. exp-prod 84.3%

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

       7. *-commutative 84.3%

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

       8. neg-mul-1 84.3%

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

       9. associate-*r* 84.3%

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

       10. metadata-eval 84.3%

           \[\leadsto \frac{1}{\frac{e^{\color{blue}{-0.5} \cdot \log x}}{y}} \]

       11. log-pow 84.3%

           \[\leadsto \frac{1}{\frac{e^{\color{blue}{\log \left({x}^{-0.5}\right)}}}{y}} \]

       12. rem-exp-log 88.6%

           \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{-0.5}}}{y}} \]

   12. Simplified 88.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{{x}^{-0.5}}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+81}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{{x}^{-0.5}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+46} \lor \neg \left(y \leq 1.82 \cdot 10^{+81}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.95e+46) (not (<= y 1.82e+81))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.95e+46) || !(y <= 1.82e+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 <= (-1.95d+46)) .or. (.not. (y <= 1.82d+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 <= -1.95e+46) || !(y <= 1.82e+81)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.95e+46) or not (y <= 1.82e+81):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.95e+46) || !(y <= 1.82e+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 <= -1.95e+46) || ~((y <= 1.82e+81)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.95e+46], N[Not[LessEqual[y, 1.82e+81]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.94999999999999997e46 or 1.82000000000000003e81 < y

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. add-sqr-sqrt 99.2%

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

      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 89.1%

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

   if -1.94999999999999997e46 < y < 1.82000000000000003e81

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 95.9%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+46} \lor \neg \left(y \leq 1.82 \cdot 10^{+81}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
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: 62.2% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 0.92) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.92) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.92d0) then
        tmp = 1.0d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.92) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.92:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.92)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.92)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.92], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 0.92000000000000004

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 59.3%

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

   if 0.92000000000000004 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 62.9%

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

      1. neg-mul-1 62.9%

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

   5. Simplified 62.9%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.92:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.1% 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 62.4%

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

4. Final simplification 62.4%

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

Alternative 7: 32.5% accurate, 107.0× 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 x around 0 28.6%

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

4. Final simplification 28.6%

   \[\leadsto 1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024039 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))