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

Percentage Accurate: 99.9% → 99.9%

Time: 7.1s

Alternatives: 8

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

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

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 2: 63.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x} \cdot y + \left(1 - x\right) \leq -2:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* (sqrt x) y) (- 1.0 x)) -2.0) (- x) (+ x 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((sqrt(x) * y) + (1.0 - x)) <= -2.0) {
		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 (((sqrt(x) * y) + (1.0d0 - x)) <= (-2.0d0)) 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 (((Math.sqrt(x) * y) + (1.0 - x)) <= -2.0) {
		tmp = -x;
	} else {
		tmp = x + 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sqrt(x) * y) + (1.0 - x)) <= -2.0:
		tmp = -x
	else:
		tmp = x + 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sqrt(x) * y) + Float64(1.0 - x)) <= -2.0)
		tmp = Float64(-x);
	else
		tmp = Float64(x + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sqrt(x) * y) + (1.0 - x)) <= -2.0)
		tmp = -x;
	else
		tmp = x + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -2.0], (-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))) < -2

   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. lower--.f64 60.8

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

   5. Simplified 60.8%

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

   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. lower-neg.f64 61.1

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

   8. Simplified 61.1%

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

   if -2 < (+.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 94.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 63.0

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

   7. Simplified 63.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x} \cdot y + \left(1 - x\right) \leq -2:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;x + 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x} \cdot y + \left(1 - x\right) \leq -2:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* (sqrt x) y) (- 1.0 x)) -2.0) (- x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((sqrt(x) * y) + (1.0 - x)) <= -2.0) {
		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 (((sqrt(x) * y) + (1.0d0 - x)) <= (-2.0d0)) 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 (((Math.sqrt(x) * y) + (1.0 - x)) <= -2.0) {
		tmp = -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sqrt(x) * y) + (1.0 - x)) <= -2.0:
		tmp = -x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(sqrt(x) * y) + Float64(1.0 - x)) <= -2.0)
		tmp = Float64(-x);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((sqrt(x) * y) + (1.0 - x)) <= -2.0)
		tmp = -x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision], -2.0], (-x), 1.0]

Derivation

1. Split input into 2 regimes

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

   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. lower--.f64 60.8

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

   5. Simplified 60.8%

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

   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. lower-neg.f64 61.1

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

   8. Simplified 61.1%

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

   if -2 < (+.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 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. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 98.2

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

   5. Simplified 98.2%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 62.5%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x} \cdot y + \left(1 - x\right) \leq -2:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+55}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt x) y 1.0)))
   (if (<= y -2.7e+45) t_0 (if (<= y 4.8e+55) (- 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 <= -2.7e+45) {
		tmp = t_0;
	} else if (y <= 4.8e+55) {
		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 <= -2.7e+45)
		tmp = t_0;
	elseif (y <= 4.8e+55)
		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, -2.7e+45], t$95$0, If[LessEqual[y, 4.8e+55], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.69999999999999984e45 or 4.7999999999999998e55 < 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. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 93.0

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

   5. Simplified 93.0%

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

   if -2.69999999999999984e45 < y < 4.7999999999999998e55

   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. lower--.f64 96.3

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

   5. Simplified 96.3%

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

Alternative 5: 91.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot y\\ \mathbf{if}\;y \leq -6.8 \cdot 10^{+121}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+56}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) y)))
   (if (<= y -6.8e+121) t_0 (if (<= y 5e+56) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (y <= -6.8e+121) {
		tmp = t_0;
	} else if (y <= 5e+56) {
		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 <= (-6.8d+121)) then
        tmp = t_0
    else if (y <= 5d+56) 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 <= -6.8e+121) {
		tmp = t_0;
	} else if (y <= 5e+56) {
		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 <= -6.8e+121:
		tmp = t_0
	elif y <= 5e+56:
		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 <= -6.8e+121)
		tmp = t_0;
	elseif (y <= 5e+56)
		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 <= -6.8e+121)
		tmp = t_0;
	elseif (y <= 5e+56)
		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, -6.8e+121], t$95$0, If[LessEqual[y, 5e+56], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.80000000000000021e121 or 5.00000000000000024e56 < 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. lower-*.f64 N/A

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

      2. lower-sqrt.f64 92.1

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

   5. Simplified 92.1%

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

   if -6.80000000000000021e121 < y < 5.00000000000000024e56

   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. lower--.f64 91.7

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

   5. Simplified 91.7%

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

Alternative 6: 98.7% 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. lower-fma.f64 N/A

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

      3. lower-sqrt.f64 98.9

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

   5. Simplified 98.9%

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

   if 1 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(-1 \cdot x + \sqrt{x} \cdot y\right)} \]
   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 99.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 99.6

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

   8. Simplified 99.6%

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

Alternative 7: 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. lower--.f64 61.6

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

5. Simplified 61.6%

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

Alternative 8: 32.5% 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 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. lower-fma.f64 N/A

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

   3. lower-sqrt.f64 67.9

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

5. Simplified 67.9%

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

6. Taylor expanded in y around 0

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

8. Simplified 30.1%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))