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

Percentage Accurate: 99.9% → 99.9%

Time: 7.4s

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

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

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: 62.1% 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. --lowering--.f64 57.2

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

   5. Simplified 57.2%

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

   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 56.3

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

   8. Simplified 56.3%

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

   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. 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%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 66.9%

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

4. Final simplification 61.3%

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

Alternative 3: 95.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \mathbf{if}\;y \leq -2 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+32}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma (sqrt x) y 1.0)))
   (if (<= y -2e+51) t_0 (if (<= y 7.5e+32) (- 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 <= -2e+51) {
		tmp = t_0;
	} else if (y <= 7.5e+32) {
		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 <= -2e+51)
		tmp = t_0;
	elseif (y <= 7.5e+32)
		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, -2e+51], t$95$0, If[LessEqual[y, 7.5e+32], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2e51 or 7.49999999999999959e32 < 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 91.7

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

   5. Simplified 91.7%

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

   if -2e51 < y < 7.49999999999999959e32

   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 96.8

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

   5. Simplified 96.8%

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

Alternative 4: 92.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot y\\ \mathbf{if}\;y \leq -7 \cdot 10^{+90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+74}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) y)))
   (if (<= y -7e+90) t_0 (if (<= y 1.4e+74) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (y <= -7e+90) {
		tmp = t_0;
	} else if (y <= 1.4e+74) {
		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 <= (-7d+90)) then
        tmp = t_0
    else if (y <= 1.4d+74) 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 <= -7e+90) {
		tmp = t_0;
	} else if (y <= 1.4e+74) {
		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 <= -7e+90:
		tmp = t_0
	elif y <= 1.4e+74:
		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 <= -7e+90)
		tmp = t_0;
	elseif (y <= 1.4e+74)
		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 <= -7e+90)
		tmp = t_0;
	elseif (y <= 1.4e+74)
		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, -7e+90], t$95$0, If[LessEqual[y, 1.4e+74], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.9999999999999997e90 or 1.40000000000000001e74 < y

   1. Initial program 99.8%

      \[\left(1 - x\right) + y \cdot \sqrt{x} \]
   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 89.9

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

   5. Simplified 89.9%

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

   if -6.9999999999999997e90 < y < 1.40000000000000001e74

   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 93.9

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

   5. Simplified 93.9%

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

Alternative 5: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.64:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{x}, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot y - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.64) (fma (sqrt x) y 1.0) (- (* (sqrt x) y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.64) {
		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 <= 0.64)
		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, 0.64], 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 < 0.640000000000000013

   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 98.2

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

   5. Simplified 98.2%

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

   if 0.640000000000000013 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \left(-1 \cdot x + \sqrt{x} \cdot y\right)} \]
   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. Taylor expanded in x around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 98.0

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

   8. Simplified 98.0%

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

Alternative 6: 62.5% 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 62.3

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

5. Simplified 62.3%

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

Alternative 7: 31.5% 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. accelerator-lowering-fma.f64 N/A

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

   3. sqrt-lowering-sqrt.f64 68.4

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

5. Simplified 68.4%

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

6. Taylor expanded in y around 0

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

8. Simplified 31.9%

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

Reproduce

herbie shell --seed 2024198 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))