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

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

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, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot y + \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (* (sqrt x) y) (- 1.0 x)))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return (Math.sqrt(x) * y) + (1.0 - x);
}

Python

def code(x, y):
	return (math.sqrt(x) * y) + (1.0 - x)

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = (sqrt(x) * y) + (1.0 - x);
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Final simplification 99.9%

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

Alternative 2: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -850000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \sqrt{x}, 1\right)\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+55}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{x} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -850000000000.0)
   (fma y (sqrt x) 1.0)
   (if (<= y 4.9e+55) (- 1.0 x) (+ 1.0 (* (sqrt x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -850000000000.0) {
		tmp = fma(y, sqrt(x), 1.0);
	} else if (y <= 4.9e+55) {
		tmp = 1.0 - x;
	} else {
		tmp = 1.0 + (sqrt(x) * y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -850000000000.0)
		tmp = fma(y, sqrt(x), 1.0);
	elseif (y <= 4.9e+55)
		tmp = Float64(1.0 - x);
	else
		tmp = Float64(1.0 + Float64(sqrt(x) * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -850000000000.0], N[(y * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y, 4.9e+55], N[(1.0 - x), $MachinePrecision], N[(1.0 + N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5e11

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if -8.5e11 < y < 4.90000000000000015e55

   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 99.5

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

   5. Applied rewrites 99.5%

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

   if 4.90000000000000015e55 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 93.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -850000000000:\\ \;\;\;\;\mathsf{fma}\left(y, \sqrt{x}, 1\right)\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+55}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;1 + \sqrt{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{x} \cdot y + \left(1 - x\right) \leq -5000:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ (* (sqrt x) y) (- 1.0 x)) -5000.0) (- x) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((sqrt(x) * y) + (1.0 - x)) <= -5000.0) {
		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)) <= (-5000.0d0)) 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)) <= -5000.0) {
		tmp = -x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((math.sqrt(x) * y) + (1.0 - x)) <= -5000.0:
		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)) <= -5000.0)
		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)) <= -5000.0)
		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], -5000.0], (-x), 1.0]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x))) < -5e3

   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. lower--.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 63.1%

      \[\leadsto -x \]

   if -5e3 < (+.f64 (-.f64 #s(literal 1 binary64) x) (*.f64 y (sqrt.f64 x)))

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. lower--.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 62.2%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{x} \cdot y + \left(1 - x\right) \leq -5000:\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(y, \sqrt{x}, 1\right)\\ \mathbf{if}\;y \leq -850000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+55}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fma y (sqrt x) 1.0)))
   (if (<= y -850000000000.0) t_0 (if (<= y 4.9e+55) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = fma(y, sqrt(x), 1.0);
	double tmp;
	if (y <= -850000000000.0) {
		tmp = t_0;
	} else if (y <= 4.9e+55) {
		tmp = 1.0 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = fma(y, sqrt(x), 1.0)
	tmp = 0.0
	if (y <= -850000000000.0)
		tmp = t_0;
	elseif (y <= 4.9e+55)
		tmp = Float64(1.0 - x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -850000000000.0], t$95$0, If[LessEqual[y, 4.9e+55], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5e11 or 4.90000000000000015e55 < 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-sqrt.f64 90.0

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

   5. Applied rewrites 90.0%

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

   if -8.5e11 < y < 4.90000000000000015e55

   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 99.5

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

   5. Applied rewrites 99.5%

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

Alternative 5: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot y\\ \mathbf{if}\;y \leq -9 \cdot 10^{+92}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+57}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) y)))
   (if (<= y -9e+92) t_0 (if (<= y 1.65e+57) (- 1.0 x) t_0))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (y <= -9e+92) {
		tmp = t_0;
	} else if (y <= 1.65e+57) {
		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 <= (-9d+92)) then
        tmp = t_0
    else if (y <= 1.65d+57) 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 <= -9e+92) {
		tmp = t_0;
	} else if (y <= 1.65e+57) {
		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 <= -9e+92:
		tmp = t_0
	elif y <= 1.65e+57:
		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 <= -9e+92)
		tmp = t_0;
	elseif (y <= 1.65e+57)
		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 <= -9e+92)
		tmp = t_0;
	elseif (y <= 1.65e+57)
		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, -9e+92], t$95$0, If[LessEqual[y, 1.65e+57], N[(1.0 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.9999999999999998e92 or 1.6500000000000001e57 < 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 90.3

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

   5. Applied rewrites 90.3%

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

   if -8.9999999999999998e92 < y < 1.6500000000000001e57

   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 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9 \cdot 10^{+92}:\\ \;\;\;\;\sqrt{x} \cdot y\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+57}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot y\\ \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) y))) (if (<= x 1.0) (+ 1.0 t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (x <= 1.0) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	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 (x <= 1.0d0) then
        tmp = 1.0d0 + t_0
    else
        tmp = t_0 - x
    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 (x <= 1.0) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * y
	tmp = 0
	if x <= 1.0:
		tmp = 1.0 + t_0
	else:
		tmp = t_0 - x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * y)
	tmp = 0.0
	if (x <= 1.0)
		tmp = Float64(1.0 + t_0);
	else
		tmp = Float64(t_0 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * y;
	tmp = 0.0;
	if (x <= 1.0)
		tmp = 1.0 + t_0;
	else
		tmp = t_0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

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

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

   5. Applied rewrites 96.8%

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

   if 1 < x

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+r- N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1 + \sqrt{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot y - x\\ \end{array} \]
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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

   5. lower--.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 8: 63.1% 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 63.6

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

5. Applied rewrites 63.6%

   \[\leadsto \color{blue}{1 - x} \]
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. lower--.f64 63.6

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

5. Applied rewrites 63.6%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 31.1%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024244 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))