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

Percentage Accurate: 99.9% → 99.9%

Time: 8.0s

Alternatives: 10

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: 95.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+56} \lor \neg \left(y \leq 1.55 \cdot 10^{+38}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -9.2e+56) (not (<= y 1.55e+38)))
   (+ 1.0 (* y (sqrt x)))
   (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -9.2e+56) || !(y <= 1.55e+38)) {
		tmp = 1.0 + (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 <= (-9.2d+56)) .or. (.not. (y <= 1.55d+38))) then
        tmp = 1.0d0 + (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 <= -9.2e+56) || !(y <= 1.55e+38)) {
		tmp = 1.0 + (y * Math.sqrt(x));
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -9.2e+56) or not (y <= 1.55e+38):
		tmp = 1.0 + (y * math.sqrt(x))
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -9.2e+56) || !(y <= 1.55e+38))
		tmp = Float64(1.0 + 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 <= -9.2e+56) || ~((y <= 1.55e+38)))
		tmp = 1.0 + (y * sqrt(x));
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -9.2e+56], N[Not[LessEqual[y, 1.55e+38]], $MachinePrecision]], N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.20000000000000058e56 or 1.55000000000000009e38 < y

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 95.0%

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

   if -9.20000000000000058e56 < y < 1.55000000000000009e38

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 96.7%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+56} \lor \neg \left(y \leq 1.55 \cdot 10^{+38}\right):\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+93} \lor \neg \left(y \leq 6.6 \cdot 10^{+70}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.95e+93) (not (<= y 6.6e+70))) (* y (sqrt x)) (- 1.0 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.95e+93) || !(y <= 6.6e+70)) {
		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+93)) .or. (.not. (y <= 6.6d+70))) 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+93) || !(y <= 6.6e+70)) {
		tmp = y * Math.sqrt(x);
	} else {
		tmp = 1.0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.95e+93) or not (y <= 6.6e+70):
		tmp = y * math.sqrt(x)
	else:
		tmp = 1.0 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.95e+93) || !(y <= 6.6e+70))
		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+93) || ~((y <= 6.6e+70)))
		tmp = y * sqrt(x);
	else
		tmp = 1.0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.95e+93], N[Not[LessEqual[y, 6.6e+70]], $MachinePrecision]], N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(1.0 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.9500000000000001e93 or 6.60000000000000033e70 < y

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 99.7%

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

   6. Taylor expanded in y around inf 93.6%

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

   if -1.9500000000000001e93 < y < 6.60000000000000033e70

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 93.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+93} \lor \neg \left(y \leq 6.6 \cdot 10^{+70}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, 1.0], N[(1.0 + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $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. 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. Taylor expanded in x around 0 98.3%

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

   if 1 < x

   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. Taylor expanded in x around inf 97.3%

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

      1. *-commutative 97.3%

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

      2. sqrt-div 97.3%

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

      3. metadata-eval 97.3%

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

      4. un-div-inv 97.3%

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

   7. Applied egg-rr 97.3%

      \[\leadsto x \cdot \left(\color{blue}{\frac{y}{\sqrt{x}}} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

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

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;x \leq 0.062:\\ \;\;\;\;1 + t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt x)))) (if (<= x 0.062) (+ 1.0 t_0) (- t_0 x))))

C

double code(double x, double y) {
	double t_0 = y * sqrt(x);
	double tmp;
	if (x <= 0.062) {
		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 = y * sqrt(x)
    if (x <= 0.062d0) 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 = y * Math.sqrt(x);
	double tmp;
	if (x <= 0.062) {
		tmp = 1.0 + t_0;
	} else {
		tmp = t_0 - x;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * math.sqrt(x)
	tmp = 0
	if x <= 0.062:
		tmp = 1.0 + t_0
	else:
		tmp = t_0 - x
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * sqrt(x))
	tmp = 0.0
	if (x <= 0.062)
		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 = y * sqrt(x);
	tmp = 0.0;
	if (x <= 0.062)
		tmp = 1.0 + t_0;
	else
		tmp = t_0 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.062], N[(1.0 + t$95$0), $MachinePrecision], N[(t$95$0 - x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.062

   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. Taylor expanded in x around 0 98.2%

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

   if 0.062 < x

   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. Taylor expanded in x around inf 97.3%

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

      1. *-commutative 97.3%

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

      2. sqrt-div 97.3%

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

      3. metadata-eval 97.3%

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

      4. un-div-inv 97.4%

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

   7. Applied egg-rr 97.4%

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

   8. Taylor expanded in x around 0 97.4%

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

      1. neg-mul-1 97.4%

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

      2. +-commutative 97.4%

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

      3. unsub-neg 97.4%

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

      4. *-commutative 97.4%

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

   10. Simplified 97.4%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.062:\\ \;\;\;\;1 + y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 7: 62.2% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.062:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 0.062) 1.0 (- x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.062) {
		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 <= 0.062d0) 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 <= 0.062) {
		tmp = 1.0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.062:
		tmp = 1.0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.062)
		tmp = 1.0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.062)
		tmp = 1.0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.062], 1.0, (-x)]

Derivation

1. Split input into 2 regimes

2. if x < 0.062

   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. Taylor expanded in y around inf 99.7%

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

   6. Taylor expanded in x around 0 98.1%

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

   7. Taylor expanded in y around 0 61.0%

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

   if 0.062 < x

   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. Taylor expanded in x around inf 97.3%

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

   6. Taylor expanded in y around 0 61.1%

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

      1. neg-mul-1 61.1%

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

   8. Simplified 61.1%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.062:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 15.3× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (/ (* y (- 1.0 x)) y))

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return (y * (1.0 - x)) / y;
}

Python

def code(x, y):
	return (y * (1.0 - x)) / y

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(y * N[(1.0 - x), $MachinePrecision]), $MachinePrecision] / y), $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. Taylor expanded in y around inf 88.9%

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

6. Taylor expanded in y around 0 52.1%

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

   1. associate-*r/ 65.3%

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

8. Applied egg-rr 65.3%

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

9. Final simplification 65.3%

   \[\leadsto \frac{y \cdot \left(1 - x\right)}{y} \]
10. Add Preprocessing

Alternative 9: 63.2% 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. 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. Taylor expanded in y around 0 63.1%

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

6. Final simplification 63.1%

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

Alternative 10: 31.9% 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. 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. Taylor expanded in y around inf 88.9%

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

6. Taylor expanded in x around 0 67.6%

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

7. Taylor expanded in y around 0 31.5%

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

8. Final simplification 31.5%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024066 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))