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

Percentage Accurate: 85.6% → 99.8%

Time: 14.7s

Alternatives: 10

Speedup: 1.9×

Specification

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * log((1.0 - y)))) - t;
}

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * math.log((1.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(z * log(Float64(1.0 - y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * log(y)) + (z * log((1.0 - y)))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Initial Program: 85.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * log((1.0 - y)))) - t;
}

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * math.log((1.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(z * log(Float64(1.0 - y)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * log(y)) + (z * log((1.0 - y)))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (fma z (log1p (- y)) (- (* x (log y)) t)))

C

double code(double x, double y, double z, double t) {
	return fma(z, log1p(-y), ((x * log(y)) - t));
}

Julia

function code(x, y, z, t)
	return fma(z, log1p(Float64(-y)), Float64(Float64(x * log(y)) - t))
end

Wolfram

code[x_, y_, z_, t_] := N[(z * N[Log[1 + (-y)], $MachinePrecision] + N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Step-by-step derivation

   1. +-commutative 85.2%

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

   2. associate--l+ 85.2%

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

   3. fma-define 85.2%

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

   4. sub-neg 85.2%

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

   5. log1p-define 99.8%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{log1p}\left(-y\right)}, x \cdot \log y - t\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right)} \]
4. Add Preprocessing

5. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), x \cdot \log y - t\right) \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y + y \cdot \left(z \cdot \left(y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right) + -1\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* x (log y))
   (* y (* z (+ (* y (- (* y -0.3333333333333333) 0.5)) -1.0))))
  t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (y * (z * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0)))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (y * (z * ((y * ((y * (-0.3333333333333333d0)) - 0.5d0)) + (-1.0d0))))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + (y * (z * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0)))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (y * (z * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(y * Float64(z * Float64(Float64(y * Float64(Float64(y * -0.3333333333333333) - 0.5)) + -1.0)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * log(y)) + (y * (z * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0)))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(y * N[(z * N[(N[(y * N[(N[(y * -0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0 99.8%

   \[\leadsto \left(x \cdot \log y + \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + -0.3333333333333333 \cdot \left(y \cdot z\right)\right)\right)}\right) - t \]

4. Taylor expanded in z around 0 99.8%

   \[\leadsto \left(x \cdot \log y + \color{blue}{y \cdot \left(z \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)}\right) - t \]

5. Final simplification 99.8%

   \[\leadsto \left(x \cdot \log y + y \cdot \left(z \cdot \left(y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right) + -1\right)\right)\right) - t \]
6. Add Preprocessing

Alternative 3: 89.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-14} \lor \neg \left(x \leq 8.6 \cdot 10^{-159}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.7e-14) (not (<= x 8.6e-159)))
   (- (* x (log y)) t)
   (- (* y (- (* y (+ (* z -0.5) (* -0.3333333333333333 (* z y)))) z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.7e-14) || !(x <= 8.6e-159)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-4.7d-14)) .or. (.not. (x <= 8.6d-159))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y * ((y * ((z * (-0.5d0)) + ((-0.3333333333333333d0) * (z * y)))) - z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.7e-14) || !(x <= 8.6e-159)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -4.7e-14) or not (x <= 8.6e-159):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.7e-14) || !(x <= 8.6e-159))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(-0.3333333333333333 * Float64(z * y)))) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -4.7e-14) || ~((x <= 8.6e-159)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.7e-14], N[Not[LessEqual[x, 8.6e-159]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.3333333333333333 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000002e-14 or 8.6e-159 < x

   1. Initial program 92.9%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 92.6%

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

   if -4.7000000000000002e-14 < x < 8.6e-159

   1. Initial program 71.5%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 67.2%

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) - t} \]
   4. Step-by-step derivation

      1. fma-neg 67.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \log \left(1 - y\right), -t\right)} \]

      2. sub-neg 67.2%

         \[\leadsto \mathsf{fma}\left(z, \log \color{blue}{\left(1 + \left(-y\right)\right)}, -t\right) \]

      3. log1p-define 94.4%

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{log1p}\left(-y\right)}, -t\right) \]

   5. Simplified 94.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), -t\right)} \]

   6. Taylor expanded in y around 0 94.4%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + -0.3333333333333333 \cdot \left(y \cdot z\right)\right)\right) - t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-14} \lor \neg \left(x \leq 8.6 \cdot 10^{-159}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y + z \cdot \left(y \cdot \left(y \cdot -0.5 + -1\right)\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (* y (+ (* y -0.5) -1.0)))) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * (y * ((y * -0.5) + -1.0)))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (z * (y * ((y * (-0.5d0)) + (-1.0d0))))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + (z * (y * ((y * -0.5) + -1.0)))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * (y * ((y * -0.5) + -1.0)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(z * Float64(y * Float64(Float64(y * -0.5) + -1.0)))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * log(y)) + (z * (y * ((y * -0.5) + -1.0)))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[(y * N[(N[(y * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0 99.7%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(y \cdot \left(-0.5 \cdot y - 1\right)\right)}\right) - t \]

4. Final simplification 99.7%

   \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \left(y \cdot -0.5 + -1\right)\right)\right) - t \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+93} \lor \neg \left(x \leq 4.2 \cdot 10^{+107}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -6.2e+93) (not (<= x 4.2e+107)))
   (* x (log y))
   (- (* y (- (* y (+ (* z -0.5) (* -0.3333333333333333 (* z y)))) z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.2e+93) || !(x <= 4.2e+107)) {
		tmp = x * log(y);
	} else {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-6.2d+93)) .or. (.not. (x <= 4.2d+107))) then
        tmp = x * log(y)
    else
        tmp = (y * ((y * ((z * (-0.5d0)) + ((-0.3333333333333333d0) * (z * y)))) - z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -6.2e+93) || !(x <= 4.2e+107)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -6.2e+93) or not (x <= 4.2e+107):
		tmp = x * math.log(y)
	else:
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -6.2e+93) || !(x <= 4.2e+107))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(-0.3333333333333333 * Float64(z * y)))) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -6.2e+93) || ~((x <= 4.2e+107)))
		tmp = x * log(y);
	else
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -6.2e+93], N[Not[LessEqual[x, 4.2e+107]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.3333333333333333 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.20000000000000038e93 or 4.1999999999999999e107 < x

   1. Initial program 97.2%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.2%

      \[\leadsto \left(x \cdot \log y + \color{blue}{-1 \cdot \left(y \cdot z\right)}\right) - t \]
   4. Step-by-step derivation

      1. associate-*r* 99.2%

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

      2. mul-1-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in x around inf 99.2%

      \[\leadsto \color{blue}{x \cdot \left(\left(\log y + -1 \cdot \frac{y \cdot z}{x}\right) - \frac{t}{x}\right)} \]
   7. Step-by-step derivation

      1. associate--l+ 99.2%

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

      2. associate-*r/ 99.2%

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

      3. div-sub 99.2%

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

      4. associate-*r* 99.2%

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

      5. mul-1-neg 99.2%

         \[\leadsto x \cdot \left(\log y + \frac{\color{blue}{\left(-y\right)} \cdot z - t}{x}\right) \]

   8. Simplified 99.2%

      \[\leadsto \color{blue}{x \cdot \left(\log y + \frac{\left(-y\right) \cdot z - t}{x}\right)} \]

   9. Taylor expanded in x around inf 76.1%

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

   if -6.20000000000000038e93 < x < 4.1999999999999999e107

   1. Initial program 79.0%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 61.7%

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) - t} \]
   4. Step-by-step derivation

      1. fma-neg 61.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \log \left(1 - y\right), -t\right)} \]

      2. sub-neg 61.7%

         \[\leadsto \mathsf{fma}\left(z, \log \color{blue}{\left(1 + \left(-y\right)\right)}, -t\right) \]

      3. log1p-define 81.1%

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{log1p}\left(-y\right)}, -t\right) \]

   5. Simplified 81.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), -t\right)} \]

   6. Taylor expanded in y around 0 81.1%

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + -0.3333333333333333 \cdot \left(y \cdot z\right)\right)\right) - t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+93} \lor \neg \left(x \leq 4.2 \cdot 10^{+107}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y - z \cdot y\right) - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (- (* x (log y)) (* z y)) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) - (z * y)) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) - (z * y)) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) - (z * y)) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) - (z * y)) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) - Float64(z * y)) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * log(y)) - (z * y)) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(z * y), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0 99.3%

   \[\leadsto \left(x \cdot \log y + \color{blue}{-1 \cdot \left(y \cdot z\right)}\right) - t \]
4. Step-by-step derivation

   1. associate-*r* 99.3%

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

   2. mul-1-neg 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in x around 0 99.3%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)} - t \]
7. Step-by-step derivation

   1. +-commutative 99.3%

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

   2. mul-1-neg 99.3%

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

   3. sub-neg 99.3%

      \[\leadsto \color{blue}{\left(x \cdot \log y - y \cdot z\right)} - t \]

8. Simplified 99.3%

   \[\leadsto \color{blue}{\left(x \cdot \log y - y \cdot z\right)} - t \]

9. Final simplification 99.3%

   \[\leadsto \left(x \cdot \log y - z \cdot y\right) - t \]
10. Add Preprocessing

Alternative 7: 57.6% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* y (- (* y (+ (* z -0.5) (* -0.3333333333333333 (* z y)))) z)) t))

C

double code(double x, double y, double z, double t) {
	return (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * ((y * ((z * (-0.5d0)) + ((-0.3333333333333333d0) * (z * y)))) - z)) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
}

Python

def code(x, y, z, t):
	return (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(-0.3333333333333333 * Float64(z * y)))) - z)) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z)) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.3333333333333333 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0 47.7%

   \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) - t} \]
4. Step-by-step derivation

   1. fma-neg 47.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \log \left(1 - y\right), -t\right)} \]

   2. sub-neg 47.7%

      \[\leadsto \mathsf{fma}\left(z, \log \color{blue}{\left(1 + \left(-y\right)\right)}, -t\right) \]

   3. log1p-define 61.1%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{log1p}\left(-y\right)}, -t\right) \]

5. Simplified 61.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), -t\right)} \]

6. Taylor expanded in y around 0 61.1%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + y \cdot \left(-0.5 \cdot z + -0.3333333333333333 \cdot \left(y \cdot z\right)\right)\right) - t} \]

7. Final simplification 61.1%

   \[\leadsto y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right) - t \]
8. Add Preprocessing

Alternative 8: 57.5% accurate, 19.2× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right) - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* y (- (* -0.5 (* z y)) z)) t))

C

double code(double x, double y, double z, double t) {
	return (y * ((-0.5 * (z * y)) - z)) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * (((-0.5d0) * (z * y)) - z)) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * ((-0.5 * (z * y)) - z)) - t;
}

Python

def code(x, y, z, t):
	return (y * ((-0.5 * (z * y)) - z)) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(y * Float64(Float64(-0.5 * Float64(z * y)) - z)) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * ((-0.5 * (z * y)) - z)) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in x around 0 47.7%

   \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right) - t} \]
4. Step-by-step derivation

   1. fma-neg 47.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \log \left(1 - y\right), -t\right)} \]

   2. sub-neg 47.7%

      \[\leadsto \mathsf{fma}\left(z, \log \color{blue}{\left(1 + \left(-y\right)\right)}, -t\right) \]

   3. log1p-define 61.1%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\mathsf{log1p}\left(-y\right)}, -t\right) \]

5. Simplified 61.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, \mathsf{log1p}\left(-y\right), -t\right)} \]

6. Taylor expanded in y around 0 61.0%

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + -0.5 \cdot \left(y \cdot z\right)\right) - t} \]

7. Final simplification 61.0%

   \[\leadsto y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right) - t \]
8. Add Preprocessing

Alternative 9: 57.2% accurate, 35.2× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(-z\right) - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (* y (- z)) t))

C

double code(double x, double y, double z, double t) {
	return (y * -z) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * -z) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (y * -z) - t;
}

Python

def code(x, y, z, t):
	return (y * -z) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(y * Float64(-z)) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * -z) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 85.2%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in y around 0 99.3%

   \[\leadsto \left(x \cdot \log y + \color{blue}{-1 \cdot \left(y \cdot z\right)}\right) - t \]
4. Step-by-step derivation

   1. associate-*r* 99.3%

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

   2. mul-1-neg 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in x around 0 60.7%

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot z\right) - t} \]
7. Step-by-step derivation

   1. associate-*r* 60.7%

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

   2. mul-1-neg 60.7%

      \[\leadsto \color{blue}{\left(-y\right)} \cdot z - t \]

8. Simplified 60.7%

   \[\leadsto \color{blue}{\left(-y\right) \cdot z - t} \]

9. Final simplification 60.7%

   \[\leadsto y \cdot \left(-z\right) - t \]
10. Add Preprocessing

Alternative 10: 42.9% accurate, 105.5× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- t))

C

double code(double x, double y, double z, double t) {
	return -t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -t
end function

Java

public static double code(double x, double y, double z, double t) {
	return -t;
}

Python

def code(x, y, z, t):
	return -t

Julia

function code(x, y, z, t)
	return Float64(-t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -t;
end

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 85.2%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
2. Add Preprocessing

3. Taylor expanded in t around inf 46.9%

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

   1. neg-mul-1 46.9%

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

5. Simplified 46.9%

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

6. Final simplification 46.9%

   \[\leadsto -t \]
7. Add Preprocessing

Developer target: 99.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (*
   (- z)
   (+
    (+ (* 0.5 (* y y)) y)
    (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y)))))
  (- t (* x (log y)))))

C

double code(double x, double y, double z, double t) {
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * log(y)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-z * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * Math.log(y)));
}

Python

def code(x, y, z, t):
	return (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * math.log(y)))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(-z) * Float64(Float64(Float64(0.5 * Float64(y * y)) + y) + Float64(Float64(0.3333333333333333 / Float64(1.0 * Float64(1.0 * 1.0))) * Float64(y * Float64(y * y))))) - Float64(t - Float64(x * log(y))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (-z * (((0.5 * (y * y)) + y) + ((0.3333333333333333 / (1.0 * (1.0 * 1.0))) * (y * (y * y))))) - (t - (x * log(y)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[((-z) * N[(N[(N[(0.5 * N[(y * y), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + N[(N[(0.3333333333333333 / N[(1.0 * N[(1.0 * 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t - N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024079 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))