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

Percentage Accurate: 85.3% → 99.8%

Time: 14.2s

Alternatives: 7

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: 85.3% 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 85.3%

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

   1. +-commutative 85.3%

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

   2. associate--l+ 85.3%

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

   3. fma-def 85.3%

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

   4. sub-neg 85.3%

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

   5. log1p-def 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: 76.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+166} \lor \neg \left(x \leq -9 \cdot 10^{+122} \lor \neg \left(x \leq -3.1 \cdot 10^{+19}\right) \land x \leq 5.2 \cdot 10^{+29}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.65e+166)
         (not (or (<= x -9e+122) (and (not (<= x -3.1e+19)) (<= x 5.2e+29)))))
   (* x (log y))
   (- (* y (- z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.65e+166) || !((x <= -9e+122) || (!(x <= -3.1e+19) && (x <= 5.2e+29)))) {
		tmp = x * log(y);
	} else {
		tmp = (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 <= (-1.65d+166)) .or. (.not. (x <= (-9d+122)) .or. (.not. (x <= (-3.1d+19))) .and. (x <= 5.2d+29))) then
        tmp = x * log(y)
    else
        tmp = (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 <= -1.65e+166) || !((x <= -9e+122) || (!(x <= -3.1e+19) && (x <= 5.2e+29)))) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.65e+166) or not ((x <= -9e+122) or (not (x <= -3.1e+19) and (x <= 5.2e+29))):
		tmp = x * math.log(y)
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.65e+166) || !((x <= -9e+122) || (!(x <= -3.1e+19) && (x <= 5.2e+29))))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.65e+166) || ~(((x <= -9e+122) || (~((x <= -3.1e+19)) && (x <= 5.2e+29)))))
		tmp = x * log(y);
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.65e+166], N[Not[Or[LessEqual[x, -9e+122], And[N[Not[LessEqual[x, -3.1e+19]], $MachinePrecision], LessEqual[x, 5.2e+29]]]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6500000000000001e166 or -8.99999999999999995e122 < x < -3.1e19 or 5.2e29 < x

   1. Initial program 97.0%

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

   3. Taylor expanded in y around 0 99.0%

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

      1. associate-*r* 99.0%

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

      2. mul-1-neg 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0 99.0%

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

      1. +-commutative 99.0%

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

      2. mul-1-neg 99.0%

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

      3. sub-neg 99.0%

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

   8. Simplified 99.0%

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

      1. prod-diff 99.0%

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

      2. *-commutative 99.0%

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

      3. fma-neg 99.0%

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

      4. prod-diff 99.0%

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

      5. *-commutative 99.0%

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

      6. fma-neg 99.0%

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

      7. associate-+l+ 99.0%

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

      8. fma-neg 99.0%

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

      9. distribute-rgt-neg-in 99.0%

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

      10. *-commutative 99.0%

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

      11. *-commutative 99.0%

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

   10. Applied egg-rr 99.0%

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

       1. distribute-rgt-neg-out 99.0%

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

       2. fma-neg 99.0%

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

       3. count-2 99.0%

          \[\leadsto \left(\left(x \cdot \log y - y \cdot z\right) + \color{blue}{2 \cdot \mathsf{fma}\left(-z, y, y \cdot z\right)}\right) - t \]

   12. Simplified 99.0%

       \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y \cdot z\right) + 2 \cdot \mathsf{fma}\left(-z, y, y \cdot z\right)\right)} - t \]

   13. Taylor expanded in x around inf 76.0%

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

   if -1.6500000000000001e166 < x < -8.99999999999999995e122 or -3.1e19 < x < 5.2e29

   1. Initial program 76.2%

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

   3. Taylor expanded in y around 0 99.5%

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

      1. associate-*r* 99.5%

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

      2. mul-1-neg 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 89.3%

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

      1. mul-1-neg 89.3%

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

      2. distribute-rgt-neg-in 89.3%

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

   8. Simplified 89.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+166} \lor \neg \left(x \leq -9 \cdot 10^{+122} \lor \neg \left(x \leq -3.1 \cdot 10^{+19}\right) \land x \leq 5.2 \cdot 10^{+29}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -29000000000 \lor \neg \left(x \leq 6.6 \cdot 10^{-156}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -29000000000.0) (not (<= x 6.6e-156)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -29000000000.0) || !(x <= 6.6e-156)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -29000000000.0) || !(x <= 6.6e-156)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -29000000000.0) or not (x <= 6.6e-156):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -29000000000.0) || !(x <= 6.6e-156))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -29000000000.0], N[Not[LessEqual[x, 6.6e-156]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9e10 or 6.5999999999999997e-156 < x

   1. Initial program 94.5%

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

   3. Taylor expanded in x around inf 93.9%

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

   if -2.9e10 < x < 6.5999999999999997e-156

   1. Initial program 71.0%

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

   3. Taylor expanded in x around 0 69.0%

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

      1. sub-neg 69.0%

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

      2. mul-1-neg 69.0%

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

      3. log1p-def 98.1%

         \[\leadsto z \cdot \color{blue}{\mathsf{log1p}\left(-1 \cdot y\right)} - t \]

      4. mul-1-neg 98.1%

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

   5. Simplified 98.1%

      \[\leadsto \color{blue}{z \cdot \mathsf{log1p}\left(-y\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -29000000000 \lor \neg \left(x \leq 6.6 \cdot 10^{-156}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -31000000000 \lor \neg \left(x \leq 7.5 \cdot 10^{-156}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -31000000000.0) (not (<= x 7.5e-156)))
   (- (* x (log y)) t)
   (- (* y (- z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -31000000000.0) || !(x <= 7.5e-156)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (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 <= (-31000000000.0d0)) .or. (.not. (x <= 7.5d-156))) then
        tmp = (x * log(y)) - t
    else
        tmp = (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 <= -31000000000.0) || !(x <= 7.5e-156)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -31000000000.0) or not (x <= 7.5e-156):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -31000000000.0) || !(x <= 7.5e-156))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(-z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -31000000000.0) || ~((x <= 7.5e-156)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -31000000000.0], N[Not[LessEqual[x, 7.5e-156]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * (-z)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e10 or 7.49999999999999959e-156 < x

   1. Initial program 94.5%

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

   3. Taylor expanded in x around inf 93.9%

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

   if -3.1e10 < x < 7.49999999999999959e-156

   1. Initial program 71.0%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 97.4%

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

      1. mul-1-neg 97.4%

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

      2. distribute-rgt-neg-in 97.4%

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

   8. Simplified 97.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -31000000000 \lor \neg \left(x \leq 7.5 \cdot 10^{-156}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% 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 85.3%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 6: 57.8% 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 85.3%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 60.4%

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

   1. mul-1-neg 60.4%

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

   2. distribute-rgt-neg-in 60.4%

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

8. Simplified 60.4%

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

9. Final simplification 60.4%

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

Alternative 7: 43.3% 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 85.3%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. Taylor expanded in t around inf 45.9%

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

    1. neg-mul-1 45.9%

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

11. Simplified 45.9%

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

12. Final simplification 45.9%

    \[\leadsto -t \]
13. 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 2024031 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- 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))