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

Percentage Accurate: 85.1% → 99.8%

Time: 13.0s

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: 85.1% 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 82.2%

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

   1. +-commutative 82.2%

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

   2. associate--l+ 82.2%

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

   3. fma-def 82.2%

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

   4. sub-neg 82.2%

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

   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. Final simplification 99.8%

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

Alternative 2: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+185} \lor \neg \left(x \leq -7.8 \cdot 10^{+115}\right) \land \left(x \leq -3600000 \lor \neg \left(x \leq 9.6 \cdot 10^{+58}\right)\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.8e+185)
         (and (not (<= x -7.8e+115))
              (or (<= x -3600000.0) (not (<= x 9.6e+58)))))
   (* x (log y))
   (- (* y (- z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.8e+185) || (!(x <= -7.8e+115) && ((x <= -3600000.0) || !(x <= 9.6e+58)))) {
		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 <= (-3.8d+185)) .or. (.not. (x <= (-7.8d+115))) .and. (x <= (-3600000.0d0)) .or. (.not. (x <= 9.6d+58))) 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 <= -3.8e+185) || (!(x <= -7.8e+115) && ((x <= -3600000.0) || !(x <= 9.6e+58)))) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.8e+185) or (not (x <= -7.8e+115) and ((x <= -3600000.0) or not (x <= 9.6e+58))):
		tmp = x * math.log(y)
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.8e+185) || (!(x <= -7.8e+115) && ((x <= -3600000.0) || !(x <= 9.6e+58))))
		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 <= -3.8e+185) || (~((x <= -7.8e+115)) && ((x <= -3600000.0) || ~((x <= 9.6e+58)))))
		tmp = x * log(y);
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.8e+185], And[N[Not[LessEqual[x, -7.8e+115]], $MachinePrecision], Or[LessEqual[x, -3600000.0], N[Not[LessEqual[x, 9.6e+58]], $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 < -3.7999999999999998e185 or -7.80000000000000012e115 < x < -3.6e6 or 9.5999999999999999e58 < x

   1. Initial program 94.0%

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

      1. +-commutative 94.0%

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

      2. associate--l+ 94.0%

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

      3. fma-def 94.0%

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

      4. sub-neg 94.0%

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

      5. log1p-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in z around 0 93.9%

      \[\leadsto \color{blue}{x \cdot \log y - t} \]
   5. Step-by-step derivation

      1. log-pow 6.6%

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

   6. Simplified 6.6%

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

   7. Taylor expanded in t around 0 6.6%

      \[\leadsto \color{blue}{\log \left({y}^{x}\right)} \]
   8. Step-by-step derivation

      1. log-pow 76.4%

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

      2. *-commutative 76.4%

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

   9. Applied egg-rr 76.4%

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

   if -3.7999999999999998e185 < x < -7.80000000000000012e115 or -3.6e6 < x < 9.5999999999999999e58

   1. Initial program 74.0%

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

   2. Taylor expanded in y around 0 99.3%

      \[\leadsto \left(x \cdot \log y + \color{blue}{-1 \cdot \left(y \cdot z\right)}\right) - t \]
   3. 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%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in x around 0 83.7%

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

      1. mul-1-neg 83.7%

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

      2. distribute-rgt-neg-in 83.7%

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

   7. Simplified 83.7%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+185} \lor \neg \left(x \leq -7.8 \cdot 10^{+115}\right) \land \left(x \leq -3600000 \lor \neg \left(x \leq 9.6 \cdot 10^{+58}\right)\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]

Alternative 3: 90.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-24} \lor \neg \left(x \leq 1.25 \cdot 10^{-75}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.6e-24) (not (<= x 1.25e-75)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.6e-24) || !(x <= 1.25e-75)) {
		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 <= -2.6e-24) || !(x <= 1.25e-75)) {
		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 <= -2.6e-24) or not (x <= 1.25e-75):
		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 <= -2.6e-24) || !(x <= 1.25e-75))
		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, -2.6e-24], N[Not[LessEqual[x, 1.25e-75]], $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 < -2.6e-24 or 1.24999999999999995e-75 < x

   1. Initial program 90.2%

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

   2. Taylor expanded in x around inf 90.1%

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

   if -2.6e-24 < x < 1.24999999999999995e-75

   1. Initial program 70.2%

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

   2. Taylor expanded in x around 0 64.2%

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

      1. sub-neg 64.2%

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

      2. mul-1-neg 64.2%

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

      3. log1p-def 93.3%

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

      4. mul-1-neg 93.3%

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

   4. Simplified 93.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-24} \lor \neg \left(x \leq 1.25 \cdot 10^{-75}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]

Alternative 4: 90.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-23} \lor \neg \left(x \leq 5.5 \cdot 10^{-81}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.05e-23) (not (<= x 5.5e-81)))
   (- (* x (log y)) t)
   (- (* y (- z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.05e-23) || !(x <= 5.5e-81)) {
		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 <= (-1.05d-23)) .or. (.not. (x <= 5.5d-81))) 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 <= -1.05e-23) || !(x <= 5.5e-81)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.05e-23) or not (x <= 5.5e-81):
		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 <= -1.05e-23) || !(x <= 5.5e-81))
		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 <= -1.05e-23) || ~((x <= 5.5e-81)))
		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, -1.05e-23], N[Not[LessEqual[x, 5.5e-81]], $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 < -1.05e-23 or 5.50000000000000026e-81 < x

   1. Initial program 90.2%

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

   2. Taylor expanded in x around inf 90.1%

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

   if -1.05e-23 < x < 5.50000000000000026e-81

   1. Initial program 70.2%

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

   2. Taylor expanded in y around 0 99.0%

      \[\leadsto \left(x \cdot \log y + \color{blue}{-1 \cdot \left(y \cdot z\right)}\right) - t \]
   3. 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%

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

   4. Simplified 99.0%

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

   5. Taylor expanded in x around 0 92.3%

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

      1. mul-1-neg 92.3%

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

      2. distribute-rgt-neg-in 92.3%

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

   7. Simplified 92.3%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-23} \lor \neg \left(x \leq 5.5 \cdot 10^{-81}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]

Alternative 5: 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 82.2%

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

2. Taylor expanded in y around 0 99.4%

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

   1. associate-*r* 99.4%

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

   2. mul-1-neg 99.4%

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

4. Simplified 99.4%

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

5. Taylor expanded in x around 0 99.4%

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

   1. log-pow 45.7%

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

   2. +-commutative 45.7%

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

   3. mul-1-neg 45.7%

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

   4. sub-neg 45.7%

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

   5. log-pow 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

   \[\leadsto \left(x \cdot \log y - z \cdot y\right) - t \]

Alternative 6: 57.4% 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 82.2%

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

2. Taylor expanded in y around 0 99.4%

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

   1. associate-*r* 99.4%

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

   2. mul-1-neg 99.4%

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

4. Simplified 99.4%

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

5. Taylor expanded in x around 0 59.7%

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

   1. mul-1-neg 59.7%

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

   2. distribute-rgt-neg-in 59.7%

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

7. Simplified 59.7%

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

8. Final simplification 59.7%

   \[\leadsto y \cdot \left(-z\right) - t \]

Alternative 7: 42.8% 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 82.2%

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

   1. +-commutative 82.2%

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

   2. associate--l+ 82.2%

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

   3. fma-def 82.2%

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

   4. sub-neg 82.2%

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

   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. Taylor expanded in t around inf 42.2%

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

   1. neg-mul-1 42.2%

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

6. Simplified 42.2%

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

7. Final simplification 42.2%

   \[\leadsto -t \]

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 2023321 
(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))