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

Percentage Accurate: 85.0% → 99.8%

Time: 16.7s

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: 85.0% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.3%

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

   1. +-commutative 89.3%

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

   2. associate--l+ 89.3%

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

   3. fma-define 89.3%

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

   4. sub-neg 89.3%

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

   5. log1p-define 99.8%

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

   6. fma-neg 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 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 89.3%

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

   1. +-commutative 89.3%

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

   2. associate--l+ 89.3%

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

   3. fma-define 89.3%

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

   4. sub-neg 89.3%

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

   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 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \log y + z \cdot \left(-0.5 \cdot {y}^{2} - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (- (* -0.5 (pow y 2.0)) y))) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * ((-0.5 * pow(y, 2.0)) - 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 * (((-0.5d0) * (y ** 2.0d0)) - y))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + (z * ((-0.5 * Math.pow(y, 2.0)) - y))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * ((-0.5 * math.pow(y, 2.0)) - y))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(z * Float64(Float64(-0.5 * (y ^ 2.0)) - y))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((x * log(y)) + (z * ((-0.5 * (y ^ 2.0)) - y))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[(N[(-0.5 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 89.3%

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

3. Taylor expanded in y around 0 99.4%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(-1 \cdot y + -0.5 \cdot {y}^{2}\right)}\right) - t \]

4. Final simplification 99.4%

   \[\leadsto \left(x \cdot \log y + z \cdot \left(-0.5 \cdot {y}^{2} - y\right)\right) - t \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-109} \lor \neg \left(x \leq 1.05 \cdot 10^{-120}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.5e-109) (not (<= x 1.05e-120)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.5e-109) || !(x <= 1.05e-120)) {
		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 <= -3.5e-109) || !(x <= 1.05e-120)) {
		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 <= -3.5e-109) or not (x <= 1.05e-120):
		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 <= -3.5e-109) || !(x <= 1.05e-120))
		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, -3.5e-109], N[Not[LessEqual[x, 1.05e-120]], $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 < -3.5e-109 or 1.05e-120 < x

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0 99.4%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(-1 \cdot y + -0.5 \cdot {y}^{2}\right)}\right) - t \]

   4. Taylor expanded in y around 0 92.7%

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

   if -3.5e-109 < x < 1.05e-120

   1. Initial program 78.2%

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

   3. Taylor expanded in x around 0 73.3%

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

      1. sub-neg 73.3%

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

      2. mul-1-neg 73.3%

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

      3. log1p-define 95.1%

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

      4. mul-1-neg 95.1%

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

   5. Simplified 95.1%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-109} \lor \neg \left(x \leq 1.05 \cdot 10^{-120}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-108} \lor \neg \left(x \leq 7 \cdot 10^{-119}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -7e-108) (not (<= x 7e-119)))
   (- (* x (log y)) t)
   (- (* y (- z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7e-108) || !(x <= 7e-119)) {
		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 <= (-7d-108)) .or. (.not. (x <= 7d-119))) 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 <= -7e-108) || !(x <= 7e-119)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -7e-108) or not (x <= 7e-119):
		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 <= -7e-108) || !(x <= 7e-119))
		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 <= -7e-108) || ~((x <= 7e-119)))
		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, -7e-108], N[Not[LessEqual[x, 7e-119]], $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 < -6.9999999999999997e-108 or 7e-119 < x

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0 99.4%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(-1 \cdot y + -0.5 \cdot {y}^{2}\right)}\right) - t \]

   4. Taylor expanded in y around 0 92.7%

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

   if -6.9999999999999997e-108 < x < 7e-119

   1. Initial program 78.2%

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

   3. Taylor expanded in x around 0 73.3%

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

      1. sub-neg 73.3%

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

      2. mul-1-neg 73.3%

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

      3. log1p-define 95.1%

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

      4. mul-1-neg 95.1%

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

   5. Simplified 95.1%

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

   6. Taylor expanded in y around 0 94.3%

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

      1. neg-mul-1 94.3%

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

      2. +-commutative 94.3%

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

      3. unsub-neg 94.3%

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

      4. mul-1-neg 94.3%

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

      5. *-commutative 94.3%

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

      6. distribute-rgt-neg-in 94.3%

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

   8. Simplified 94.3%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-108} \lor \neg \left(x \leq 7 \cdot 10^{-119}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+57} \lor \neg \left(x \leq 1.2 \cdot 10^{+135}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5e+57) (not (<= x 1.2e+135))) (* x (log y)) (- (* y (- z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5e+57) || !(x <= 1.2e+135)) {
		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 <= (-5d+57)) .or. (.not. (x <= 1.2d+135))) 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 <= -5e+57) || !(x <= 1.2e+135)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5e+57) or not (x <= 1.2e+135):
		tmp = x * math.log(y)
	else:
		tmp = (y * -z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5e+57) || !(x <= 1.2e+135))
		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 <= -5e+57) || ~((x <= 1.2e+135)))
		tmp = x * log(y);
	else
		tmp = (y * -z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5e+57], N[Not[LessEqual[x, 1.2e+135]], $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 < -4.99999999999999972e57 or 1.19999999999999999e135 < x

   1. Initial program 99.6%

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

   3. Taylor expanded in y around 0 99.6%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(-1 \cdot y + -0.5 \cdot {y}^{2}\right)}\right) - t \]

   4. Taylor expanded in x around inf 80.4%

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

   if -4.99999999999999972e57 < x < 1.19999999999999999e135

   1. Initial program 83.6%

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

   3. Taylor expanded in x around 0 66.6%

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

      1. sub-neg 66.6%

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

      2. mul-1-neg 66.6%

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

      3. log1p-define 82.9%

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

      4. mul-1-neg 82.9%

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

   5. Simplified 82.9%

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

   6. Taylor expanded in y around 0 81.6%

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

      1. neg-mul-1 81.6%

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

      2. +-commutative 81.6%

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

      3. unsub-neg 81.6%

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

      4. mul-1-neg 81.6%

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

      5. *-commutative 81.6%

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

      6. distribute-rgt-neg-in 81.6%

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

   8. Simplified 81.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+57} \lor \neg \left(x \leq 1.2 \cdot 10^{+135}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 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 89.3%

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

3. Taylor expanded in y around 0 99.0%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)} - t \]
4. 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. unsub-neg 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 8: 44.2% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+231}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z -5.2e+231) (* y (- z)) (- t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.2e+231) {
		tmp = y * -z;
	} else {
		tmp = -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 (z <= (-5.2d+231)) then
        tmp = y * -z
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.2e+231) {
		tmp = y * -z;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -5.2e+231:
		tmp = y * -z
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.2e+231)
		tmp = Float64(y * Float64(-z));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -5.2e+231)
		tmp = y * -z;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.2e+231], N[(y * (-z)), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if z < -5.1999999999999997e231

   1. Initial program 40.7%

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

   3. Taylor expanded in x around 0 23.8%

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

      1. sub-neg 23.8%

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

      2. mul-1-neg 23.8%

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

      3. log1p-define 83.5%

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

      4. mul-1-neg 83.5%

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

   5. Simplified 83.5%

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

   6. Taylor expanded in y around 0 83.5%

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

      1. neg-mul-1 83.5%

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

      2. +-commutative 83.5%

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

      3. unsub-neg 83.5%

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

      4. mul-1-neg 83.5%

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

      5. *-commutative 83.5%

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

      6. distribute-rgt-neg-in 83.5%

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

   8. Simplified 83.5%

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

   9. Taylor expanded in z around inf 62.6%

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

       1. mul-1-neg 62.6%

          \[\leadsto \color{blue}{-y \cdot z} \]

       2. distribute-rgt-neg-in 62.6%

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

   11. Simplified 62.6%

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

   if -5.1999999999999997e231 < z

   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 99.3%

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(-1 \cdot y + -0.5 \cdot {y}^{2}\right)}\right) - t \]

   4. Taylor expanded in t around inf 50.0%

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

      1. neg-mul-1 50.0%

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

   6. Simplified 50.0%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+231}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.5% 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 89.3%

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

3. Taylor expanded in x around 0 49.4%

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

   1. sub-neg 49.4%

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

   2. mul-1-neg 49.4%

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

   3. log1p-define 59.8%

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

   4. mul-1-neg 59.8%

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

5. Simplified 59.8%

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

6. Taylor expanded in y around 0 59.0%

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

   1. neg-mul-1 59.0%

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

   2. +-commutative 59.0%

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

   3. unsub-neg 59.0%

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

   4. mul-1-neg 59.0%

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

   5. *-commutative 59.0%

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

   6. distribute-rgt-neg-in 59.0%

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

8. Simplified 59.0%

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

9. Final simplification 59.0%

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

Alternative 10: 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 89.3%

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

3. Taylor expanded in y around 0 99.4%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(-1 \cdot y + -0.5 \cdot {y}^{2}\right)}\right) - t \]

4. Taylor expanded in t around inf 48.1%

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

   1. neg-mul-1 48.1%

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

6. Simplified 48.1%

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

7. Final simplification 48.1%

   \[\leadsto -t \]
8. 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 2024053 
(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))