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

Percentage Accurate: 85.7% → 99.0%

Time: 14.0s

Alternatives: 9

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: 85.7% 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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in y around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. fma-define 99.8%

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

   3. mul-1-neg 99.8%

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

   4. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 88.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;t \leq -6.9 \cdot 10^{-240} \lor \neg \left(t \leq 1.35 \cdot 10^{-75}\right):\\ \;\;\;\;t\_1 - t\\ \mathbf{else}:\\ \;\;\;\;t\_1 - y \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= t -6.9e-240) (not (<= t 1.35e-75))) (- t_1 t) (- t_1 (* y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((t <= -6.9e-240) || !(t <= 1.35e-75)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (y * z);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if ((t <= (-6.9d-240)) .or. (.not. (t <= 1.35d-75))) then
        tmp = t_1 - t
    else
        tmp = t_1 - (y * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if ((t <= -6.9e-240) || !(t <= 1.35e-75)) {
		tmp = t_1 - t;
	} else {
		tmp = t_1 - (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (t <= -6.9e-240) or not (t <= 1.35e-75):
		tmp = t_1 - t
	else:
		tmp = t_1 - (y * z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if ((t <= -6.9e-240) || !(t <= 1.35e-75))
		tmp = Float64(t_1 - t);
	else
		tmp = Float64(t_1 - Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((t <= -6.9e-240) || ~((t <= 1.35e-75)))
		tmp = t_1 - t;
	else
		tmp = t_1 - (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t, -6.9e-240], N[Not[LessEqual[t, 1.35e-75]], $MachinePrecision]], N[(t$95$1 - t), $MachinePrecision], N[(t$95$1 - N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.9000000000000001e-240 or 1.3499999999999999e-75 < t

   1. Initial program 91.1%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.8%

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

      3. mul-1-neg 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in y around 0 91.1%

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

   if -6.9000000000000001e-240 < t < 1.3499999999999999e-75

   1. Initial program 66.0%

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

   3. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-define 99.7%

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

      3. mul-1-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in t around 0 94.0%

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

      1. +-commutative 94.0%

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

      2. neg-mul-1 94.0%

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

      3. unsub-neg 94.0%

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

   8. Simplified 94.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.9 \cdot 10^{-240} \lor \neg \left(t \leq 1.35 \cdot 10^{-75}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-82} \lor \neg \left(x \leq 0.0062\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.5e-82) (not (<= x 0.0062)))
   (- (* x (log y)) t)
   (- (* y (- z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.5e-82) || !(x <= 0.0062)) {
		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 <= (-2.5d-82)) .or. (.not. (x <= 0.0062d0))) 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 <= -2.5e-82) || !(x <= 0.0062)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * -z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.5e-82) or not (x <= 0.0062):
		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 <= -2.5e-82) || !(x <= 0.0062))
		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 <= -2.5e-82) || ~((x <= 0.0062)))
		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, -2.5e-82], N[Not[LessEqual[x, 0.0062]], $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 < -2.4999999999999999e-82 or 0.00619999999999999978 < 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 y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-define 99.7%

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

      3. mul-1-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around 0 94.5%

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

   if -2.4999999999999999e-82 < x < 0.00619999999999999978

   1. Initial program 71.4%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. mul-1-neg 99.9%

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

      4. *-commutative 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in x around 0 88.2%

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

      1. neg-mul-1 88.2%

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

      2. distribute-rgt-neg-in 88.2%

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

   8. Simplified 88.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-82} \lor \neg \left(x \leq 0.0062\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+70} \lor \neg \left(x \leq 5.5 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot z\right) \cdot -0.5 - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.3e+70) (not (<= x 5.5e+27)))
   (* x (log y))
   (- (* y (- (* (* y z) -0.5) z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.3e+70) || !(x <= 5.5e+27)) {
		tmp = x * log(y);
	} else {
		tmp = (y * (((y * z) * -0.5) - 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 <= (-2.3d+70)) .or. (.not. (x <= 5.5d+27))) then
        tmp = x * log(y)
    else
        tmp = (y * (((y * z) * (-0.5d0)) - 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 <= -2.3e+70) || !(x <= 5.5e+27)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * (((y * z) * -0.5) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.3e+70) or not (x <= 5.5e+27):
		tmp = x * math.log(y)
	else:
		tmp = (y * (((y * z) * -0.5) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.3e+70) || !(x <= 5.5e+27))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(Float64(Float64(y * z) * -0.5) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.3e+70) || ~((x <= 5.5e+27)))
		tmp = x * log(y);
	else
		tmp = (y * (((y * z) * -0.5) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.3e+70], N[Not[LessEqual[x, 5.5e+27]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(N[(y * z), $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.29999999999999994e70 or 5.49999999999999966e27 < x

   1. Initial program 96.3%

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

   3. Taylor expanded in y around 0 99.6%

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

      1. +-commutative 99.6%

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

      2. fma-define 99.6%

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

      3. mul-1-neg 99.6%

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

      4. *-commutative 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around inf 75.8%

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

   if -2.29999999999999994e70 < x < 5.49999999999999966e27

   1. Initial program 76.6%

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

   3. Taylor expanded in x around 0 60.2%

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

      1. sub-neg 60.2%

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

      2. log1p-define 82.7%

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

   5. Simplified 82.7%

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

   6. Taylor expanded in y around 0 82.7%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.3 \cdot 10^{+70} \lor \neg \left(x \leq 5.5 \cdot 10^{+27}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot z\right) \cdot -0.5 - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) - (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 = ((x * log(y)) - (y * z)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in y around 0 99.8%

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

   1. associate-*r* 99.8%

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

   2. mul-1-neg 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 6: 48.2% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.9 \cdot 10^{-240} \lor \neg \left(t \leq 1.1 \cdot 10^{-75}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -6.9e-240) (not (<= t 1.1e-75))) (- t) (* y (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -6.9e-240) || !(t <= 1.1e-75)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	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 ((t <= (-6.9d-240)) .or. (.not. (t <= 1.1d-75))) then
        tmp = -t
    else
        tmp = y * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -6.9e-240) || !(t <= 1.1e-75)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -6.9e-240) or not (t <= 1.1e-75):
		tmp = -t
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -6.9e-240) || !(t <= 1.1e-75))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -6.9e-240) || ~((t <= 1.1e-75)))
		tmp = -t;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -6.9e-240], N[Not[LessEqual[t, 1.1e-75]], $MachinePrecision]], (-t), N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.9000000000000001e-240 or 1.10000000000000003e-75 < t

   1. Initial program 91.1%

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

   3. Taylor expanded in y around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. fma-define 99.8%

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

      3. mul-1-neg 99.8%

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

      4. *-commutative 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in t around inf 56.8%

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

      1. neg-mul-1 56.8%

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

   8. Simplified 56.8%

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

   if -6.9000000000000001e-240 < t < 1.10000000000000003e-75

   1. Initial program 66.0%

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

   3. Taylor expanded in y around 0 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-define 99.7%

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

      3. mul-1-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in y around inf 35.1%

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

      1. neg-mul-1 35.1%

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

      2. distribute-rgt-neg-in 35.1%

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

   8. Simplified 35.1%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.9 \cdot 10^{-240} \lor \neg \left(t \leq 1.1 \cdot 10^{-75}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.9% 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 84.8%

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

3. Taylor expanded in y around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. fma-define 99.8%

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

   3. mul-1-neg 99.8%

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

   4. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0 58.6%

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

   1. neg-mul-1 58.6%

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

   2. distribute-rgt-neg-in 58.6%

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

8. Simplified 58.6%

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

Alternative 8: 43.9% 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 84.8%

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

3. Taylor expanded in y around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. fma-define 99.8%

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

   3. mul-1-neg 99.8%

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

   4. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in t around inf 44.3%

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

   1. neg-mul-1 44.3%

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

8. Simplified 44.3%

   \[\leadsto \color{blue}{-t} \]
9. Add Preprocessing

Alternative 9: 2.2% accurate, 211.0× 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 t
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in y around 0 99.8%

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

   1. +-commutative 99.8%

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

   2. fma-define 99.8%

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

   3. mul-1-neg 99.8%

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

   4. *-commutative 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in t around inf 44.3%

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

   1. neg-mul-1 44.3%

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

8. Simplified 44.3%

   \[\leadsto \color{blue}{-t} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 21.7%

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

   2. sqrt-unprod 10.1%

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

   3. sqr-neg 10.1%

      \[\leadsto \sqrt{\color{blue}{t \cdot t}} \]

   4. sqrt-unprod 1.2%

      \[\leadsto \color{blue}{\sqrt{t} \cdot \sqrt{t}} \]

   5. add-sqr-sqrt 2.4%

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

   6. *-un-lft-identity 2.4%

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

10. Applied egg-rr 2.4%

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

    1. *-lft-identity 2.4%

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

12. Simplified 2.4%

    \[\leadsto \color{blue}{t} \]
13. Add Preprocessing

Developer Target 1: 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 2024141 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

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