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

Percentage Accurate: 85.7% → 99.8%

Time: 15.1s

Alternatives: 8

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: 99.8% accurate, 0.7× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. +-commutative 87.7%

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

   2. fma-def 87.7%

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

   3. sub-neg 87.7%

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

   4. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

2. Taylor expanded in y around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. mul-1-neg 99.5%

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

   3. unsub-neg 99.5%

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

   4. associate-*r* 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l* 99.5%

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

4. Applied egg-rr 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

2. Taylor expanded in y around 0 99.5%

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

3. Taylor expanded in z around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. *-commutative 99.5%

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

   3. mul-1-neg 99.5%

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

   4. unsub-neg 99.5%

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

   5. *-commutative 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 86.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+199}:\\ \;\;\;\;t_1 - z \cdot y\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+115} \lor \neg \left(z \leq 4 \cdot 10^{+231}\right):\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= z -5.2e+199)
     (- t_1 (* z y))
     (if (or (<= z -5.8e+115) (not (<= z 4e+231)))
       (- (* z (- y)) t)
       (- t_1 t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (z <= -5.2e+199) {
		tmp = t_1 - (z * y);
	} else if ((z <= -5.8e+115) || !(z <= 4e+231)) {
		tmp = (z * -y) - t;
	} else {
		tmp = t_1 - 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) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (z <= (-5.2d+199)) then
        tmp = t_1 - (z * y)
    else if ((z <= (-5.8d+115)) .or. (.not. (z <= 4d+231))) then
        tmp = (z * -y) - t
    else
        tmp = t_1 - t
    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 (z <= -5.2e+199) {
		tmp = t_1 - (z * y);
	} else if ((z <= -5.8e+115) || !(z <= 4e+231)) {
		tmp = (z * -y) - t;
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if z <= -5.2e+199:
		tmp = t_1 - (z * y)
	elif (z <= -5.8e+115) or not (z <= 4e+231):
		tmp = (z * -y) - t
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (z <= -5.2e+199)
		tmp = Float64(t_1 - Float64(z * y));
	elseif ((z <= -5.8e+115) || !(z <= 4e+231))
		tmp = Float64(Float64(z * Float64(-y)) - t);
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (z <= -5.2e+199)
		tmp = t_1 - (z * y);
	elseif ((z <= -5.8e+115) || ~((z <= 4e+231)))
		tmp = (z * -y) - t;
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.2e+199], N[(t$95$1 - N[(z * y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -5.8e+115], N[Not[LessEqual[z, 4e+231]], $MachinePrecision]], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.2000000000000003e199

   1. Initial program 53.8%

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

   2. Taylor expanded in y around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. mul-1-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 99.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)} - t \]
   6. 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. mul-1-neg 99.9%

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

      3. sub-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in t around 0 94.6%

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

   if -5.2000000000000003e199 < z < -5.80000000000000009e115 or 4.0000000000000002e231 < z

   1. Initial program 53.2%

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

   2. Taylor expanded in x around 0 46.2%

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

   3. Taylor expanded in y around 0 89.6%

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

      1. associate-*r* 89.6%

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

      2. mul-1-neg 89.6%

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

   5. Simplified 89.6%

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

   if -5.80000000000000009e115 < z < 4.0000000000000002e231

   1. Initial program 96.8%

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

   2. Taylor expanded in x around inf 96.5%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+199}:\\ \;\;\;\;x \cdot \log y - z \cdot y\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{+115} \lor \neg \left(z \leq 4 \cdot 10^{+231}\right):\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]

Alternative 5: 89.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-180} \lor \neg \left(x \leq 8 \cdot 10^{-24}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.05e-180) (not (<= x 8e-24)))
   (- (* x (log y)) t)
   (- (* z (- y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.05e-180) || !(x <= 8e-24)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * -y) - 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-180)) .or. (.not. (x <= 8d-24))) then
        tmp = (x * log(y)) - t
    else
        tmp = (z * -y) - 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-180) || !(x <= 8e-24)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * -y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.05e-180) or not (x <= 8e-24):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * -y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.05e-180) || !(x <= 8e-24))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * Float64(-y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.05e-180) || ~((x <= 8e-24)))
		tmp = (x * log(y)) - t;
	else
		tmp = (z * -y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.05e-180], N[Not[LessEqual[x, 8e-24]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * (-y)), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0499999999999999e-180 or 7.99999999999999939e-24 < x

   1. Initial program 93.2%

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

   2. Taylor expanded in x around inf 93.2%

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

   if -1.0499999999999999e-180 < x < 7.99999999999999939e-24

   1. Initial program 75.9%

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

   2. Taylor expanded in x around 0 68.4%

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

   3. Taylor expanded in y around 0 91.3%

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

      1. associate-*r* 91.3%

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

      2. mul-1-neg 91.3%

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

   5. Simplified 91.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-180} \lor \neg \left(x \leq 8 \cdot 10^{-24}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right) - t\\ \end{array} \]

Alternative 6: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

2. Taylor expanded in y around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. mul-1-neg 99.5%

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

   3. unsub-neg 99.5%

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

   4. associate-*r* 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l* 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in y 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. +-commutative 99.4%

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

   2. mul-1-neg 99.4%

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

   3. sub-neg 99.4%

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

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 7: 58.4% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

2. Taylor expanded in x around 0 47.7%

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

3. Taylor expanded in y around 0 59.2%

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

   1. associate-*r* 59.2%

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

   2. mul-1-neg 59.2%

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

5. Simplified 59.2%

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

6. Final simplification 59.2%

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

Alternative 8: 44.2% 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 87.7%

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

   1. +-commutative 87.7%

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

   2. fma-def 87.7%

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

   3. sub-neg 87.7%

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

   4. log1p-def 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in t around inf 47.2%

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

   1. mul-1-neg 47.2%

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

6. Simplified 47.2%

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

7. Final simplification 47.2%

   \[\leadsto -t \]

Developer target: 99.5% 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 2023301 
(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))