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

Percentage Accurate: 85.7% → 99.8%

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: 99.8% 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 86.8%

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

   1. +-commutative 86.8%

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

   2. associate--l+ 86.8%

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

   3. fma-define 86.8%

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

   4. sub-neg 86.8%

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

   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. Add Preprocessing

Alternative 2: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-18} \lor \neg \left(x \leq 1.95 \cdot 10^{-137}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right) + -1\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.55e-18) (not (<= x 1.95e-137)))
   (- (* x (log y)) t)
   (-
    (* z (* y (+ (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)) -1.0)))
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.55e-18) || !(x <= 1.95e-137)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - 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 <= (-1.55d-18)) .or. (.not. (x <= 1.95d-137))) then
        tmp = (x * log(y)) - t
    else
        tmp = (z * (y * ((y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0)) + (-1.0d0)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.55e-18) || !(x <= 1.95e-137)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.55e-18) or not (x <= 1.95e-137):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.55e-18) || !(x <= 1.95e-137))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.55e-18) || ~((x <= 1.95e-137)))
		tmp = (x * log(y)) - t;
	else
		tmp = (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.55e-18], N[Not[LessEqual[x, 1.95e-137]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[(y * N[(N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000003e-18 or 1.95e-137 < x

   1. Initial program 95.2%

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

      1. +-commutative 95.2%

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

      2. associate--l+ 95.2%

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

      3. fma-define 95.2%

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

      4. sub-neg 95.2%

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

      5. log1p-define 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 99.6%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(-\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right), x \cdot \log y - t\right) \]

      2. distribute-rgt-neg-in 99.6%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(-\sqrt[3]{y}\right)}\right), x \cdot \log y - t\right) \]

      3. pow2 99.6%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \left(-\sqrt[3]{y}\right)\right), x \cdot \log y - t\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(-\sqrt[3]{y}\right)}\right), x \cdot \log y - t\right) \]

   7. Taylor expanded in z around 0 95.2%

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

   if -1.55000000000000003e-18 < x < 1.95e-137

   1. Initial program 73.0%

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

   3. Taylor expanded in x around 0 67.0%

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

   4. Taylor expanded in y around 0 92.5%

      \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-18} \lor \neg \left(x \leq 1.95 \cdot 10^{-137}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right) + -1\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+65} \lor \neg \left(x \leq 6.6 \cdot 10^{-18}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right) + -1\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.55e+65) (not (<= x 6.6e-18)))
   (* x (log y))
   (-
    (* z (* y (+ (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)) -1.0)))
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.55e+65) || !(x <= 6.6e-18)) {
		tmp = x * log(y);
	} else {
		tmp = (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - 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 <= (-1.55d+65)) .or. (.not. (x <= 6.6d-18))) then
        tmp = x * log(y)
    else
        tmp = (z * (y * ((y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0)) + (-1.0d0)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.55e+65) || !(x <= 6.6e-18)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.55e+65) or not (x <= 6.6e-18):
		tmp = x * math.log(y)
	else:
		tmp = (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.55e+65) || !(x <= 6.6e-18))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(z * Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.55e+65) || ~((x <= 6.6e-18)))
		tmp = x * log(y);
	else
		tmp = (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.55e+65], N[Not[LessEqual[x, 6.6e-18]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(z * N[(y * N[(N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.54999999999999995e65 or 6.6000000000000003e-18 < x

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. associate--l+ 99.7%

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

      3. fma-define 99.7%

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

      4. sub-neg 99.7%

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

      5. log1p-define 99.7%

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

   3. Simplified 99.7%

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

      1. add-cube-cbrt 99.7%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(-\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right), x \cdot \log y - t\right) \]

      2. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(-\sqrt[3]{y}\right)}\right), x \cdot \log y - t\right) \]

      3. pow2 99.7%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \left(-\sqrt[3]{y}\right)\right), x \cdot \log y - t\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(-\sqrt[3]{y}\right)}\right), x \cdot \log y - t\right) \]

   7. Taylor expanded in x around inf 76.3%

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

   if -1.54999999999999995e65 < x < 6.6000000000000003e-18

   1. Initial program 76.5%

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

   3. Taylor expanded in x around 0 61.8%

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

   4. Taylor expanded in y around 0 84.1%

      \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+65} \lor \neg \left(x \leq 6.6 \cdot 10^{-18}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right) + -1\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in y around 0 99.5%

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

   1. associate--l+ 99.5%

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

   2. +-commutative 99.5%

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

   3. mul-1-neg 99.5%

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

   4. unsub-neg 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 57.5% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right) + -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (* z (* y (+ (* y (- (* y (- (* y -0.25) 0.3333333333333333)) 0.5)) -1.0)))
  t))

C

double code(double x, double y, double z, double t) {
	return (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - 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 * ((y * ((y * ((y * (-0.25d0)) - 0.3333333333333333d0)) - 0.5d0)) + (-1.0d0)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t;
}

Python

def code(x, y, z, t):
	return (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(z * Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (z * (y * ((y * ((y * ((y * -0.25) - 0.3333333333333333)) - 0.5)) + -1.0))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z * N[(y * N[(N[(y * N[(N[(y * N[(N[(y * -0.25), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in x around 0 44.8%

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

4. Taylor expanded in y around 0 57.4%

   \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(-0.25 \cdot y - 0.3333333333333333\right) - 0.5\right) - 1\right)\right)} - t \]

5. Final simplification 57.4%

   \[\leadsto z \cdot \left(y \cdot \left(y \cdot \left(y \cdot \left(y \cdot -0.25 - 0.3333333333333333\right) - 0.5\right) + -1\right)\right) - t \]
6. Add Preprocessing

Alternative 6: 57.4% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ z \cdot \left(y \cdot \left(y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right) + -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* z (* y (+ (* y (- (* y -0.3333333333333333) 0.5)) -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return (z * (y * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0))) - 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 * ((y * ((y * (-0.3333333333333333d0)) - 0.5d0)) + (-1.0d0)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (z * (y * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0))) - t;
}

Python

def code(x, y, z, t):
	return (z * (y * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(z * Float64(y * Float64(Float64(y * Float64(Float64(y * -0.3333333333333333) - 0.5)) + -1.0))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (z * (y * ((y * ((y * -0.3333333333333333) - 0.5)) + -1.0))) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(z * N[(y * N[(N[(y * N[(N[(y * -0.3333333333333333), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.8%

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

3. Taylor expanded in x around 0 44.8%

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

4. Taylor expanded in y around 0 57.4%

   \[\leadsto z \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(-0.3333333333333333 \cdot y - 0.5\right) - 1\right)\right)} - t \]

5. Final simplification 57.4%

   \[\leadsto z \cdot \left(y \cdot \left(y \cdot \left(y \cdot -0.3333333333333333 - 0.5\right) + -1\right)\right) - t \]
6. Add Preprocessing

Alternative 7: 47.7% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-140} \lor \neg \left(t \leq 1.06 \cdot 10^{-193}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.7e-140) (not (<= t 1.06e-193))) (- t) (* z (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.7e-140) || !(t <= 1.06e-193)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	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 <= (-3.7d-140)) .or. (.not. (t <= 1.06d-193))) then
        tmp = -t
    else
        tmp = z * -y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.7e-140) || !(t <= 1.06e-193)) {
		tmp = -t;
	} else {
		tmp = z * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.7e-140) or not (t <= 1.06e-193):
		tmp = -t
	else:
		tmp = z * -y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.7e-140) || !(t <= 1.06e-193))
		tmp = Float64(-t);
	else
		tmp = Float64(z * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -3.7e-140) || ~((t <= 1.06e-193)))
		tmp = -t;
	else
		tmp = z * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.7e-140], N[Not[LessEqual[t, 1.06e-193]], $MachinePrecision]], (-t), N[(z * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -3.69999999999999977e-140 or 1.06e-193 < t

   1. Initial program 90.2%

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

      1. +-commutative 90.2%

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

      2. associate--l+ 90.2%

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

      3. fma-define 90.2%

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

      4. sub-neg 90.2%

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

      5. log1p-define 99.9%

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

   3. Simplified 99.9%

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

      1. add-cube-cbrt 99.7%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(-\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right), x \cdot \log y - t\right) \]

      2. distribute-rgt-neg-in 99.7%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(-\sqrt[3]{y}\right)}\right), x \cdot \log y - t\right) \]

      3. pow2 99.7%

         \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \left(-\sqrt[3]{y}\right)\right), x \cdot \log y - t\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(-\sqrt[3]{y}\right)}\right), x \cdot \log y - t\right) \]

   7. Taylor expanded in t around inf 55.2%

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

      1. neg-mul-1 55.2%

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

   9. Simplified 55.2%

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

   if -3.69999999999999977e-140 < t < 1.06e-193

   1. Initial program 73.7%

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

   3. Taylor expanded in z around inf 4.8%

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

      1. sub-neg 4.8%

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

      2. log1p-define 29.2%

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

   5. Simplified 29.2%

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

   6. Taylor expanded in y around 0 27.6%

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

      1. associate-*r* 27.6%

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

      2. neg-mul-1 27.6%

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

   8. Simplified 27.6%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{-140} \lor \neg \left(t \leq 1.06 \cdot 10^{-193}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% 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 86.8%

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

3. Taylor expanded in x around 0 44.8%

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

4. Taylor expanded in y around 0 57.3%

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

   1. associate-*r* 57.3%

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

   2. neg-mul-1 57.3%

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

6. Simplified 57.3%

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

7. Final simplification 57.3%

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

Alternative 9: 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 86.8%

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

   1. +-commutative 86.8%

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

   2. associate--l+ 86.8%

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

   3. fma-define 86.8%

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

   4. sub-neg 86.8%

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

   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. Step-by-step derivation

   1. add-cube-cbrt 99.6%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(-\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right), x \cdot \log y - t\right) \]

   2. distribute-rgt-neg-in 99.6%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(-\sqrt[3]{y}\right)}\right), x \cdot \log y - t\right) \]

   3. pow2 99.6%

      \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2}} \cdot \left(-\sqrt[3]{y}\right)\right), x \cdot \log y - t\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \mathsf{fma}\left(z, \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{2} \cdot \left(-\sqrt[3]{y}\right)}\right), x \cdot \log y - t\right) \]

7. Taylor expanded in t around inf 44.4%

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

   1. neg-mul-1 44.4%

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

9. Simplified 44.4%

   \[\leadsto \color{blue}{-t} \]
10. 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 2024137 
(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))