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

Percentage Accurate: 85.1% → 99.8%

Time: 15.7s

Alternatives: 10

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * log((1.0 - y)))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (z * log((1.0d0 - y)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + (z * Math.log((1.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * math.log((1.0 - y)))) - t

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * log((1.0 - y)))) - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (z * log((1.0d0 - y)))) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + (z * Math.log((1.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * math.log((1.0 - y)))) - t

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. +-commutative 85.2%

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

   2. associate--l+ 85.2%

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

   3. fma-define 85.2%

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

   4. sub-neg 85.2%

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

   5. log1p-define 99.8%

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

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: 89.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-186} \lor \neg \left(x \leq 2.9 \cdot 10^{-78}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot -0.5 + -1\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.8e-186) (not (<= x 2.9e-78)))
   (- (* x (log y)) t)
   (- (* y (* z (+ (* y -0.5) -1.0))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.8e-186) || !(x <= 2.9e-78)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y * (z * ((y * -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 <= (-5.8d-186)) .or. (.not. (x <= 2.9d-78))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y * (z * ((y * (-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 <= -5.8e-186) || !(x <= 2.9e-78)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * (z * ((y * -0.5) + -1.0))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5.8e-186) or not (x <= 2.9e-78):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y * (z * ((y * -0.5) + -1.0))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.8e-186) || !(x <= 2.9e-78))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(z * Float64(Float64(y * -0.5) + -1.0))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5.8e-186) || ~((x <= 2.9e-78)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y * (z * ((y * -0.5) + -1.0))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.8e-186], N[Not[LessEqual[x, 2.9e-78]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(z * N[(N[(y * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.80000000000000038e-186 or 2.9000000000000001e-78 < x

   1. Initial program 91.1%

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

      1. fma-define 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in y around 0 89.8%

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

   if -5.80000000000000038e-186 < x < 2.9000000000000001e-78

   1. Initial program 71.0%

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

   3. Taylor expanded in y around 0 99.0%

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

   4. Taylor expanded in x around 0 96.1%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-186} \lor \neg \left(x \leq 2.9 \cdot 10^{-78}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot -0.5 + -1\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in y around 0 99.2%

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

4. Final simplification 99.2%

   \[\leadsto \left(x \cdot \log y + z \cdot \left(y \cdot \left(y \cdot -0.5 + -1\right)\right)\right) - t \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+89} \lor \neg \left(x \leq 43\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot -0.5 + -1\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -7.2e+89) (not (<= x 43.0)))
   (* x (log y))
   (- (* y (* z (+ (* y -0.5) -1.0))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -7.2e+89) || !(x <= 43.0)) {
		tmp = x * log(y);
	} else {
		tmp = (y * (z * ((y * -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 <= (-7.2d+89)) .or. (.not. (x <= 43.0d0))) then
        tmp = x * log(y)
    else
        tmp = (y * (z * ((y * (-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 <= -7.2e+89) || !(x <= 43.0)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * (z * ((y * -0.5) + -1.0))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -7.2e+89) or not (x <= 43.0):
		tmp = x * math.log(y)
	else:
		tmp = (y * (z * ((y * -0.5) + -1.0))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -7.2e+89) || !(x <= 43.0))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(z * Float64(Float64(y * -0.5) + -1.0))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -7.2e+89) || ~((x <= 43.0)))
		tmp = x * log(y);
	else
		tmp = (y * (z * ((y * -0.5) + -1.0))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -7.2e+89], N[Not[LessEqual[x, 43.0]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(z * N[(N[(y * -0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2e89 or 43 < x

   1. Initial program 94.7%

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

      1. +-commutative 94.7%

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

      2. associate--l+ 94.7%

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

      3. fma-define 94.7%

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

      4. sub-neg 94.7%

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

      5. log1p-define 99.6%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 99.5%

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

      2. distribute-rgt-neg-in 99.5%

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

      3. pow2 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 78.7%

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

   if -7.2e89 < x < 43

   1. Initial program 76.7%

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

   3. Taylor expanded in y around 0 99.4%

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

   4. Taylor expanded in x around 0 82.0%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+89} \lor \neg \left(x \leq 43\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot -0.5 + -1\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. fma-define 85.2%

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

3. Simplified 85.2%

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

5. Taylor expanded in y around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. mul-1-neg 98.9%

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

   3. unsub-neg 98.9%

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

7. Simplified 98.9%

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

8. Final simplification 98.9%

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

Alternative 6: 47.6% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-58} \lor \neg \left(t \leq 4.2 \cdot 10^{+31}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.8e-58) (not (<= t 4.2e+31))) (- t) (* y (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.8e-58) || !(t <= 4.2e+31)) {
		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 <= (-1.8d-58)) .or. (.not. (t <= 4.2d+31))) 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 <= -1.8e-58) || !(t <= 4.2e+31)) {
		tmp = -t;
	} else {
		tmp = y * -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.8e-58) or not (t <= 4.2e+31):
		tmp = -t
	else:
		tmp = y * -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.8e-58) || !(t <= 4.2e+31))
		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 <= -1.8e-58) || ~((t <= 4.2e+31)))
		tmp = -t;
	else
		tmp = y * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.8e-58], N[Not[LessEqual[t, 4.2e+31]], $MachinePrecision]], (-t), N[(y * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.80000000000000005e-58 or 4.19999999999999958e31 < t

   1. Initial program 96.2%

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

      1. fma-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in t around inf 67.0%

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

      1. neg-mul-1 67.0%

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

   7. Simplified 67.0%

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

   if -1.80000000000000005e-58 < t < 4.19999999999999958e31

   1. Initial program 75.6%

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

      1. fma-define 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in y around 0 98.2%

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

      1. +-commutative 98.2%

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

      2. mul-1-neg 98.2%

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

      3. unsub-neg 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in t around inf 71.0%

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

      1. associate-/l* 70.2%

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

      2. associate-/l* 58.7%

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

   10. Simplified 58.7%

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

   11. Taylor expanded in y around inf 27.1%

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

       1. mul-1-neg 27.1%

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

       2. distribute-rgt-neg-out 27.1%

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

   13. Simplified 27.1%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-58} \lor \neg \left(t \leq 4.2 \cdot 10^{+31}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.8% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

3. Taylor expanded in y around 0 99.2%

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

4. Taylor expanded in x around 0 52.7%

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

5. Final simplification 52.7%

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

Alternative 8: 57.5% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

   1. fma-define 85.2%

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

3. Simplified 85.2%

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

5. Taylor expanded in y around 0 98.9%

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

   1. +-commutative 98.9%

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

   2. mul-1-neg 98.9%

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

   3. unsub-neg 98.9%

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

7. Simplified 98.9%

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

8. Taylor expanded in x around 0 52.6%

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

   1. mul-1-neg 52.6%

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

   2. *-commutative 52.6%

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

   3. distribute-rgt-neg-in 52.6%

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

10. Simplified 52.6%

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

11. Final simplification 52.6%

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

Alternative 9: 42.9% 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 85.2%

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

   1. fma-define 85.2%

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

3. Simplified 85.2%

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

5. Taylor expanded in t around inf 37.7%

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

   1. neg-mul-1 37.7%

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

7. Simplified 37.7%

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

Alternative 10: 2.3% 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 85.2%

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

   1. fma-define 85.2%

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

3. Simplified 85.2%

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

5. Taylor expanded in t around inf 37.7%

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

   1. neg-mul-1 37.7%

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

7. Simplified 37.7%

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

   1. neg-sub0 37.7%

      \[\leadsto \color{blue}{0 - t} \]

   2. sub-neg 37.7%

      \[\leadsto \color{blue}{0 + \left(-t\right)} \]

   3. add-sqr-sqrt 18.1%

      \[\leadsto 0 + \color{blue}{\sqrt{-t} \cdot \sqrt{-t}} \]

   4. sqrt-unprod 12.0%

      \[\leadsto 0 + \color{blue}{\sqrt{\left(-t\right) \cdot \left(-t\right)}} \]

   5. sqr-neg 12.0%

      \[\leadsto 0 + \sqrt{\color{blue}{t \cdot t}} \]

   6. sqrt-unprod 1.3%

      \[\leadsto 0 + \color{blue}{\sqrt{t} \cdot \sqrt{t}} \]

   7. add-sqr-sqrt 2.3%

      \[\leadsto 0 + \color{blue}{t} \]

9. Applied egg-rr 2.3%

   \[\leadsto \color{blue}{0 + t} \]
10. Step-by-step derivation

    1. +-lft-identity 2.3%

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

11. Simplified 2.3%

    \[\leadsto \color{blue}{t} \]
12. 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 2024185 
(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))