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

Percentage Accurate: 85.6% → 99.8%

Time: 13.5s

Alternatives: 11

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: 85.6% 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 85.5%

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

   1. +-commutative 85.5%

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

   2. associate--l+ 85.5%

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

   3. fma-define 85.5%

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

   4. sub-neg 85.5%

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

   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: 89.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -51 \lor \neg \left(x \leq 1.8 \cdot 10^{-157}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -51.0) (not (<= x 1.8e-157)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -51.0) || !(x <= 1.8e-157)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -51.0) || !(x <= 1.8e-157)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -51.0) or not (x <= 1.8e-157):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -51.0) || !(x <= 1.8e-157))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -51.0], N[Not[LessEqual[x, 1.8e-157]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -51 or 1.8e-157 < x

   1. Initial program 93.2%

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

      1. +-commutative 93.2%

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

      2. associate--l+ 93.2%

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

      3. fma-define 93.2%

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

      4. sub-neg 93.2%

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

      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. Taylor expanded in z around 0 92.1%

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

   if -51 < x < 1.8e-157

   1. Initial program 72.4%

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

      1. +-commutative 72.4%

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

      2. associate--l+ 72.4%

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

      3. fma-define 72.4%

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

      4. sub-neg 72.4%

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

      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. Taylor expanded in x around 0 65.1%

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

      1. sub-neg 65.1%

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

      2. log1p-define 92.7%

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

   7. Simplified 92.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -51 \lor \neg \left(x \leq 1.8 \cdot 10^{-157}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (y * (((-0.5d0) * (z * y)) - z))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in y around 0 99.4%

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

4. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (y * (z * ((-1.0d0) + (y * (-0.5d0)))))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in y around 0 99.4%

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

4. Taylor expanded in z around 0 99.3%

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

5. Final simplification 99.3%

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

Alternative 5: 65.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+25} \lor \neg \left(x \leq 1.25 \cdot 10^{+78}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.4e+25) (not (<= x 1.25e+78))) (* x (log y)) (- t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e+25) || !(x <= 1.25e+78)) {
		tmp = x * log(y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2.4d+25)) .or. (.not. (x <= 1.25d+78))) then
        tmp = x * log(y)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e+25) || !(x <= 1.25e+78)) {
		tmp = x * Math.log(y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.4e+25) or not (x <= 1.25e+78):
		tmp = x * math.log(y)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.4e+25) || !(x <= 1.25e+78))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.4e+25) || ~((x <= 1.25e+78)))
		tmp = x * log(y);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.4e+25], N[Not[LessEqual[x, 1.25e+78]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999996e25 or 1.24999999999999996e78 < x

   1. Initial program 95.7%

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

      1. +-commutative 95.7%

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

      2. associate--l+ 95.7%

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

      3. fma-define 95.7%

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

      4. sub-neg 95.7%

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

      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. Taylor expanded in x around inf 95.6%

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

      1. associate--l+ 95.6%

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

      2. div-sub 95.6%

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

      3. sub-neg 95.6%

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

      4. log1p-define 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 74.7%

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

   if -2.39999999999999996e25 < x < 1.24999999999999996e78

   1. Initial program 76.9%

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

      1. +-commutative 76.9%

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

      2. associate--l+ 76.9%

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

      3. fma-define 76.9%

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

      4. sub-neg 76.9%

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

      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. Taylor expanded in t around inf 59.4%

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

      1. neg-mul-1 59.4%

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

   7. Simplified 59.4%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+25} \lor \neg \left(x \leq 1.25 \cdot 10^{+78}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.9 \cdot 10^{+254}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z 3.9e+254) (- (* x (log y)) t) (* y (- (* -0.5 (* z y)) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.9e+254) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = y * ((-0.5 * (z * 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 (z <= 3.9d+254) then
        tmp = (x * log(y)) - t
    else
        tmp = y * (((-0.5d0) * (z * y)) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 3.9e+254) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = y * ((-0.5 * (z * y)) - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 3.9e+254:
		tmp = (x * math.log(y)) - t
	else:
		tmp = y * ((-0.5 * (z * y)) - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 3.9e+254)
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(y * Float64(Float64(-0.5 * Float64(z * y)) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 3.9e+254)
		tmp = (x * log(y)) - t;
	else
		tmp = y * ((-0.5 * (z * y)) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 3.9e+254], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(y * N[(N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.9000000000000001e254

   1. Initial program 88.0%

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

      1. +-commutative 88.0%

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

      2. associate--l+ 88.0%

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

      3. fma-define 88.0%

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

      4. sub-neg 88.0%

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

      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. Taylor expanded in z around 0 86.2%

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

   if 3.9000000000000001e254 < z

   1. Initial program 35.2%

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

      1. +-commutative 35.2%

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

      2. associate--l+ 35.2%

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

      3. fma-define 35.2%

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

      4. sub-neg 35.2%

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

      5. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 2.1%

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

      1. sub-neg 2.1%

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

      2. log1p-define 68.6%

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

   7. Simplified 68.6%

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

   8. Taylor expanded in y around 0 68.6%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.9 \cdot 10^{+254}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.1% 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 85.5%

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

   1. +-commutative 85.5%

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

   2. associate--l+ 85.5%

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

   3. fma-define 85.5%

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

   4. sub-neg 85.5%

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

   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. Taylor expanded in y around 0 98.7%

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

   1. +-commutative 98.7%

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

   2. mul-1-neg 98.7%

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

   3. unsub-neg 98.7%

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

7. Simplified 98.7%

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

8. Final simplification 98.7%

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

Alternative 8: 49.0% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-65} \lor \neg \left(t \leq 2.25 \cdot 10^{-119}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + y \cdot \left(z \cdot -0.3333333333333333 + \left(z \cdot y\right) \cdot -0.25\right)\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.4e-65) (not (<= t 2.25e-119)))
   (- t)
   (*
    y
    (-
     (* y (+ (* z -0.5) (* y (+ (* z -0.3333333333333333) (* (* z y) -0.25)))))
     z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.4e-65) || !(t <= 2.25e-119)) {
		tmp = -t;
	} else {
		tmp = y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + ((z * y) * -0.25))))) - 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 <= (-4.4d-65)) .or. (.not. (t <= 2.25d-119))) then
        tmp = -t
    else
        tmp = y * ((y * ((z * (-0.5d0)) + (y * ((z * (-0.3333333333333333d0)) + ((z * y) * (-0.25d0)))))) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.4e-65) || !(t <= 2.25e-119)) {
		tmp = -t;
	} else {
		tmp = y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + ((z * y) * -0.25))))) - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -4.4e-65) or not (t <= 2.25e-119):
		tmp = -t
	else:
		tmp = y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + ((z * y) * -0.25))))) - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.4e-65) || !(t <= 2.25e-119))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(y * Float64(Float64(z * -0.3333333333333333) + Float64(Float64(z * y) * -0.25))))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4.4e-65) || ~((t <= 2.25e-119)))
		tmp = -t;
	else
		tmp = y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + ((z * y) * -0.25))))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.4e-65], N[Not[LessEqual[t, 2.25e-119]], $MachinePrecision]], (-t), N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(y * N[(N[(z * -0.3333333333333333), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.40000000000000042e-65 or 2.2500000000000001e-119 < t

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. associate--l+ 94.5%

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

      3. fma-define 94.5%

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

      4. sub-neg 94.5%

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

      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. Taylor expanded in t around inf 57.7%

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

      1. neg-mul-1 57.7%

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

   7. Simplified 57.7%

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

   if -4.40000000000000042e-65 < t < 2.2500000000000001e-119

   1. Initial program 68.1%

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

      1. +-commutative 68.1%

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

      2. associate--l+ 68.1%

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

      3. fma-define 68.1%

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

      4. sub-neg 68.1%

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

      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. Taylor expanded in z around inf 5.9%

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

      1. sub-neg 5.9%

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

      2. log1p-define 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in y around 0 35.7%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-65} \lor \neg \left(t \leq 2.25 \cdot 10^{-119}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + y \cdot \left(z \cdot -0.3333333333333333 + \left(z \cdot y\right) \cdot -0.25\right)\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.9% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-65} \lor \neg \left(t \leq 2.15 \cdot 10^{-119}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + \left(z \cdot y\right) \cdot -0.3333333333333333\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.75e-65) (not (<= t 2.15e-119)))
   (- t)
   (* y (- (* y (+ (* z -0.5) (* (* z y) -0.3333333333333333))) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.75e-65) || !(t <= 2.15e-119)) {
		tmp = -t;
	} else {
		tmp = y * ((y * ((z * -0.5) + ((z * y) * -0.3333333333333333))) - 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 <= (-1.75d-65)) .or. (.not. (t <= 2.15d-119))) then
        tmp = -t
    else
        tmp = y * ((y * ((z * (-0.5d0)) + ((z * y) * (-0.3333333333333333d0)))) - z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.75e-65) || !(t <= 2.15e-119)) {
		tmp = -t;
	} else {
		tmp = y * ((y * ((z * -0.5) + ((z * y) * -0.3333333333333333))) - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.75e-65) or not (t <= 2.15e-119):
		tmp = -t
	else:
		tmp = y * ((y * ((z * -0.5) + ((z * y) * -0.3333333333333333))) - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.75e-65) || !(t <= 2.15e-119))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(Float64(z * y) * -0.3333333333333333))) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.75e-65) || ~((t <= 2.15e-119)))
		tmp = -t;
	else
		tmp = y * ((y * ((z * -0.5) + ((z * y) * -0.3333333333333333))) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.75e-65], N[Not[LessEqual[t, 2.15e-119]], $MachinePrecision]], (-t), N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(N[(z * y), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.75000000000000002e-65 or 2.15e-119 < t

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. associate--l+ 94.5%

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

      3. fma-define 94.5%

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

      4. sub-neg 94.5%

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

      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. Taylor expanded in t around inf 57.7%

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

      1. neg-mul-1 57.7%

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

   7. Simplified 57.7%

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

   if -1.75000000000000002e-65 < t < 2.15e-119

   1. Initial program 68.1%

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

      1. +-commutative 68.1%

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

      2. associate--l+ 68.1%

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

      3. fma-define 68.1%

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

      4. sub-neg 68.1%

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

      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. Taylor expanded in z around inf 5.9%

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

      1. sub-neg 5.9%

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

      2. log1p-define 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in y around 0 35.7%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{-65} \lor \neg \left(t \leq 2.15 \cdot 10^{-119}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + \left(z \cdot y\right) \cdot -0.3333333333333333\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.9% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-64} \lor \neg \left(t \leq 1.05 \cdot 10^{-118}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -8.8e-64) (not (<= t 1.05e-118)))
   (- t)
   (* y (- (* -0.5 (* z y)) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -8.8e-64) || !(t <= 1.05e-118)) {
		tmp = -t;
	} else {
		tmp = y * ((-0.5 * (z * 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 <= (-8.8d-64)) .or. (.not. (t <= 1.05d-118))) then
        tmp = -t
    else
        tmp = y * (((-0.5d0) * (z * 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 <= -8.8e-64) || !(t <= 1.05e-118)) {
		tmp = -t;
	} else {
		tmp = y * ((-0.5 * (z * y)) - z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -8.8e-64) or not (t <= 1.05e-118):
		tmp = -t
	else:
		tmp = y * ((-0.5 * (z * y)) - z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -8.8e-64) || !(t <= 1.05e-118))
		tmp = Float64(-t);
	else
		tmp = Float64(y * Float64(Float64(-0.5 * Float64(z * y)) - z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -8.8e-64) || ~((t <= 1.05e-118)))
		tmp = -t;
	else
		tmp = y * ((-0.5 * (z * y)) - z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -8.8e-64], N[Not[LessEqual[t, 1.05e-118]], $MachinePrecision]], (-t), N[(y * N[(N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.7999999999999998e-64 or 1.05e-118 < t

   1. Initial program 94.5%

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

      1. +-commutative 94.5%

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

      2. associate--l+ 94.5%

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

      3. fma-define 94.5%

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

      4. sub-neg 94.5%

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

      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. Taylor expanded in t around inf 57.7%

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

      1. neg-mul-1 57.7%

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

   7. Simplified 57.7%

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

   if -8.7999999999999998e-64 < t < 1.05e-118

   1. Initial program 68.1%

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

      1. +-commutative 68.1%

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

      2. associate--l+ 68.1%

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

      3. fma-define 68.1%

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

      4. sub-neg 68.1%

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

      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. Taylor expanded in z around inf 5.9%

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

      1. sub-neg 5.9%

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

      2. log1p-define 36.4%

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

   7. Simplified 36.4%

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

   8. Taylor expanded in y around 0 35.5%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-64} \lor \neg \left(t \leq 1.05 \cdot 10^{-118}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-0.5 \cdot \left(z \cdot y\right) - z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.7% 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 85.5%

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

   1. +-commutative 85.5%

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

   2. associate--l+ 85.5%

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

   3. fma-define 85.5%

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

   4. sub-neg 85.5%

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

   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. Taylor expanded in t around inf 41.3%

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

   1. neg-mul-1 41.3%

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

7. Simplified 41.3%

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

Developer target: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-z * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024086 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

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