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

Percentage Accurate: 85.7% → 99.8%

Time: 14.8s

Alternatives: 14

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

   1. +-commutative 88.3%

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

   2. fma-define 88.3%

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

   3. sub-neg 88.3%

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

   4. log1p-define 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (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 * ((y * ((z * (-0.5d0)) + ((-0.3333333333333333d0) * (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 * ((y * ((z * -0.5) + (-0.3333333333333333 * (z * y)))) - z))) - t;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

3. Taylor expanded in y around 0 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \left(x \cdot \log y + y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(z \cdot y\right)\right) - z\right)\right) - t \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

3. Taylor expanded in y around 0 99.4%

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

4. Final simplification 99.4%

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

Alternative 4: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-55} \lor \neg \left(x \leq 7.5 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(z + y \cdot \left(\left(z \cdot y\right) \cdot 0.3333333333333333 + z \cdot 0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.05e-55) (not (<= x 7.5e-19)))
   (- (* x (log y)) t)
   (- (* (- y) (+ z (* y (+ (* (* z y) 0.3333333333333333) (* z 0.5))))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.05e-55) || !(x <= 7.5e-19)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (-y * (z + (y * (((z * y) * 0.3333333333333333) + (z * 0.5))))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-1.05d-55)) .or. (.not. (x <= 7.5d-19))) then
        tmp = (x * log(y)) - t
    else
        tmp = (-y * (z + (y * (((z * y) * 0.3333333333333333d0) + (z * 0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.05e-55) || !(x <= 7.5e-19)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (-y * (z + (y * (((z * y) * 0.3333333333333333) + (z * 0.5))))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.05e-55) or not (x <= 7.5e-19):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (-y * (z + (y * (((z * y) * 0.3333333333333333) + (z * 0.5))))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.05e-55) || !(x <= 7.5e-19))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(-y) * Float64(z + Float64(y * Float64(Float64(Float64(z * y) * 0.3333333333333333) + Float64(z * 0.5))))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.05e-55) || ~((x <= 7.5e-19)))
		tmp = (x * log(y)) - t;
	else
		tmp = (-y * (z + (y * (((z * y) * 0.3333333333333333) + (z * 0.5))))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.05e-55], N[Not[LessEqual[x, 7.5e-19]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[((-y) * N[(z + N[(y * N[(N[(N[(z * y), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0500000000000001e-55 or 7.49999999999999957e-19 < x

   1. Initial program 95.3%

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

   3. Taylor expanded in y around 0 98.8%

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

      1. +-commutative 98.8%

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

      2. mul-1-neg 98.8%

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

      3. unsub-neg 98.8%

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

   5. Simplified 98.8%

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

   6. Taylor expanded in x around inf 94.1%

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

   if -1.0500000000000001e-55 < x < 7.49999999999999957e-19

   1. Initial program 78.9%

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

   3. Taylor expanded in x around 0 72.1%

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

      1. sub-neg 72.1%

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

      2. log1p-define 92.6%

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

   5. Simplified 92.6%

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

   6. Taylor expanded in y around 0 92.6%

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

   7. Taylor expanded in z around -inf 92.6%

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z \cdot \left(1 + y \cdot \left(0.5 + 0.3333333333333333 \cdot y\right)\right)\right)\right)} - t \]
   8. Step-by-step derivation

      1. mul-1-neg 92.6%

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

      2. associate-*r* 92.6%

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

      3. *-commutative 92.6%

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

   9. Simplified 92.6%

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

   10. Taylor expanded in y around 0 92.6%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-55} \lor \neg \left(x \leq 7.5 \cdot 10^{-19}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot \left(z + y \cdot \left(\left(z \cdot y\right) \cdot 0.3333333333333333 + z \cdot 0.5\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% 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 88.3%

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

3. Taylor expanded in y around 0 99.3%

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

4. Final simplification 99.3%

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

Alternative 6: 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 88.3%

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

3. Taylor expanded in y around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 7: 56.7% accurate, 9.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * ((y * ((z * (-0.5d0)) + (y * ((z * (-0.3333333333333333d0)) + ((z * y) * (-0.25d0)))))) - z)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

3. Taylor expanded in x around 0 48.8%

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

   1. sub-neg 48.8%

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

   2. log1p-define 59.9%

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

5. Simplified 59.9%

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

6. Taylor expanded in y around 0 59.6%

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

7. Final simplification 59.6%

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

Alternative 8: 56.6% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

3. Taylor expanded in x around 0 48.8%

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

   1. sub-neg 48.8%

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

   2. log1p-define 59.9%

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

5. Simplified 59.9%

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

6. Taylor expanded in y around 0 59.6%

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

7. Taylor expanded in z around -inf 59.5%

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z \cdot \left(1 + y \cdot \left(0.5 + 0.3333333333333333 \cdot y\right)\right)\right)\right)} - t \]
8. Step-by-step derivation

   1. mul-1-neg 59.5%

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

   2. associate-*r* 59.5%

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

   3. *-commutative 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in y around 0 59.6%

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

11. Final simplification 59.6%

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

Alternative 9: 56.6% accurate, 14.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

3. Taylor expanded in x around 0 48.8%

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

   1. sub-neg 48.8%

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

   2. log1p-define 59.9%

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

5. Simplified 59.9%

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

6. Taylor expanded in y around 0 59.6%

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

7. Taylor expanded in z around 0 59.5%

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

8. Final simplification 59.5%

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

Alternative 10: 56.6% accurate, 14.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

3. Taylor expanded in x around 0 48.8%

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

   1. sub-neg 48.8%

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

   2. log1p-define 59.9%

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

5. Simplified 59.9%

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

6. Taylor expanded in y around 0 59.6%

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

7. Taylor expanded in z around -inf 59.5%

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z \cdot \left(1 + y \cdot \left(0.5 + 0.3333333333333333 \cdot y\right)\right)\right)\right)} - t \]
8. Step-by-step derivation

   1. mul-1-neg 59.5%

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

   2. associate-*r* 59.5%

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

   3. *-commutative 59.5%

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

9. Simplified 59.5%

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

10. Final simplification 59.5%

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

Alternative 11: 56.5% accurate, 17.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

3. Taylor expanded in x around 0 48.8%

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

   1. sub-neg 48.8%

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

   2. log1p-define 59.9%

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

5. Simplified 59.9%

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

6. Taylor expanded in y around 0 59.6%

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

7. Taylor expanded in z around -inf 59.5%

   \[\leadsto \color{blue}{-1 \cdot \left(y \cdot \left(z \cdot \left(1 + y \cdot \left(0.5 + 0.3333333333333333 \cdot y\right)\right)\right)\right)} - t \]
8. Step-by-step derivation

   1. mul-1-neg 59.5%

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

   2. associate-*r* 59.5%

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

   3. *-commutative 59.5%

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

9. Simplified 59.5%

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

10. Taylor expanded in y around 0 59.4%

    \[\leadsto \left(-\color{blue}{y \cdot \left(z + 0.5 \cdot \left(y \cdot z\right)\right)}\right) - t \]
11. Step-by-step derivation

    1. *-commutative 59.4%

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

12. Simplified 59.4%

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

13. Final simplification 59.4%

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

Alternative 12: 56.5% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

3. Taylor expanded in x around 0 48.8%

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

   1. sub-neg 48.8%

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

   2. log1p-define 59.9%

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

5. Simplified 59.9%

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

6. Taylor expanded in y around 0 59.6%

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

7. Taylor expanded in y around 0 59.4%

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

   1. associate-*r* 59.4%

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

   2. *-commutative 59.4%

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

   3. distribute-rgt-out 59.4%

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

9. Simplified 59.4%

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

10. Final simplification 59.4%

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

Alternative 13: 56.2% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

3. Taylor expanded in y around 0 98.8%

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

   1. +-commutative 98.8%

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

   2. mul-1-neg 98.8%

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

   3. unsub-neg 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in x around 0 58.8%

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

   1. associate-*r* 58.8%

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

   2. mul-1-neg 58.8%

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

8. Simplified 58.8%

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

9. Final simplification 58.8%

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

Alternative 14: 42.2% accurate, 105.5× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 88.3%

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

   1. *-commutative 88.3%

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

   2. add-cube-cbrt 87.5%

      \[\leadsto \left(\log y \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} + z \cdot \log \left(1 - y\right)\right) - t \]

   3. associate-*r* 87.5%

      \[\leadsto \left(\color{blue}{\left(\log y \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}} + z \cdot \log \left(1 - y\right)\right) - t \]

   4. fma-define 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right), \sqrt[3]{x}, z \cdot \log \left(1 - y\right)\right)} - t \]

   5. pow2 87.5%

      \[\leadsto \mathsf{fma}\left(\log y \cdot \color{blue}{{\left(\sqrt[3]{x}\right)}^{2}}, \sqrt[3]{x}, z \cdot \log \left(1 - y\right)\right) - t \]

   6. *-commutative 87.5%

      \[\leadsto \mathsf{fma}\left(\log y \cdot {\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, \color{blue}{\log \left(1 - y\right) \cdot z}\right) - t \]

   7. sub-neg 87.5%

      \[\leadsto \mathsf{fma}\left(\log y \cdot {\left(\sqrt[3]{x}\right)}^{2}, \sqrt[3]{x}, \log \color{blue}{\left(1 + \left(-y\right)\right)} \cdot z\right) - t \]

   8. log1p-undefine 99.1%

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

   9. add-sqr-sqrt 0.0%

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

   10. sqrt-unprod 85.6%

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

   11. sqr-neg 85.6%

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

   12. sqrt-unprod 85.4%

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

   13. add-sqr-sqrt 85.4%

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

4. Applied egg-rr 85.4%

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

5. Taylor expanded in y around inf 70.3%

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

   1. distribute-lft-out 70.3%

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

   2. mul-1-neg 70.3%

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

   3. distribute-rgt-out 70.3%

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

   4. distribute-lft-neg-in 70.3%

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

   5. log-rec 70.3%

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

   6. remove-double-neg 70.3%

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

   7. +-commutative 70.3%

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

7. Simplified 70.3%

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

8. Taylor expanded in t around inf 47.1%

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

   1. neg-mul-1 47.1%

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

10. Simplified 47.1%

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

11. Final simplification 47.1%

    \[\leadsto -t \]
12. 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 2024067 
(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))