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

Percentage Accurate: 85.2% → 99.6%

Time: 15.6s

Alternatives: 10

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% 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: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

3. Taylor expanded in y around 0 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 85.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+191} \lor \neg \left(z \leq 7.6 \cdot 10^{+42}\right):\\ \;\;\;\;t\_1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (or (<= z -2.5e+191) (not (<= z 7.6e+42))) (- t_1 (* y z)) (- t_1 t))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if ((z <= -2.5e+191) || !(z <= 7.6e+42)) {
		tmp = t_1 - (y * z);
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if ((z <= (-2.5d+191)) .or. (.not. (z <= 7.6d+42))) then
        tmp = t_1 - (y * z)
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if ((z <= -2.5e+191) || !(z <= 7.6e+42)) {
		tmp = t_1 - (y * z);
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if (z <= -2.5e+191) or not (z <= 7.6e+42):
		tmp = t_1 - (y * z)
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if ((z <= -2.5e+191) || !(z <= 7.6e+42))
		tmp = Float64(t_1 - Float64(y * z));
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if ((z <= -2.5e+191) || ~((z <= 7.6e+42)))
		tmp = t_1 - (y * z);
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[z, -2.5e+191], N[Not[LessEqual[z, 7.6e+42]], $MachinePrecision]], N[(t$95$1 - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5000000000000001e191 or 7.5999999999999997e42 < z

   1. Initial program 53.1%

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

      1. +-rgt-identity 53.1%

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

      2. +-rgt-identity 53.1%

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

      3. +-commutative 53.1%

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

      4. fma-define 53.1%

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

      5. sub-neg 53.1%

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

      6. 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. Taylor expanded in y around 0 99.6%

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

      1. fma-define 99.6%

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

      2. fma-define 99.6%

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

      3. associate-*r* 99.6%

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

      4. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around 0 98.7%

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

      1. associate-*r* 98.7%

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

      2. mul-1-neg 98.7%

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

   10. Simplified 98.7%

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

   11. Taylor expanded in t around 0 78.0%

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

       1. +-commutative 78.0%

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

       2. neg-mul-1 78.0%

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

       3. unsub-neg 78.0%

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

   13. Simplified 78.0%

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

   if -2.5000000000000001e191 < z < 7.5999999999999997e42

   1. Initial program 95.1%

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

      1. +-rgt-identity 95.1%

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

      2. +-rgt-identity 95.1%

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

      3. +-commutative 95.1%

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

      4. fma-define 95.1%

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

      5. sub-neg 95.1%

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

      6. 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. Taylor expanded in z around 0 94.8%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+191} \lor \neg \left(z \leq 7.6 \cdot 10^{+42}\right):\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00016 \lor \neg \left(x \leq 1.9 \cdot 10^{-95}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot z\right) \cdot -0.5 - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.00016) (not (<= x 1.9e-95)))
   (- (* x (log y)) t)
   (- (* y (- (* (* y z) -0.5) z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.00016) || !(x <= 1.9e-95)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y * (((y * z) * -0.5) - z)) - 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 <= (-0.00016d0)) .or. (.not. (x <= 1.9d-95))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y * (((y * z) * (-0.5d0)) - z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.00016) || !(x <= 1.9e-95)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * (((y * z) * -0.5) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -0.00016) or not (x <= 1.9e-95):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y * (((y * z) * -0.5) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -0.00016) || !(x <= 1.9e-95))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(Float64(Float64(y * z) * -0.5) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -0.00016) || ~((x <= 1.9e-95)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y * (((y * z) * -0.5) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -0.00016], N[Not[LessEqual[x, 1.9e-95]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(N[(N[(y * z), $MachinePrecision] * -0.5), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.60000000000000013e-4 or 1.8999999999999999e-95 < x

   1. Initial program 91.3%

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

      1. +-rgt-identity 91.3%

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

      2. +-rgt-identity 91.3%

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

      3. +-commutative 91.3%

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

      4. fma-define 91.3%

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

      5. sub-neg 91.3%

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

      6. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 90.4%

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

   if -1.60000000000000013e-4 < x < 1.8999999999999999e-95

   1. Initial program 73.2%

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

      1. +-rgt-identity 73.2%

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

      2. +-rgt-identity 73.2%

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

      3. +-commutative 73.2%

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

      4. fma-define 73.2%

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

      5. sub-neg 73.2%

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

      6. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 63.0%

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

      1. fmm-def 63.0%

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

      2. sub-neg 63.0%

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

      3. log1p-define 89.8%

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

   7. Simplified 89.8%

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

   8. Taylor expanded in y around 0 89.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00016 \lor \neg \left(x \leq 1.9 \cdot 10^{-95}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(y \cdot z\right) \cdot -0.5 - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+85} \lor \neg \left(x \leq 9.5 \cdot 10^{+33}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(y \cdot z\right)\right) - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.3e+85) (not (<= x 9.5e+33)))
   (* x (log y))
   (- (* y (- (* y (+ (* z -0.5) (* -0.3333333333333333 (* y z)))) z)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.3e+85) || !(x <= 9.5e+33)) {
		tmp = x * log(y);
	} else {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (y * z)))) - z)) - 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 <= (-3.3d+85)) .or. (.not. (x <= 9.5d+33))) then
        tmp = x * log(y)
    else
        tmp = (y * ((y * ((z * (-0.5d0)) + ((-0.3333333333333333d0) * (y * z)))) - z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.3e+85) || !(x <= 9.5e+33)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (y * z)))) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -3.3e+85) or not (x <= 9.5e+33):
		tmp = x * math.log(y)
	else:
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (y * z)))) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.3e+85) || !(x <= 9.5e+33))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(-0.3333333333333333 * Float64(y * z)))) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -3.3e+85) || ~((x <= 9.5e+33)))
		tmp = x * log(y);
	else
		tmp = (y * ((y * ((z * -0.5) + (-0.3333333333333333 * (y * z)))) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.3e+85], N[Not[LessEqual[x, 9.5e+33]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(-0.3333333333333333 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2999999999999999e85 or 9.5000000000000003e33 < 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. +-rgt-identity 93.2%

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

      2. +-rgt-identity 93.2%

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

      3. +-commutative 93.2%

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

      4. fma-define 93.2%

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

      5. sub-neg 93.2%

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

      6. log1p-define 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 99.6%

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

      1. fma-define 99.6%

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

      2. fma-define 99.6%

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

      3. associate-*r* 99.6%

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

      4. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 81.2%

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

   if -3.2999999999999999e85 < x < 9.5000000000000003e33

   1. Initial program 77.3%

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

      1. +-rgt-identity 77.3%

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

      2. +-rgt-identity 77.3%

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

      3. +-commutative 77.3%

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

      4. fma-define 77.3%

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

      5. sub-neg 77.3%

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

      6. log1p-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 62.6%

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

      1. fmm-def 62.6%

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

      2. sub-neg 62.6%

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

      3. log1p-define 85.1%

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

   7. Simplified 85.1%

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

   8. Taylor expanded in y around 0 85.1%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+85} \lor \neg \left(x \leq 9.5 \cdot 10^{+33}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + -0.3333333333333333 \cdot \left(y \cdot z\right)\right) - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. +-rgt-identity 83.4%

      \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

   2. +-rgt-identity 83.4%

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

   3. +-commutative 83.4%

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

   4. fma-define 83.4%

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

   5. sub-neg 83.4%

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

   6. 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. Taylor expanded in y around 0 99.8%

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

   1. fma-define 99.8%

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

   2. fma-define 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

7. Simplified 99.8%

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

8. Taylor expanded in y around 0 99.4%

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

   1. +-commutative 99.4%

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

   2. mul-1-neg 99.4%

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

   3. unsub-neg 99.4%

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

10. Simplified 99.4%

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

Alternative 6: 57.9% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. +-rgt-identity 83.4%

      \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

   2. +-rgt-identity 83.4%

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

   3. +-commutative 83.4%

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

   4. fma-define 83.4%

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

   5. sub-neg 83.4%

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

   6. 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. Taylor expanded in x around 0 43.6%

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

   1. fmm-def 43.6%

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

   2. sub-neg 43.6%

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

   3. log1p-define 59.9%

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

7. Simplified 59.9%

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

8. Taylor expanded in y around 0 59.9%

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

9. Final simplification 59.9%

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

Alternative 7: 57.8% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.4%

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

   1. +-rgt-identity 83.4%

      \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

   2. +-rgt-identity 83.4%

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

   3. +-commutative 83.4%

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

   4. fma-define 83.4%

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

   5. sub-neg 83.4%

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

   6. 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. Taylor expanded in x around 0 43.6%

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

   1. fmm-def 43.6%

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

   2. sub-neg 43.6%

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

   3. log1p-define 59.9%

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

7. Simplified 59.9%

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

8. Taylor expanded in y around 0 59.8%

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

9. Final simplification 59.8%

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

Alternative 8: 57.5% 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 83.4%

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

   1. +-rgt-identity 83.4%

      \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

   2. +-rgt-identity 83.4%

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

   3. +-commutative 83.4%

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

   4. fma-define 83.4%

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

   5. sub-neg 83.4%

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

   6. 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. Taylor expanded in x around 0 43.6%

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

   1. fmm-def 43.6%

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

   2. sub-neg 43.6%

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

   3. log1p-define 59.9%

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

7. Simplified 59.9%

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

8. Taylor expanded in y around 0 59.5%

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

   1. associate-*r* 59.5%

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

   2. mul-1-neg 59.5%

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

10. Simplified 59.5%

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

11. Final simplification 59.5%

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

Alternative 9: 43.0% 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 83.4%

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

   1. +-rgt-identity 83.4%

      \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

   2. +-rgt-identity 83.4%

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

   3. +-commutative 83.4%

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

   4. fma-define 83.4%

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

   5. sub-neg 83.4%

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

   6. 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. Taylor expanded in t around inf 42.7%

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

   1. neg-mul-1 42.7%

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

7. Simplified 42.7%

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

Alternative 10: 2.3% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 83.4%

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

   1. +-rgt-identity 83.4%

      \[\leadsto \color{blue}{\left(\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) + 0\right)} - t \]

   2. +-rgt-identity 83.4%

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

   3. +-commutative 83.4%

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

   4. fma-define 83.4%

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

   5. sub-neg 83.4%

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

   6. 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. Taylor expanded in t around inf 42.7%

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

   1. neg-mul-1 42.7%

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

7. Simplified 42.7%

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

   1. neg-sub0 42.7%

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

   2. sub-neg 42.7%

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

   3. add-sqr-sqrt 22.3%

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

   4. sqrt-unprod 13.3%

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

   5. sqr-neg 13.3%

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

   6. sqrt-unprod 1.0%

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

   7. add-sqr-sqrt 2.2%

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

9. Applied egg-rr 2.2%

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

    1. +-lft-identity 2.2%

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

11. Simplified 2.2%

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

Developer Target 1: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024165 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

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