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

Percentage Accurate: 85.7% → 99.8%

Time: 16.0s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

   1. +-commutative 86.1%

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

   2. associate--l+ 86.1%

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

   3. fma-define 86.1%

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

   4. sub-neg 86.1%

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

   5. log1p-define 99.8%

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

3. Simplified 99.8%

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

Alternative 2: 90.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-120} \lor \neg \left(x \leq 1.25 \cdot 10^{-25}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.5e-120) (not (<= x 1.25e-25)))
   (- (* x (log y)) t)
   (- (* z (log1p (- y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.5e-120) || !(x <= 1.25e-25)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (z * log1p(-y)) - t;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.5e-120) || !(x <= 1.25e-25)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (z * Math.log1p(-y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.5e-120) or not (x <= 1.25e-25):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (z * math.log1p(-y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.5e-120) || !(x <= 1.25e-25))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(z * log1p(Float64(-y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.5e-120], N[Not[LessEqual[x, 1.25e-25]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(z * N[Log[1 + (-y)], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.50000000000000005e-120 or 1.2499999999999999e-25 < x

   1. Initial program 94.1%

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

      1. *-commutative 94.1%

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

      2. add-sqr-sqrt 40.9%

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

      3. associate-*r* 40.9%

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

      4. fma-define 40.9%

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

      5. sub-neg 40.9%

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

      6. log1p-undefine 41.9%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 40.4%

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

      9. sqr-neg 40.4%

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

      10. sqrt-unprod 40.4%

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

      11. add-sqr-sqrt 40.4%

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

   4. Applied egg-rr 40.4%

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

   5. Taylor expanded in y around 0 93.1%

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

   if -1.50000000000000005e-120 < x < 1.2499999999999999e-25

   1. Initial program 72.9%

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

   3. Taylor expanded in x around 0 62.7%

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

      1. sub-neg 62.7%

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

      2. log1p-define 89.7%

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

   5. Simplified 89.7%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-120} \lor \neg \left(x \leq 1.25 \cdot 10^{-25}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \mathsf{log1p}\left(-y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-115} \lor \neg \left(x \leq 4.2 \cdot 10^{-18}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + y \cdot \left(z \cdot -0.3333333333333333 + -0.25 \cdot \left(z \cdot y\right)\right)\right) - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.12e-115) (not (<= x 4.2e-18)))
   (- (* x (log y)) t)
   (-
    (*
     y
     (-
      (*
       y
       (+ (* z -0.5) (* y (+ (* z -0.3333333333333333) (* -0.25 (* z y))))))
      z))
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.12e-115) || !(x <= 4.2e-18)) {
		tmp = (x * log(y)) - t;
	} else {
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - 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 <= (-1.12d-115)) .or. (.not. (x <= 4.2d-18))) then
        tmp = (x * log(y)) - t
    else
        tmp = (y * ((y * ((z * (-0.5d0)) + (y * ((z * (-0.3333333333333333d0)) + ((-0.25d0) * (z * y)))))) - 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 <= -1.12e-115) || !(x <= 4.2e-18)) {
		tmp = (x * Math.log(y)) - t;
	} else {
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.12e-115) or not (x <= 4.2e-18):
		tmp = (x * math.log(y)) - t
	else:
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.12e-115) || !(x <= 4.2e-18))
		tmp = Float64(Float64(x * log(y)) - t);
	else
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(y * Float64(Float64(z * -0.3333333333333333) + Float64(-0.25 * Float64(z * y)))))) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.12e-115) || ~((x <= 4.2e-18)))
		tmp = (x * log(y)) - t;
	else
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.12e-115], N[Not[LessEqual[x, 4.2e-18]], $MachinePrecision]], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(y * N[(N[(z * -0.3333333333333333), $MachinePrecision] + N[(-0.25 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1199999999999999e-115 or 4.19999999999999999e-18 < x

   1. Initial program 94.1%

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

      1. *-commutative 94.1%

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

      2. add-sqr-sqrt 40.9%

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

      3. associate-*r* 40.9%

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

      4. fma-define 40.9%

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

      5. sub-neg 40.9%

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

      6. log1p-undefine 41.9%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 40.4%

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

      9. sqr-neg 40.4%

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

      10. sqrt-unprod 40.4%

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

      11. add-sqr-sqrt 40.4%

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

   4. Applied egg-rr 40.4%

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

   5. Taylor expanded in y around 0 93.1%

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

   if -1.1199999999999999e-115 < x < 4.19999999999999999e-18

   1. Initial program 72.9%

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

   3. Taylor expanded in x around 0 62.7%

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

      1. sub-neg 62.7%

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

      2. log1p-define 89.7%

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

   5. Simplified 89.7%

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

   6. Taylor expanded in y around 0 88.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-115} \lor \neg \left(x \leq 4.2 \cdot 10^{-18}\right):\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + y \cdot \left(z \cdot -0.3333333333333333 + -0.25 \cdot \left(z \cdot y\right)\right)\right) - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in y around 0 99.0%

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

4. Final simplification 99.0%

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

Alternative 5: 78.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+64} \lor \neg \left(x \leq 3.4 \cdot 10^{+47}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + y \cdot \left(z \cdot -0.3333333333333333 + -0.25 \cdot \left(z \cdot y\right)\right)\right) - z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -5.2e+64) (not (<= x 3.4e+47)))
   (* x (log y))
   (-
    (*
     y
     (-
      (*
       y
       (+ (* z -0.5) (* y (+ (* z -0.3333333333333333) (* -0.25 (* z y))))))
      z))
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -5.2e+64) || !(x <= 3.4e+47)) {
		tmp = x * log(y);
	} else {
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - 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 <= (-5.2d+64)) .or. (.not. (x <= 3.4d+47))) then
        tmp = x * log(y)
    else
        tmp = (y * ((y * ((z * (-0.5d0)) + (y * ((z * (-0.3333333333333333d0)) + ((-0.25d0) * (z * y)))))) - 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 <= -5.2e+64) || !(x <= 3.4e+47)) {
		tmp = x * Math.log(y);
	} else {
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -5.2e+64) or not (x <= 3.4e+47):
		tmp = x * math.log(y)
	else:
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -5.2e+64) || !(x <= 3.4e+47))
		tmp = Float64(x * log(y));
	else
		tmp = Float64(Float64(y * Float64(Float64(y * Float64(Float64(z * -0.5) + Float64(y * Float64(Float64(z * -0.3333333333333333) + Float64(-0.25 * Float64(z * y)))))) - z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -5.2e+64) || ~((x <= 3.4e+47)))
		tmp = x * log(y);
	else
		tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -5.2e+64], N[Not[LessEqual[x, 3.4e+47]], $MachinePrecision]], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(y * N[(N[(z * -0.5), $MachinePrecision] + N[(y * N[(N[(z * -0.3333333333333333), $MachinePrecision] + N[(-0.25 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999994e64 or 3.3999999999999998e47 < x

   1. Initial program 97.0%

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

      1. +-commutative 97.0%

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

      2. associate--l+ 97.0%

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

      3. fma-define 97.0%

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

      4. sub-neg 97.0%

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

      5. log1p-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in t around 0 76.8%

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

      1. +-commutative 76.8%

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

      2. fma-define 76.8%

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

      3. sub-neg 76.8%

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

      4. log1p-define 79.6%

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

   7. Simplified 79.6%

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

   8. Taylor expanded in z around 0 76.1%

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

   if -5.19999999999999994e64 < x < 3.3999999999999998e47

   1. Initial program 77.8%

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

   3. Taylor expanded in x around 0 61.9%

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

      1. sub-neg 61.9%

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

      2. log1p-define 83.7%

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

   5. Simplified 83.7%

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

   6. Taylor expanded in y around 0 82.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+64} \lor \neg \left(x \leq 3.4 \cdot 10^{+47}\right):\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(y \cdot \left(z \cdot -0.5 + y \cdot \left(z \cdot -0.3333333333333333 + -0.25 \cdot \left(z \cdot y\right)\right)\right) - z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.3% 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 86.1%

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

3. Taylor expanded in y around 0 98.1%

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

   1. associate-*r* 98.1%

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

   2. mul-1-neg 98.1%

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

5. Simplified 98.1%

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

6. Final simplification 98.1%

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

Alternative 7: 57.1% accurate, 9.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (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 = (y * ((y * ((z * (-0.5d0)) + (y * ((z * (-0.3333333333333333d0)) + ((-0.25d0) * (z * y)))))) - 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) + (-0.25 * (z * y)))))) - z)) - t;
}

Python

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = (y * ((y * ((z * -0.5) + (y * ((z * -0.3333333333333333) + (-0.25 * (z * y)))))) - 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[(-0.25 * N[(z * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in x around 0 44.0%

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

   1. sub-neg 44.0%

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

   2. log1p-define 57.7%

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

5. Simplified 57.7%

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

6. Taylor expanded in y around 0 57.2%

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

7. Final simplification 57.2%

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

Alternative 8: 57.1% 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 86.1%

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

3. Taylor expanded in x around 0 44.0%

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

   1. sub-neg 44.0%

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

   2. log1p-define 57.7%

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

5. Simplified 57.7%

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

6. Taylor expanded in y around 0 57.1%

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

7. Taylor expanded in z around 0 57.1%

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

8. Final simplification 57.1%

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

Alternative 9: 57.0% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in x around 0 44.0%

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

   1. sub-neg 44.0%

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

   2. log1p-define 57.7%

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

5. Simplified 57.7%

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

6. Taylor expanded in y around 0 56.9%

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

7. Final simplification 56.9%

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

Alternative 10: 57.0% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in y around 0 99.0%

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

4. Taylor expanded in x around 0 56.9%

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

5. Final simplification 56.9%

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

Alternative 11: 56.7% accurate, 35.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in x around 0 44.0%

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

   1. sub-neg 44.0%

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

   2. log1p-define 57.7%

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

5. Simplified 57.7%

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

6. Taylor expanded in y around 0 56.0%

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

   1. associate-*r* 56.0%

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

   2. mul-1-neg 56.0%

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

8. Simplified 56.0%

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

9. Final simplification 56.0%

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

Alternative 12: 42.7% accurate, 105.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

3. Taylor expanded in x around 0 44.0%

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

   1. sub-neg 44.0%

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

   2. log1p-define 57.7%

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

5. Simplified 57.7%

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

6. Taylor expanded in z around 0 41.5%

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

   1. neg-mul-1 41.5%

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

8. Simplified 41.5%

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

Alternative 13: 2.2% 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 86.1%

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

3. Taylor expanded in x around 0 44.0%

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

   1. sub-neg 44.0%

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

   2. log1p-define 57.7%

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

5. Simplified 57.7%

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

6. Taylor expanded in z around 0 41.5%

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

   1. neg-mul-1 41.5%

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

8. Simplified 41.5%

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

   1. add-sqr-sqrt 21.1%

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

   2. sqrt-unprod 13.9%

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

   3. sqr-neg 13.9%

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

   4. sqrt-unprod 1.2%

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

   5. add-sqr-sqrt 2.2%

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

   6. *-un-lft-identity 2.2%

      \[\leadsto \color{blue}{1 \cdot t} \]

10. Applied egg-rr 2.2%

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

    1. *-lft-identity 2.2%

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

12. Simplified 2.2%

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

Developer Target 1: 99.7% 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 2024144 
(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))