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

Percentage Accurate: 85.1% → 99.5%

Time: 12.6s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(x \cdot \log y + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right)} - t \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\color{blue}{\log y \cdot x} + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   2. remove-double-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)} \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   3. mul-1-neg N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \log y}\right)\right) \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(\mathsf{neg}\left(\log y\right)\right)\right)} \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   5. log-rec N/A

      \[\leadsto \left(\left(-1 \cdot \color{blue}{\log \left(\frac{1}{y}\right)}\right) \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

   6. lower-fma.f64 N/A

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

   7. log-rec N/A

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

   8. distribute-rgt-neg-in N/A

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

   9. mul-1-neg N/A

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

   10. remove-double-neg N/A

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

   11. lower-log.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. +-commutative N/A

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

   15. associate-*r* N/A

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

   16. distribute-rgt-out N/A

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

   17. lower-*.f64 N/A

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

   18. lower-fma.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 2: 90.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-104} \lor \neg \left(x \leq 1.96 \cdot 10^{-131}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \mathsf{fma}\left(-0.5, y, -1\right)\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.25e-104) (not (<= x 1.96e-131)))
   (fma (log y) x (- t))
   (- (* (* z (fma -0.5 y -1.0)) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.25e-104) || !(x <= 1.96e-131)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = ((z * fma(-0.5, y, -1.0)) * y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.25e-104) || !(x <= 1.96e-131))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = Float64(Float64(Float64(z * fma(-0.5, y, -1.0)) * y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.25e-104], N[Not[LessEqual[x, 1.96e-131]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[(N[(N[(z * N[(-0.5 * y + -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2499999999999999e-104 or 1.95999999999999992e-131 < x

   1. Initial program 88.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot \log y - t} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

      8. fp-cancel-sub-sign-inv N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. log-rec N/A

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

      15. mul-1-neg N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 88.2%

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

   if -2.2499999999999999e-104 < x < 1.95999999999999992e-131

   1. Initial program 68.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 94.2%

      \[\leadsto \left(-y\right) \cdot z - t \]

   9. Taylor expanded in y around 0

      \[\leadsto y \cdot \color{blue}{\left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)} - t \]
   10. Step-by-step derivation

   11. Applied rewrites 94.3%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.25 \cdot 10^{-104} \lor \neg \left(x \leq 1.96 \cdot 10^{-131}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot \mathsf{fma}\left(-0.5, y, -1\right)\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+128} \lor \neg \left(x \leq 5.4 \cdot 10^{+74}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.9e+128) (not (<= x 5.4e+74)))
   (* (log y) x)
   (-
    (* (* (- (* (- (* (- (* -0.25 y) 0.3333333333333333) y) 0.5) y) 1.0) y) z)
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.9e+128) || !(x <= 5.4e+74)) {
		tmp = log(y) * x;
	} else {
		tmp = ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * 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 <= (-1.9d+128)) .or. (.not. (x <= 5.4d+74))) then
        tmp = log(y) * x
    else
        tmp = (((((((((-0.25d0) * y) - 0.3333333333333333d0) * y) - 0.5d0) * y) - 1.0d0) * 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 <= -1.9e+128) || !(x <= 5.4e+74)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1.9e+128) or not (x <= 5.4e+74):
		tmp = math.log(y) * x
	else:
		tmp = ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.9e+128) || !(x <= 5.4e+74))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1.9e+128) || ~((x <= 5.4e+74)))
		tmp = log(y) * x;
	else
		tmp = ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.9e+128], N[Not[LessEqual[x, 5.4e+74]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(N[(-0.25 * y), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.89999999999999995e128 or 5.3999999999999996e74 < x

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(x \cdot \log y + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right)} - t \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\color{blue}{\log y \cdot x} + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

      2. remove-double-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)} \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

      3. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \log y}\right)\right) \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(-1 \cdot \left(\mathsf{neg}\left(\log y\right)\right)\right)} \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

      5. log-rec N/A

         \[\leadsto \left(\left(-1 \cdot \color{blue}{\log \left(\frac{1}{y}\right)}\right) \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right) - t \]

      6. lower-fma.f64 N/A

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

      7. log-rec N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-log.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. +-commutative N/A

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

      15. associate-*r* N/A

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

      16. distribute-rgt-out N/A

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

      17. lower-*.f64 N/A

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

      18. lower-fma.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 80.2

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

   8. Applied rewrites 80.2%

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

   if -1.89999999999999995e128 < x < 5.3999999999999996e74

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower--.f64 57.8

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

   5. Applied rewrites 57.8%

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

   6. Taylor expanded in y around 0

      \[\leadsto \left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right) \cdot z - t \]
   7. Step-by-step derivation

   8. Applied rewrites 79.7%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+128} \lor \neg \left(x \leq 5.4 \cdot 10^{+74}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. *-commutative N/A

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

   9. cancel-sign-sub N/A

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

   10. distribute-lft-neg-out N/A

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

   11. *-commutative N/A

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

   12. associate--l- N/A

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

   13. lower--.f64 N/A

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

5. Applied rewrites 99.5%

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

Alternative 5: 58.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (* (* (- (* (- (* (- (* -0.25 y) 0.3333333333333333) y) 0.5) y) 1.0) y) z)
  t))

C

double code(double x, double y, double z, double t) {
	return ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * 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 = (((((((((-0.25d0) * y) - 0.3333333333333333d0) * y) - 0.5d0) * y) - 1.0d0) * y) * z) - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t;
}

Python

def code(x, y, z, t):
	return ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = ((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y) * z) - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(-0.25 * y), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower--.f64 45.3

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

5. Applied rewrites 45.3%

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

6. Taylor expanded in y around 0

   \[\leadsto \left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right) \cdot z - t \]
7. Step-by-step derivation

8. Applied rewrites 62.4%

   \[\leadsto \left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y\right) \cdot z - t \]
9. Add Preprocessing

Alternative 6: 58.6% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower--.f64 45.3

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

5. Applied rewrites 45.3%

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

6. Taylor expanded in y around 0

   \[\leadsto \left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right) \cdot z - t \]
7. Step-by-step derivation

8. Applied rewrites 62.4%

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

Alternative 7: 58.5% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower--.f64 45.3

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

5. Applied rewrites 45.3%

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

6. Taylor expanded in y around 0

   \[\leadsto \left(y \cdot \left(\frac{-1}{2} \cdot y - 1\right)\right) \cdot z - t \]
7. Step-by-step derivation

8. Applied rewrites 62.4%

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

Alternative 8: 49.2% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-31} \lor \neg \left(t \leq 1.4 \cdot 10^{-48}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.8e-31) (not (<= t 1.4e-48))) (- t) (- (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.8e-31) || !(t <= 1.4e-48)) {
		tmp = -t;
	} else {
		tmp = -(z * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-3.8d-31)) .or. (.not. (t <= 1.4d-48))) then
        tmp = -t
    else
        tmp = -(z * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.8e-31) || !(t <= 1.4e-48)) {
		tmp = -t;
	} else {
		tmp = -(z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.8e-31) or not (t <= 1.4e-48):
		tmp = -t
	else:
		tmp = -(z * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.8e-31) || !(t <= 1.4e-48))
		tmp = Float64(-t);
	else
		tmp = Float64(-Float64(z * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -3.8e-31) || ~((t <= 1.4e-48)))
		tmp = -t;
	else
		tmp = -(z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.8e-31], N[Not[LessEqual[t, 1.4e-48]], $MachinePrecision]], (-t), (-N[(z * y), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if t < -3.8e-31 or 1.40000000000000002e-48 < t

   1. Initial program 93.1%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

      2. lower-neg.f64 70.9

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

   5. Applied rewrites 70.9%

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

   if -3.8e-31 < t < 1.40000000000000002e-48

   1. Initial program 69.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. distribute-lft-neg-out N/A

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

      11. *-commutative N/A

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

      12. associate--l- N/A

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

      13. lower--.f64 N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.3%

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

   9. Taylor expanded in y around inf

      \[\leadsto -y \cdot z \]
   10. Step-by-step derivation

   11. Applied rewrites 32.6%

       \[\leadsto -z \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-31} \lor \neg \left(t \leq 1.4 \cdot 10^{-48}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.5% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-log.f64 N/A

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

   4. lower--.f64 45.3

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

5. Applied rewrites 45.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 62.4%

   \[\leadsto \left(-y\right) \cdot z - t \]

9. Taylor expanded in y around 0

   \[\leadsto y \cdot \color{blue}{\left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)} - t \]
10. Step-by-step derivation

11. Applied rewrites 62.4%

    \[\leadsto \left(z \cdot \mathsf{fma}\left(-0.5, y, -1\right)\right) \cdot \color{blue}{y} - t \]
12. Add Preprocessing

Alternative 10: 58.2% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. *-commutative N/A

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

   9. cancel-sign-sub N/A

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

   10. distribute-lft-neg-out N/A

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

   11. *-commutative N/A

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

   12. associate--l- N/A

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

   13. lower--.f64 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 62.4%

   \[\leadsto -\mathsf{fma}\left(z, y, t\right) \]
9. Add Preprocessing

Alternative 11: 43.6% accurate, 73.3× 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 82.2%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(t\right)} \]

   2. lower-neg.f64 44.6

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

5. Applied rewrites 44.6%

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

Developer Target 1: 99.6% accurate, 1.3× 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 2024338 
(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))