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

Percentage Accurate: 85.1% → 99.6%

Time: 12.9s

Alternatives: 14

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left({\left(\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)\right)}^{-1}\right)}^{-1} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (pow (pow (fma (log1p (- y)) z (fma (log y) x (- t))) -1.0) -1.0))

C

double code(double x, double y, double z, double t) {
	return pow(pow(fma(log1p(-y), z, fma(log(y), x, -t)), -1.0), -1.0);
}

Julia

function code(x, y, z, t)
	return (fma(log1p(Float64(-y)), z, fma(log(y), x, Float64(-t))) ^ -1.0) ^ -1.0
end

Wolfram

code[x_, y_, z_, t_] := N[Power[N[Power[N[(N[Log[1 + (-y)], $MachinePrecision] * z + N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 84.6%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 84.4

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)}}} \]

5. Final simplification 99.7%

   \[\leadsto {\left({\left(\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \mathsf{fma}\left(\log y, x, -t\right)\right)\right)}^{-1}\right)}^{-1} \]
6. Add Preprocessing

Alternative 2: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.6%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. mul-1-neg N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. lower-fma.f64 N/A

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

   8. log-rec N/A

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

   9. mul-1-neg N/A

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

   10. mul-1-neg N/A

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

   11. mul-1-neg N/A

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

   12. remove-double-neg N/A

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

   13. lower-log.f64 N/A

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

   14. *-commutative N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

5. Applied rewrites 99.4%

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

Alternative 3: 90.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-44} \lor \neg \left(x \leq 5.2 \cdot 10^{-105}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.5e-44) (not (<= x 5.2e-105)))
   (fma (log y) x (- t))
   (- (* (log1p (- y)) z) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.5e-44) || !(x <= 5.2e-105)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = (log1p(-y) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.5e-44) || !(x <= 5.2e-105))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = Float64(Float64(log1p(Float64(-y)) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.5e-44], N[Not[LessEqual[x, 5.2e-105]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[(N[(N[Log[1 + (-y)], $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.50000000000000019e-44 or 5.1999999999999997e-105 < x

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. lower-fma.f64 N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-log.f64 N/A

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

      15. lower-neg.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if -2.50000000000000019e-44 < x < 5.1999999999999997e-105

   1. Initial program 68.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} - t \]
   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. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 93.2

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

   5. Applied rewrites 93.2%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-44} \lor \neg \left(x \leq 5.2 \cdot 10^{-105}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-y\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-44} \lor \neg \left(x \leq 5.2 \cdot 10^{-105}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5 \cdot z\right), y, -z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.5e-44) (not (<= x 5.2e-105)))
   (fma (log y) x (- t))
   (-
    (*
     (fma (fma (* z (fma -0.25 y -0.3333333333333333)) y (* -0.5 z)) y (- z))
     y)
    t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.5e-44) || !(x <= 5.2e-105)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = (fma(fma((z * fma(-0.25, y, -0.3333333333333333)), y, (-0.5 * z)), y, -z) * y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.5e-44) || !(x <= 5.2e-105))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = Float64(Float64(fma(fma(Float64(z * fma(-0.25, y, -0.3333333333333333)), y, Float64(-0.5 * z)), y, Float64(-z)) * y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.5e-44], N[Not[LessEqual[x, 5.2e-105]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[(N[(N[(N[(N[(z * N[(-0.25 * y + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * y + N[(-0.5 * z), $MachinePrecision]), $MachinePrecision] * y + (-z)), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.50000000000000019e-44 or 5.1999999999999997e-105 < x

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. remove-double-neg N/A

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

      4. mul-1-neg N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. lower-fma.f64 N/A

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. mul-1-neg N/A

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-log.f64 N/A

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

      15. lower-neg.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if -2.50000000000000019e-44 < x < 5.1999999999999997e-105

   1. Initial program 68.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{z \cdot \log \left(1 - y\right)} - t \]
   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. sub-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-neg.f64 93.2

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

   5. Applied rewrites 93.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 93.0%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-44} \lor \neg \left(x \leq 5.2 \cdot 10^{-105}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5 \cdot z\right), y, -z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.6%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. *-commutative N/A

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

   3. remove-double-neg N/A

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

   4. mul-1-neg N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. log-rec N/A

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

   8. *-commutative N/A

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

   9. associate-*r* N/A

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

   10. lower-fma.f64 N/A

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

   11. log-rec N/A

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

   12. mul-1-neg N/A

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

   13. mul-1-neg N/A

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

   14. mul-1-neg N/A

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

   15. remove-double-neg N/A

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

   16. lower-log.f64 N/A

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

   17. mul-1-neg N/A

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

   18. distribute-lft-neg-in N/A

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

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

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

   20. *-commutative N/A

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

   21. lower-*.f64 N/A

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

   22. lower-neg.f64 98.9

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

5. Applied rewrites 98.9%

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

Alternative 6: 99.1% 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 84.6%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

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

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

   7. neg-mul-1 N/A

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

   8. mul-1-neg N/A

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

   9. log-rec N/A

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

   10. associate--l- N/A

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

   11. lower--.f64 N/A

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

5. Applied rewrites 98.9%

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

Alternative 7: 57.6% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.6%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 59.3

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

5. Applied rewrites 59.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 59.2%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(z \cdot \mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5 \cdot z\right), y, -z\right) \cdot \color{blue}{y} - t \]
9. Add Preprocessing

Alternative 8: 57.6% accurate, 6.9× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (* (* (fma (fma (fma -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 ((fma(fma(fma(-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(fma(fma(fma(-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[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 84.6%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 59.3

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

5. Applied rewrites 59.3%

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

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 59.2%

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

Alternative 9: 57.6% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.6%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 59.3

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

5. Applied rewrites 59.3%

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

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 59.1%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t \]
9. Step-by-step derivation

10. Applied rewrites 59.1%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right) \cdot y, y, -y\right) \cdot z - t \]
11. Add Preprocessing

Alternative 10: 57.6% accurate, 8.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.6%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 59.3

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

5. Applied rewrites 59.3%

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

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 59.1%

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

Alternative 11: 48.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-42} \lor \neg \left(t \leq 3.3 \cdot 10^{-103}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -3.3e-42) (not (<= t 3.3e-103))) (- t) (- (* z y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -3.3e-42) || !(t <= 3.3e-103)) {
		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 <= (-3.3d-42)) .or. (.not. (t <= 3.3d-103))) 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 <= -3.3e-42) || !(t <= 3.3e-103)) {
		tmp = -t;
	} else {
		tmp = -(z * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -3.3e-42) or not (t <= 3.3e-103):
		tmp = -t
	else:
		tmp = -(z * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -3.3e-42) || !(t <= 3.3e-103))
		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 <= -3.3e-42) || ~((t <= 3.3e-103)))
		tmp = -t;
	else
		tmp = -(z * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -3.3e-42], N[Not[LessEqual[t, 3.3e-103]], $MachinePrecision]], (-t), (-N[(z * y), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if t < -3.3000000000000002e-42 or 3.2999999999999999e-103 < 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 64.7

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

   5. Applied rewrites 64.7%

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

   if -3.3000000000000002e-42 < t < 3.2999999999999999e-103

   1. Initial program 71.9%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

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

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

      7. neg-mul-1 N/A

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

      8. mul-1-neg N/A

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

      9. log-rec N/A

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

      10. associate--l- N/A

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

      11. lower--.f64 N/A

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

   5. Applied rewrites 98.6%

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

   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 38.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 30.9%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-42} \lor \neg \left(t \leq 3.3 \cdot 10^{-103}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;-z \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.5% 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 84.6%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. sub-neg N/A

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

   4. lower-log1p.f64 N/A

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

   5. lower-neg.f64 59.3

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

5. Applied rewrites 59.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 58.8%

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

Alternative 13: 57.1% 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 84.6%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. remove-double-neg N/A

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

   5. mul-1-neg N/A

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

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

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

   7. neg-mul-1 N/A

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

   8. mul-1-neg N/A

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

   9. log-rec N/A

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

   10. associate--l- N/A

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

   11. lower--.f64 N/A

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

5. Applied rewrites 98.9%

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

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 58.4%

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

Alternative 14: 42.5% 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 84.6%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 43.0

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

5. Applied rewrites 43.0%

   \[\leadsto \color{blue}{-t} \]
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 2024321 
(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))