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

Percentage Accurate: 85.7% → 99.8%

Time: 12.6s

Alternatives: 14

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-log1p.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-log1p.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\color{blue}{y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)}, z, -t\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 3: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-log1p.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{fma}\left(\log y, x, \mathsf{fma}\left(\color{blue}{y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)}, z, -t\right)\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. +-commutative N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. lower-*.f64 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-neg.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. distribute-rgt-out N/A

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

   10. +-commutative N/A

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

   11. metadata-eval N/A

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

   12. sub-neg N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. sub-neg N/A

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

   16. metadata-eval N/A

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

   17. lower-fma.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. lower-log.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 89.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -9.6e-144)
   (fma (log y) x (- t))
   (if (<= x 3.3e-91) (fma (log1p (- y)) z (- t)) (- (* x (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -9.6e-144) {
		tmp = fma(log(y), x, -t);
	} else if (x <= 3.3e-91) {
		tmp = fma(log1p(-y), z, -t);
	} else {
		tmp = (x * log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -9.6e-144)
		tmp = fma(log(y), x, Float64(-t));
	elseif (x <= 3.3e-91)
		tmp = fma(log1p(Float64(-y)), z, Float64(-t));
	else
		tmp = Float64(Float64(x * log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -9.6e-144], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], If[LessEqual[x, 3.3e-91], N[(N[Log[1 + (-y)], $MachinePrecision] * z + (-t)), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.59999999999999975e-144

   1. Initial program 93.1%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-log1p.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.8

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

   7. Applied rewrites 92.8%

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

   if -9.59999999999999975e-144 < x < 3.30000000000000011e-91

   1. Initial program 75.0%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-neg.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 3.30000000000000011e-91 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -9.6e-144)
   (fma (log y) x (- t))
   (if (<= x 3.3e-91)
     (fma (* (fma -0.5 y -1.0) z) y (- t))
     (- (* x (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -9.6e-144) {
		tmp = fma(log(y), x, -t);
	} else if (x <= 3.3e-91) {
		tmp = fma((fma(-0.5, y, -1.0) * z), y, -t);
	} else {
		tmp = (x * log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -9.6e-144)
		tmp = fma(log(y), x, Float64(-t));
	elseif (x <= 3.3e-91)
		tmp = fma(Float64(fma(-0.5, y, -1.0) * z), y, Float64(-t));
	else
		tmp = Float64(Float64(x * log(y)) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -9.6e-144], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], If[LessEqual[x, 3.3e-91], N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * z), $MachinePrecision] * y + (-t)), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.59999999999999975e-144

   1. Initial program 93.1%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-log.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. lower-log1p.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-neg.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 92.8

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

   7. Applied rewrites 92.8%

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

   if -9.59999999999999975e-144 < x < 3.30000000000000011e-91

   1. Initial program 75.0%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-neg.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 96.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 97.1%

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

   if 3.30000000000000011e-91 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 94.3

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

   5. Applied rewrites 94.3%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-144}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -9.6 \cdot 10^{-144}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -9.6e-144)
     t_1
     (if (<= x 3.3e-91) (fma (* (fma -0.5 y -1.0) z) y (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -9.6e-144) {
		tmp = t_1;
	} else if (x <= 3.3e-91) {
		tmp = fma((fma(-0.5, y, -1.0) * z), y, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - t)
	tmp = 0.0
	if (x <= -9.6e-144)
		tmp = t_1;
	elseif (x <= 3.3e-91)
		tmp = fma(Float64(fma(-0.5, y, -1.0) * z), y, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -9.6e-144], t$95$1, If[LessEqual[x, 3.3e-91], N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * z), $MachinePrecision] * y + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.59999999999999975e-144 or 3.30000000000000011e-91 < x

   1. Initial program 94.1%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 93.5

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

   5. Applied rewrites 93.5%

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

   if -9.59999999999999975e-144 < x < 3.30000000000000011e-91

   1. Initial program 75.0%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-neg.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 96.7%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 97.1%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-144}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-91}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -9.8 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -9.8e-11)
     t_1
     (if (<= x 7.8e+133) (fma (* (fma -0.5 y -1.0) z) y (- t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -9.8e-11) {
		tmp = t_1;
	} else if (x <= 7.8e+133) {
		tmp = fma((fma(-0.5, y, -1.0) * z), y, -t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -9.8e-11)
		tmp = t_1;
	elseif (x <= 7.8e+133)
		tmp = fma(Float64(fma(-0.5, y, -1.0) * z), y, Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -9.8e-11], t$95$1, If[LessEqual[x, 7.8e+133], N[(N[(N[(-0.5 * y + -1.0), $MachinePrecision] * z), $MachinePrecision] * y + (-t)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.7999999999999998e-11 or 7.80000000000000028e133 < x

   1. Initial program 96.3%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. 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 81.0

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

   5. Applied rewrites 81.0%

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

   if -9.7999999999999998e-11 < x < 7.80000000000000028e133

   1. Initial program 81.9%

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

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-neg.f64 83.3

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 83.3%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.8 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{+133}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-0.5, y, -1\right) \cdot z, y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-log.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-neg N/A

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

   14. lower-log1p.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower-neg.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. distribute-neg-out N/A

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

   4. lower-neg.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 99.3

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

7. Applied rewrites 99.3%

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

Alternative 10: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

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

   5. lower--.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-log.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Final simplification 99.3%

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

Alternative 11: 48.6% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.55 \cdot 10^{-101}:\\ \;\;\;\;-t\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-103}:\\ \;\;\;\;-z \cdot y\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.55e-101) (- t) (if (<= t 1.65e-103) (- (* z y)) (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.55e-101) {
		tmp = -t;
	} else if (t <= 1.65e-103) {
		tmp = -(z * y);
	} else {
		tmp = -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 (t <= (-3.55d-101)) then
        tmp = -t
    else if (t <= 1.65d-103) then
        tmp = -(z * y)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.55e-101) {
		tmp = -t;
	} else if (t <= 1.65e-103) {
		tmp = -(z * y);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.55e-101:
		tmp = -t
	elif t <= 1.65e-103:
		tmp = -(z * y)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.55e-101)
		tmp = Float64(-t);
	elseif (t <= 1.65e-103)
		tmp = Float64(-Float64(z * y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.55e-101)
		tmp = -t;
	elseif (t <= 1.65e-103)
		tmp = -(z * y);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.55e-101], (-t), If[LessEqual[t, 1.65e-103], (-N[(z * y), $MachinePrecision]), (-t)]]

Derivation

1. Split input into 2 regimes

2. if t < -3.55e-101 or 1.64999999999999995e-103 < t

   1. Initial program 93.8%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 62.7

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

   5. Applied rewrites 62.7%

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

   if -3.55e-101 < t < 1.64999999999999995e-103

   1. Initial program 76.1%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. lower-log1p.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-neg.f64 31.6

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

   5. Applied rewrites 31.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 31.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 27.0%

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

Alternative 12: 57.7% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. lower-log1p.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-neg.f64 56.3

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

5. Applied rewrites 56.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 56.2%

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

9. Taylor expanded in y around 0

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

11. Applied rewrites 56.3%

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

12. Final simplification 56.3%

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

Alternative 13: 57.3% 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 88.0%

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

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. lower-log1p.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-neg.f64 56.3

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

5. Applied rewrites 56.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 56.2%

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

Alternative 14: 43.2% 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 88.0%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 44.5

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

5. Applied rewrites 44.5%

   \[\leadsto \color{blue}{-t} \]
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 2024268 
(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))