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

Percentage Accurate: 85.5% → 99.8%

Time: 13.2s

Alternatives: 16

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: 85.5% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. *-lft-identity N/A

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

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

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

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

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

   13. *-lft-identity N/A

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

   14. lower-log1p.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower--.f64 99.8

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (-
  (+
   (* x (log y))
   (*
    (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 ((x * log(y)) + (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(Float64(x * log(y)) + 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[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 80.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

      \[\leadsto \left(x \cdot \log y + \color{blue}{\left(-1 \cdot z + y \cdot \left(\frac{-1}{2} \cdot z + y \cdot \left(\frac{-1}{3} \cdot z + \frac{-1}{4} \cdot \left(y \cdot z\right)\right)\right)\right) \cdot y}\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(x \cdot \log y + \color{blue}{\left(-1 \cdot z + y \cdot \left(\frac{-1}{2} \cdot z + y \cdot \left(\frac{-1}{3} \cdot z + \frac{-1}{4} \cdot \left(y \cdot z\right)\right)\right)\right) \cdot y}\right) - t \]

5. Applied rewrites 99.7%

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

Alternative 3: 99.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * (((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 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 * ((((((((-0.25d0) * y) - 0.3333333333333333d0) * y) - 0.5d0) * y) - 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 * (((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y))) - t;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   2. lower-*.f64 N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y\right)}\right) - t \]

   3. lower--.f64 N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} \cdot y\right)\right) - t \]

   4. *-commutative N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\left(\color{blue}{\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot y} - 1\right) \cdot y\right)\right) - t \]

   5. lower-*.f64 N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\left(\color{blue}{\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot y} - 1\right) \cdot y\right)\right) - t \]

   6. lower--.f64 N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\left(\color{blue}{\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)} \cdot y - 1\right) \cdot y\right)\right) - t \]

   7. *-commutative N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) \cdot y} - \frac{1}{2}\right) \cdot y - 1\right) \cdot y\right)\right) - t \]

   8. lower-*.f64 N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) \cdot y} - \frac{1}{2}\right) \cdot y - 1\right) \cdot y\right)\right) - t \]

   9. lower--.f64 N/A

      \[\leadsto \left(x \cdot \log y + z \cdot \left(\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)} \cdot y - \frac{1}{2}\right) \cdot y - 1\right) \cdot y\right)\right) - t \]

   10. lower-*.f64 99.7

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

5. Applied rewrites 99.7%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y\right)}\right) - t \]
6. Add Preprocessing

Alternative 4: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

5. Applied rewrites 99.6%

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

Alternative 5: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

   8. lower-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. remove-double-neg N/A

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

   11. mul-1-neg N/A

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

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

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

   13. mul-1-neg N/A

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

   14. log-rec N/A

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

   15. mul-1-neg N/A

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

   16. *-rgt-identity N/A

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

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

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

5. Applied rewrites 99.4%

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

Alternative 6: 89.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{-21} \lor \neg \left(x \leq 3.4 \cdot 10^{-138}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.42e-21) (not (<= x 3.4e-138)))
   (fma (log y) x (- t))
   (fma (log1p (- y)) z (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1.42e-21) || !(x <= 3.4e-138)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = fma(log1p(-y), z, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.42e-21) || !(x <= 3.4e-138))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = fma(log1p(Float64(-y)), z, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.42e-21], N[Not[LessEqual[x, 3.4e-138]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[(N[Log[1 + (-y)], $MachinePrecision] * z + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.42e-21 or 3.4000000000000001e-138 < x

   1. Initial program 90.9%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. log-rec N/A

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

      15. mul-1-neg N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 90.0%

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

   if -1.42e-21 < x < 3.4000000000000001e-138

   1. Initial program 66.8%

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

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. *-lft-identity N/A

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

      14. lower-log1p.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower--.f64 99.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, \color{blue}{-1 \cdot t}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 91.0

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

   7. Applied rewrites 91.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{-21} \lor \neg \left(x \leq 3.4 \cdot 10^{-138}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{-21} \lor \neg \left(x \leq 3.4 \cdot 10^{-138}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y, z, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1.42e-21) (not (<= x 3.4e-138)))
   (fma (log y) x (- t))
   (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) {
	double tmp;
	if ((x <= -1.42e-21) || !(x <= 3.4e-138)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = fma((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y), z, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1.42e-21) || !(x <= 3.4e-138))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y), z, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1.42e-21], N[Not[LessEqual[x, 3.4e-138]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(-0.25 * y), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.42e-21 or 3.4000000000000001e-138 < x

   1. Initial program 90.9%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. log-rec N/A

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

      15. mul-1-neg N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 90.0%

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

   if -1.42e-21 < x < 3.4000000000000001e-138

   1. Initial program 66.8%

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

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. *-lft-identity N/A

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

      14. lower-log1p.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower--.f64 99.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 89.9

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

   10. Applied rewrites 89.9%

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

   11. Taylor expanded in y around 0

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

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y}, z, -t\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y}, z, -t\right) \]

       3. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} \cdot y, z, -t\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot y} - 1\right) \cdot y, z, -t\right) \]

       5. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot y} - 1\right) \cdot y, z, -t\right) \]

       6. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)} \cdot y - 1\right) \cdot y, z, -t\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) \cdot y} - \frac{1}{2}\right) \cdot y - 1\right) \cdot y, z, -t\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) \cdot y} - \frac{1}{2}\right) \cdot y - 1\right) \cdot y, z, -t\right) \]

       9. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)} \cdot y - \frac{1}{2}\right) \cdot y - 1\right) \cdot y, z, -t\right) \]

       10. lower-*.f64 91.0

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

   13. Applied rewrites 91.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{-21} \lor \neg \left(x \leq 3.4 \cdot 10^{-138}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y, z, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. *-lft-identity N/A

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

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

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

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

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

   13. *-lft-identity N/A

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

   14. lower-log1p.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower--.f64 99.8

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.0

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

7. Applied rewrites 99.0%

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

8. Final simplification 99.0%

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

Alternative 9: 99.0% 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 80.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. *-commutative N/A

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

   9. cancel-sign-sub N/A

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

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

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

   11. *-commutative N/A

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

   12. associate--l- N/A

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

   13. lower--.f64 N/A

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

5. Applied rewrites 99.0%

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

Alternative 10: 58.1% accurate, 5.8× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. *-lft-identity N/A

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

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

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

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

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

   13. *-lft-identity N/A

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

   14. lower-log1p.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower--.f64 99.8

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 55.4

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

10. Applied rewrites 55.4%

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

11. Taylor expanded in y around 0

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

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y}, z, -t\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right) \cdot y}, z, -t\right) \]

    3. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)} \cdot y, z, -t\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot y} - 1\right) \cdot y, z, -t\right) \]

    5. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) \cdot y} - 1\right) \cdot y, z, -t\right) \]

    6. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\color{blue}{\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)} \cdot y - 1\right) \cdot y, z, -t\right) \]

    7. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) \cdot y} - \frac{1}{2}\right) \cdot y - 1\right) \cdot y, z, -t\right) \]

    8. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) \cdot y} - \frac{1}{2}\right) \cdot y - 1\right) \cdot y, z, -t\right) \]

    9. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\left(\color{blue}{\left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)} \cdot y - \frac{1}{2}\right) \cdot y - 1\right) \cdot y, z, -t\right) \]

    10. lower-*.f64 56.1

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

13. Applied rewrites 56.1%

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

14. Final simplification 56.1%

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

Alternative 11: 58.0% accurate, 7.3× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. *-lft-identity N/A

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

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

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

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

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

   13. *-lft-identity N/A

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

   14. lower-log1p.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower--.f64 99.8

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 55.4

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

10. Applied rewrites 55.4%

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

11. Taylor expanded in y around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower--.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. lower--.f64 N/A

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

    7. lower-*.f64 55.9

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

13. Applied rewrites 55.9%

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

14. Final simplification 55.9%

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

Alternative 12: 57.9% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. *-lft-identity N/A

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

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

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

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

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

   13. *-lft-identity N/A

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

   14. lower-log1p.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower--.f64 99.8

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 55.4

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

10. Applied rewrites 55.4%

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

11. Taylor expanded in y around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower--.f64 N/A

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

    4. lower-*.f64 55.7

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

13. Applied rewrites 55.7%

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

14. Final simplification 55.7%

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

Alternative 13: 57.9% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.7%

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

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

   8. lower-*.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. remove-double-neg N/A

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

   11. mul-1-neg N/A

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

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

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

   13. mul-1-neg N/A

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

   14. log-rec N/A

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

   15. mul-1-neg N/A

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

   16. *-rgt-identity N/A

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

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

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 55.7%

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

Alternative 14: 48.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-49} \lor \neg \left(t \leq 6.2 \cdot 10^{-63}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.02e-49) (not (<= t 6.2e-63))) (- t) (* (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.02e-49) || !(t <= 6.2e-63)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	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 <= (-1.02d-49)) .or. (.not. (t <= 6.2d-63))) then
        tmp = -t
    else
        tmp = -y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.02e-49) || !(t <= 6.2e-63)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.02e-49) or not (t <= 6.2e-63):
		tmp = -t
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.02e-49) || !(t <= 6.2e-63))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.02e-49) || ~((t <= 6.2e-63)))
		tmp = -t;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.02e-49], N[Not[LessEqual[t, 6.2e-63]], $MachinePrecision]], (-t), N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.02000000000000009e-49 or 6.19999999999999968e-63 < t

   1. Initial program 91.3%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 57.4

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

   5. Applied rewrites 57.4%

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

   if -1.02000000000000009e-49 < t < 6.19999999999999968e-63

   1. Initial program 68.0%

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

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

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

      11. *-commutative N/A

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

      12. associate--l- N/A

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

      13. lower--.f64 N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 33.4%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-49} \lor \neg \left(t \leq 6.2 \cdot 10^{-63}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.6% 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 80.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. *-commutative N/A

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

   9. cancel-sign-sub N/A

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

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

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

   11. *-commutative N/A

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

   12. associate--l- N/A

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

   13. lower--.f64 N/A

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

5. Applied rewrites 99.0%

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

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 55.4%

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

Alternative 16: 43.4% 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 80.7%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 36.7

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

5. Applied rewrites 36.7%

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

Developer Target 1: 99.5% 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 2024329 
(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))