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

Percentage Accurate: 85.1% → 99.6%

Time: 14.2s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (z * (y * ((-1.0d0) + (y * ((-0.5d0) + (y * ((-0.3333333333333333d0) + (y * (-0.25d0)))))))))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right)\right), t\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)\right)\right), t\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), t\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right)\right)\right), t\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(-1 + y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \frac{-1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{2} + y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)\right)\right)\right)\right)\right)\right), t\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y + \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

   14. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} + \frac{-1}{4} \cdot y\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{3}, \left(\frac{-1}{4} \cdot y\right)\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{3}, \left(y \cdot \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

   17. *-lowering-*.f64 99.3%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{3}, \mathsf{*.f64}\left(y, \frac{-1}{4}\right)\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

5. Simplified 99.3%

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

Alternative 2: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x * log(y)) + (z * (y * ((-1.0d0) + (y * ((-0.5d0) + (y * (-0.3333333333333333d0)))))))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right)\right), t\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right)\right), t\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), t\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + -1\right)\right)\right)\right), t\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(-1 + y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right), t\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right)\right)\right)\right), t\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \frac{-1}{2}\right)\right)\right)\right)\right)\right), t\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \left(\frac{-1}{2} + \frac{-1}{3} \cdot y\right)\right)\right)\right)\right)\right), t\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{-1}{3} \cdot y\right)\right)\right)\right)\right)\right)\right), t\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \left(y \cdot \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right), t\right) \]

   12. *-lowering-*.f64 99.2%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right)\right)\right)\right), t\right) \]

5. Simplified 99.2%

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

Alternative 3: 89.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -1700000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-144}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-1 + y \cdot \left(-0.5 + y \cdot \left(-0.3333333333333333 + y \cdot -0.25\right)\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -1700000.0)
     t_1
     (if (<= x 3.6e-144)
       (-
        (*
         z
         (*
          y
          (+ -1.0 (* y (+ -0.5 (* y (+ -0.3333333333333333 (* y -0.25))))))))
        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 <= -1700000.0) {
		tmp = t_1;
	} else if (x <= 3.6e-144) {
		tmp = (z * (y * (-1.0 + (y * (-0.5 + (y * (-0.3333333333333333 + (y * -0.25)))))))) - t;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (x * log(y)) - t
    if (x <= (-1700000.0d0)) then
        tmp = t_1
    else if (x <= 3.6d-144) then
        tmp = (z * (y * ((-1.0d0) + (y * ((-0.5d0) + (y * ((-0.3333333333333333d0) + (y * (-0.25d0))))))))) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - t;
	double tmp;
	if (x <= -1700000.0) {
		tmp = t_1;
	} else if (x <= 3.6e-144) {
		tmp = (z * (y * (-1.0 + (y * (-0.5 + (y * (-0.3333333333333333 + (y * -0.25)))))))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -1700000.0:
		tmp = t_1
	elif x <= 3.6e-144:
		tmp = (z * (y * (-1.0 + (y * (-0.5 + (y * (-0.3333333333333333 + (y * -0.25)))))))) - 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 <= -1700000.0)
		tmp = t_1;
	elseif (x <= 3.6e-144)
		tmp = Float64(Float64(z * Float64(y * Float64(-1.0 + Float64(y * Float64(-0.5 + Float64(y * Float64(-0.3333333333333333 + Float64(y * -0.25)))))))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - t;
	tmp = 0.0;
	if (x <= -1700000.0)
		tmp = t_1;
	elseif (x <= 3.6e-144)
		tmp = (z * (y * (-1.0 + (y * (-0.5 + (y * (-0.3333333333333333 + (y * -0.25)))))))) - t;
	else
		tmp = t_1;
	end
	tmp_2 = 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, -1700000.0], t$95$1, If[LessEqual[x, 3.6e-144], N[(N[(z * N[(y * N[(-1.0 + N[(y * N[(-0.5 + N[(y * N[(-0.3333333333333333 + N[(y * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e6 or 3.6e-144 < x

   1. Initial program 96.2%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

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

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

      4. mul-1-neg N/A

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

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right), \color{blue}{t}\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(x \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), t\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), t\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y\right), t\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), t\right) \]

      11. log-lowering-log.f64 94.5%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), t\right) \]

   5. Simplified 94.5%

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

   if -1.7e6 < x < 3.6e-144

   1. Initial program 70.3%

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

   3. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \log \left(1 - y\right)\right), \color{blue}{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \log \left(1 - y\right)\right), t\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\left(1 - y\right)\right)\right), t\right) \]

      4. --lowering--.f64 62.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, y\right)\right)\right), t\right) \]

   5. Simplified 62.4%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right), t\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)\right), t\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right)\right), t\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right)\right), t\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y + \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{-1}{4} \cdot y\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      14. *-lowering-*.f64 92.1%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   8. Simplified 92.1%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1700000:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-144}:\\ \;\;\;\;z \cdot \left(y \cdot \left(-1 + y \cdot \left(-0.5 + y \cdot \left(-0.3333333333333333 + y \cdot -0.25\right)\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(x \cdot \log y + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right)}, t\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\log y \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right), t\right) \]

   2. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right) \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right), t\right) \]

   3. log-rec N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right) \cdot x + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right), t\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right) \cdot x\right)\right) + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right), t\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(x \cdot \log \left(\frac{1}{y}\right)\right)\right) + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right), t\right) \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right) + y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right), t\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right) + -1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right), t\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\left(y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right), \left(-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right)\right), t\right) \]

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 5: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{+24}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 55000000000000:\\ \;\;\;\;z \cdot \left(y \cdot \left(-1 + y \cdot \left(-0.5 + y \cdot \left(-0.3333333333333333 + y \cdot -0.25\right)\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -3.8e+24)
     t_1
     (if (<= x 55000000000000.0)
       (-
        (*
         z
         (*
          y
          (+ -1.0 (* y (+ -0.5 (* y (+ -0.3333333333333333 (* y -0.25))))))))
        t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -3.8e+24) {
		tmp = t_1;
	} else if (x <= 55000000000000.0) {
		tmp = (z * (y * (-1.0 + (y * (-0.5 + (y * (-0.3333333333333333 + (y * -0.25)))))))) - t;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-3.8d+24)) then
        tmp = t_1
    else if (x <= 55000000000000.0d0) then
        tmp = (z * (y * ((-1.0d0) + (y * ((-0.5d0) + (y * ((-0.3333333333333333d0) + (y * (-0.25d0))))))))) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -3.8e+24) {
		tmp = t_1;
	} else if (x <= 55000000000000.0) {
		tmp = (z * (y * (-1.0 + (y * (-0.5 + (y * (-0.3333333333333333 + (y * -0.25)))))))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -3.8e+24:
		tmp = t_1
	elif x <= 55000000000000.0:
		tmp = (z * (y * (-1.0 + (y * (-0.5 + (y * (-0.3333333333333333 + (y * -0.25)))))))) - 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 <= -3.8e+24)
		tmp = t_1;
	elseif (x <= 55000000000000.0)
		tmp = Float64(Float64(z * Float64(y * Float64(-1.0 + Float64(y * Float64(-0.5 + Float64(y * Float64(-0.3333333333333333 + Float64(y * -0.25)))))))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -3.8e+24)
		tmp = t_1;
	elseif (x <= 55000000000000.0)
		tmp = (z * (y * (-1.0 + (y * (-0.5 + (y * (-0.3333333333333333 + (y * -0.25)))))))) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.8e+24], t$95$1, If[LessEqual[x, 55000000000000.0], N[(N[(z * N[(y * N[(-1.0 + N[(y * N[(-0.5 + N[(y * N[(-0.3333333333333333 + N[(y * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.80000000000000015e24 or 5.5e13 < x

   1. Initial program 97.2%

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

   3. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\log y}\right) \]

      2. log-lowering-log.f64 71.1%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right) \]

   5. Simplified 71.1%

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

   if -3.80000000000000015e24 < x < 5.5e13

   1. Initial program 76.5%

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

         \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \log \left(1 - y\right)\right), \color{blue}{t}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \log \left(1 - y\right)\right), t\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\left(1 - y\right)\right)\right), t\right) \]

      4. --lowering--.f64 68.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, y\right)\right)\right), t\right) \]

   5. Simplified 68.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right), t\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)\right), t\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right)\right), t\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right)\right), t\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y + \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{-1}{4} \cdot y\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

      14. *-lowering-*.f64 90.7%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   8. Simplified 90.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 55000000000000:\\ \;\;\;\;z \cdot \left(y \cdot \left(-1 + y \cdot \left(-0.5 + y \cdot \left(-0.3333333333333333 + y \cdot -0.25\right)\right)\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)}, t\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y + -1 \cdot \left(y \cdot z\right)\right), t\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y + \left(\mathsf{neg}\left(y \cdot z\right)\right)\right), t\right) \]

   3. unsub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y - y \cdot z\right), t\right) \]

   4. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right) - y \cdot z\right), t\right) \]

   5. log-rec N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right) - y \cdot z\right), t\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(x \cdot \log \left(\frac{1}{y}\right)\right)\right) - y \cdot z\right), t\right) \]

   7. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right) - y \cdot z\right), t\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right), \left(y \cdot z\right)\right), t\right) \]

   9. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(x \cdot \log \left(\frac{1}{y}\right)\right)\right), \left(y \cdot z\right)\right), t\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), \left(y \cdot z\right)\right), t\right) \]

   11. log-rec N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), \left(y \cdot z\right)\right), t\right) \]

   12. remove-double-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \log y\right), \left(y \cdot z\right)\right), t\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), \left(y \cdot z\right)\right), t\right) \]

   14. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \left(y \cdot z\right)\right), t\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \left(z \cdot y\right)\right), t\right) \]

   16. *-lowering-*.f64 98.6%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, y\right)\right), t\right) \]

5. Simplified 98.6%

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

6. Final simplification 98.6%

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

Alternative 7: 58.0% accurate, 11.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (z * (y * ((-1.0d0) + (y * ((-0.5d0) + (y * ((-0.3333333333333333d0) + (y * (-0.25d0))))))))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

      \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \log \left(1 - y\right)\right), \color{blue}{t}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \log \left(1 - y\right)\right), t\right) \]

   3. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\left(1 - y\right)\right)\right), t\right) \]

   4. --lowering--.f64 48.4%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, y\right)\right)\right), t\right) \]

5. Simplified 48.4%

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

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)}\right), t\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)\right)\right), t\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right) + -1\right)\right)\right), t\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) - \frac{1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right)\right), t\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right) + \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y - \frac{1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{4} \cdot y + \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   12. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{-1}{4} \cdot y\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   14. *-lowering-*.f64 60.5%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \frac{-1}{4}\right), \frac{-1}{3}\right)\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

8. Simplified 60.5%

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

9. Final simplification 60.5%

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

Alternative 8: 49.2% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-64}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -6.5e-64) (- 0.0 t) (if (<= t 6e-123) (- 0.0 (* y z)) (- 0.0 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.5e-64) {
		tmp = 0.0 - t;
	} else if (t <= 6e-123) {
		tmp = 0.0 - (y * z);
	} else {
		tmp = 0.0 - 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 <= (-6.5d-64)) then
        tmp = 0.0d0 - t
    else if (t <= 6d-123) then
        tmp = 0.0d0 - (y * z)
    else
        tmp = 0.0d0 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -6.5e-64) {
		tmp = 0.0 - t;
	} else if (t <= 6e-123) {
		tmp = 0.0 - (y * z);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -6.5e-64:
		tmp = 0.0 - t
	elif t <= 6e-123:
		tmp = 0.0 - (y * z)
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -6.5e-64)
		tmp = Float64(0.0 - t);
	elseif (t <= 6e-123)
		tmp = Float64(0.0 - Float64(y * z));
	else
		tmp = Float64(0.0 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -6.5e-64)
		tmp = 0.0 - t;
	elseif (t <= 6e-123)
		tmp = 0.0 - (y * z);
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -6.5e-64], N[(0.0 - t), $MachinePrecision], If[LessEqual[t, 6e-123], N[(0.0 - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(0.0 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.5000000000000004e-64 or 5.99999999999999968e-123 < t

   1. Initial program 93.4%

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

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{t} \]

      3. --lowering--.f64 64.5%

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{t}\right) \]

   5. Simplified 64.5%

      \[\leadsto \color{blue}{0 - t} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 64.5%

         \[\leadsto \mathsf{neg.f64}\left(t\right) \]

   7. Applied egg-rr 64.5%

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

   if -6.5000000000000004e-64 < t < 5.99999999999999968e-123

   1. Initial program 70.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)}, t\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y + -1 \cdot \left(y \cdot z\right)\right), t\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y + \left(\mathsf{neg}\left(y \cdot z\right)\right)\right), t\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \log y - y \cdot z\right), t\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right) - y \cdot z\right), t\right) \]

      5. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right) - y \cdot z\right), t\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(x \cdot \log \left(\frac{1}{y}\right)\right)\right) - y \cdot z\right), t\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right) - y \cdot z\right), t\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(x \cdot \log \left(\frac{1}{y}\right)\right)\right), \left(y \cdot z\right)\right), t\right) \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(x \cdot \log \left(\frac{1}{y}\right)\right)\right), \left(y \cdot z\right)\right), t\right) \]

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

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\log \left(\frac{1}{y}\right)\right)\right)\right), \left(y \cdot z\right)\right), t\right) \]

      11. log-rec N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log y\right)\right)\right)\right)\right), \left(y \cdot z\right)\right), t\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\left(x \cdot \log y\right), \left(y \cdot z\right)\right), t\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \log y\right), \left(y \cdot z\right)\right), t\right) \]

      14. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \left(y \cdot z\right)\right), t\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \left(z \cdot y\right)\right), t\right) \]

      16. *-lowering-*.f64 98.0%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, y\right)\right), t\right) \]

   5. Simplified 98.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \color{blue}{\left(-1 \cdot y\right)}\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(y\right)\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(0 - \color{blue}{y}\right)\right) \]

      6. --lowering--.f64 31.7%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(0, \color{blue}{y}\right)\right) \]

   8. Simplified 31.7%

      \[\leadsto \color{blue}{z \cdot \left(0 - y\right)} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(z, \left(\mathsf{neg}\left(y\right)\right)\right) \]

      2. neg-lowering-neg.f64 31.7%

         \[\leadsto \mathsf{*.f64}\left(z, \mathsf{neg.f64}\left(y\right)\right) \]

   10. Applied egg-rr 31.7%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-64}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-123}:\\ \;\;\;\;0 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.9% accurate, 14.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (z * (y * ((-1.0d0) + (y * ((-0.5d0) + (y * (-0.3333333333333333d0))))))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

      \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \log \left(1 - y\right)\right), \color{blue}{t}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \log \left(1 - y\right)\right), t\right) \]

   3. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\left(1 - y\right)\right)\right), t\right) \]

   4. --lowering--.f64 48.4%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, y\right)\right)\right), t\right) \]

5. Simplified 48.4%

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

6. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right), t\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)\right), t\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right) \]

   3. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) + -1\right)\right)\right), t\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), -1\right)\right)\right), t\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{3} \cdot y + \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{-1}{3} \cdot y\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\left(y \cdot \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

   10. *-lowering-*.f64 60.4%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\mathsf{*.f64}\left(y, \frac{-1}{3}\right), \frac{-1}{2}\right)\right), -1\right)\right)\right), t\right) \]

8. Simplified 60.4%

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

9. Final simplification 60.4%

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

Alternative 10: 58.0% accurate, 14.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (y * ((((-0.5d0) + (y * (-0.3333333333333333d0))) * (y * z)) - z)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

      \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \log \left(1 - y\right)\right), \color{blue}{t}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \log \left(1 - y\right)\right), t\right) \]

   3. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\left(1 - y\right)\right)\right), t\right) \]

   4. --lowering--.f64 48.4%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, y\right)\right)\right), t\right) \]

5. Simplified 48.4%

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

6. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

      \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(z\right)\right) + y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{3} \cdot \left(y \cdot z\right)\right)\right) - t \]

   2. +-commutative N/A

      \[\leadsto y \cdot \left(y \cdot \left(\frac{-1}{2} \cdot z + \frac{-1}{3} \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(z\right)\right)\right) - t \]

   3. fma-undefine N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. distribute-rgt-in N/A

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

   7. metadata-eval N/A

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

   8. sub-neg N/A

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

   9. fmm-def N/A

      \[\leadsto y \cdot \left(y \cdot \left(z \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right) - z\right) - t \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(y \cdot \left(z \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right)\right) - z\right)\right), \color{blue}{t}\right) \]

8. Simplified 60.4%

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

9. Final simplification 60.4%

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

Alternative 11: 57.8% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.7%

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

      \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \log \left(1 - y\right)\right), \color{blue}{t}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \log \left(1 - y\right)\right), t\right) \]

   3. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\left(1 - y\right)\right)\right), t\right) \]

   4. --lowering--.f64 48.4%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, y\right)\right)\right), t\right) \]

5. Simplified 48.4%

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

6. Taylor expanded in y around 0

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right), \color{blue}{t}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right), t\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\left(\mathsf{neg}\left(z\right)\right) + \frac{-1}{2} \cdot \left(y \cdot z\right)\right)\right), t\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{2} \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(z\right)\right)\right)\right), t\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \left(\frac{-1}{2} \cdot \left(y \cdot z\right) - z\right)\right), t\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\frac{-1}{2} \cdot \left(y \cdot z\right)\right), z\right)\right), t\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\left(\left(y \cdot z\right) \cdot \frac{-1}{2}\right), z\right)\right), t\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(y \cdot z\right), \frac{-1}{2}\right), z\right)\right), t\right) \]

   9. *-lowering-*.f64 60.2%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, z\right), \frac{-1}{2}\right), z\right)\right), t\right) \]

8. Simplified 60.2%

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

9. Final simplification 60.2%

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

Alternative 12: 57.5% accurate, 30.1× speedup

Math

\[\begin{array}{l} \\ 0 - \left(t + y \cdot z\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 0.0 (+ t (* y z))))

C

double code(double x, double y, double z, double t) {
	return 0.0 - (t + (y * z));
}

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 = 0.0d0 - (t + (y * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0 - (t + (y * z));
}

Python

def code(x, y, z, t):
	return 0.0 - (t + (y * z))

Julia

function code(x, y, z, t)
	return Float64(0.0 - Float64(t + Float64(y * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0 - (t + (y * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(0.0 - N[(t + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.7%

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

      \[\leadsto \mathsf{\_.f64}\left(\left(z \cdot \log \left(1 - y\right)\right), \color{blue}{t}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \log \left(1 - y\right)\right), t\right) \]

   3. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\left(1 - y\right)\right)\right), t\right) \]

   4. --lowering--.f64 48.4%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(1, y\right)\right)\right), t\right) \]

5. Simplified 48.4%

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

6. Taylor expanded in y around 0

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(-1 \cdot \left(y \cdot z\right)\right), \color{blue}{t}\right) \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\mathsf{neg}\left(y \cdot z\right)\right), t\right) \]

   3. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(0 - y \cdot z\right), t\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot z\right)\right), t\right) \]

   5. *-lowering-*.f64 59.9%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, z\right)\right), t\right) \]

8. Simplified 59.9%

   \[\leadsto \color{blue}{\left(0 - y \cdot z\right) - t} \]

9. Final simplification 59.9%

   \[\leadsto 0 - \left(t + y \cdot z\right) \]
10. Add Preprocessing

Alternative 13: 42.8% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ 0 - t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- 0.0 t))

C

double code(double x, double y, double z, double t) {
	return 0.0 - t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = 0.0d0 - t
end function

Java

public static double code(double x, double y, double z, double t) {
	return 0.0 - t;
}

Python

def code(x, y, z, t):
	return 0.0 - t

Julia

function code(x, y, z, t)
	return Float64(0.0 - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = 0.0 - t;
end

Wolfram

code[x_, y_, z_, t_] := N[(0.0 - t), $MachinePrecision]

Derivation

1. Initial program 86.7%

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

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

   2. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{t} \]

   3. --lowering--.f64 46.7%

      \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{t}\right) \]

5. Simplified 46.7%

   \[\leadsto \color{blue}{0 - t} \]
6. Step-by-step derivation

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 46.7%

      \[\leadsto \mathsf{neg.f64}\left(t\right) \]

7. Applied egg-rr 46.7%

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

8. Final simplification 46.7%

   \[\leadsto 0 - t \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (-z * (((0.5d0 * (y * y)) + y) + ((0.3333333333333333d0 / (1.0d0 * (1.0d0 * 1.0d0))) * (y * (y * y))))) - (t - (x * log(y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024155 
(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))