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

Percentage Accurate: 85.0% → 99.8%

Time: 15.5s

Alternatives: 14

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% 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

\[\begin{array}{l} \\ \left(x \cdot \log y + z \cdot \mathsf{log1p}\left(0 - y\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* x (log y)) (* z (log1p (- 0.0 y)))) t))

C

double code(double x, double y, double z, double t) {
	return ((x * log(y)) + (z * log1p((0.0 - y)))) - t;
}

Java

public static double code(double x, double y, double z, double t) {
	return ((x * Math.log(y)) + (z * Math.log1p((0.0 - y)))) - t;
}

Python

def code(x, y, z, t):
	return ((x * math.log(y)) + (z * math.log1p((0.0 - y)))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(x * log(y)) + Float64(z * log1p(Float64(0.0 - y)))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(z * N[Log[1 + N[(0.0 - y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 87.4%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \left(\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   3. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

   5. --lowering--.f64 99.7%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

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

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

   5. *-commutative N/A

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

   6. mul-1-neg N/A

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

   7. +-commutative N/A

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

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

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

5. Simplified 99.2%

   \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(z \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) - z\right) + x \cdot \log y\right)} - t \]
6. Step-by-step derivation

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

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

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

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

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   13. neg-sub0 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right), y\right), \left(0 - z \cdot y\right)\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), t\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right), y\right), \mathsf{\_.f64}\left(0, \left(z \cdot y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), t\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right), y\right), \mathsf{\_.f64}\left(0, \left(y \cdot z\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), t\right) \]

   16. *-lowering-*.f64 99.2%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right), y\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, z\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), t\right) \]

7. Applied egg-rr 99.2%

   \[\leadsto \left(\color{blue}{\left(\left(z \cdot \left(y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right)\right) \cdot y + \left(0 - y \cdot z\right)\right)} + x \cdot \log y\right) - t \]

8. Final simplification 99.2%

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

Alternative 3: 99.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

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

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

   5. *-commutative N/A

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

   6. mul-1-neg N/A

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

   7. +-commutative N/A

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

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

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 90.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - t\\ \mathbf{if}\;t \leq -2 \cdot 10^{-73}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-56}:\\ \;\;\;\;t\_1 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 t)))
   (if (<= t -2e-73) t_2 (if (<= t 5.8e-56) (- t_1 (* y z)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - t;
	double tmp;
	if (t <= -2e-73) {
		tmp = t_2;
	} else if (t <= 5.8e-56) {
		tmp = t_1 - (y * z);
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = x * log(y)
    t_2 = t_1 - t
    if (t <= (-2d-73)) then
        tmp = t_2
    else if (t <= 5.8d-56) then
        tmp = t_1 - (y * z)
    else
        tmp = t_2
    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 t_2 = t_1 - t;
	double tmp;
	if (t <= -2e-73) {
		tmp = t_2;
	} else if (t <= 5.8e-56) {
		tmp = t_1 - (y * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	t_2 = t_1 - t
	tmp = 0
	if t <= -2e-73:
		tmp = t_2
	elif t <= 5.8e-56:
		tmp = t_1 - (y * z)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - t)
	tmp = 0.0
	if (t <= -2e-73)
		tmp = t_2;
	elseif (t <= 5.8e-56)
		tmp = Float64(t_1 - Float64(y * z));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	t_2 = t_1 - t;
	tmp = 0.0;
	if (t <= -2e-73)
		tmp = t_2;
	elseif (t <= 5.8e-56)
		tmp = t_1 - (y * z);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - t), $MachinePrecision]}, If[LessEqual[t, -2e-73], t$95$2, If[LessEqual[t, 5.8e-56], N[(t$95$1 - N[(y * z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.99999999999999999e-73 or 5.79999999999999982e-56 < t

   1. Initial program 94.9%

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

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

      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 93.7%

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

   5. Simplified 93.7%

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

   if -1.99999999999999999e-73 < t < 5.79999999999999982e-56

   1. Initial program 74.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)}, t\right) \]
   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. Taylor expanded in t around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 89.9%

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

   8. Simplified 89.9%

      \[\leadsto \color{blue}{x \cdot \log y - z \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-56}:\\ \;\;\;\;x \cdot \log y - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-190}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right) + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -3.6e-190)
     t_1
     (if (<= x 9.5e-116)
       (- (* y (* z (+ (* y (+ -0.5 (* y -0.3333333333333333))) -1.0))) 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 <= -3.6e-190) {
		tmp = t_1;
	} else if (x <= 9.5e-116) {
		tmp = (y * (z * ((y * (-0.5 + (y * -0.3333333333333333))) + -1.0))) - 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 <= (-3.6d-190)) then
        tmp = t_1
    else if (x <= 9.5d-116) then
        tmp = (y * (z * ((y * ((-0.5d0) + (y * (-0.3333333333333333d0)))) + (-1.0d0)))) - 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 <= -3.6e-190) {
		tmp = t_1;
	} else if (x <= 9.5e-116) {
		tmp = (y * (z * ((y * (-0.5 + (y * -0.3333333333333333))) + -1.0))) - 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 <= -3.6e-190:
		tmp = t_1
	elif x <= 9.5e-116:
		tmp = (y * (z * ((y * (-0.5 + (y * -0.3333333333333333))) + -1.0))) - 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 <= -3.6e-190)
		tmp = t_1;
	elseif (x <= 9.5e-116)
		tmp = Float64(Float64(y * Float64(z * Float64(Float64(y * Float64(-0.5 + Float64(y * -0.3333333333333333))) + -1.0))) - 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 <= -3.6e-190)
		tmp = t_1;
	elseif (x <= 9.5e-116)
		tmp = (y * (z * ((y * (-0.5 + (y * -0.3333333333333333))) + -1.0))) - 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, -3.6e-190], t$95$1, If[LessEqual[x, 9.5e-116], N[(N[(y * N[(z * N[(N[(y * N[(-0.5 + N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000007e-190 or 9.4999999999999998e-116 < x

   1. Initial program 92.9%

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

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

      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 91.9%

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

   5. Simplified 91.9%

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

   if -3.60000000000000007e-190 < x < 9.4999999999999998e-116

   1. Initial program 67.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

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

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

   5. Simplified 98.7%

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

   6. Taylor expanded in x around 0

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

      1. --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 88.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-190}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right) + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \left(\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   3. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

   5. --lowering--.f64 99.7%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(z, \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

4. Applied egg-rr 99.7%

   \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

5. Taylor expanded in y around 0

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

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

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

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

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

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

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

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

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-out N/A

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

   8. metadata-eval N/A

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

   9. sub-neg N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.2%

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

7. Simplified 99.2%

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

Alternative 7: 76.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+159}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right) + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -8.5e+102)
     t_1
     (if (<= x 8.5e+159)
       (- (* y (* z (+ (* y (+ -0.5 (* y -0.3333333333333333))) -1.0))) t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -8.5e+102) {
		tmp = t_1;
	} else if (x <= 8.5e+159) {
		tmp = (y * (z * ((y * (-0.5 + (y * -0.3333333333333333))) + -1.0))) - 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 <= (-8.5d+102)) then
        tmp = t_1
    else if (x <= 8.5d+159) then
        tmp = (y * (z * ((y * ((-0.5d0) + (y * (-0.3333333333333333d0)))) + (-1.0d0)))) - 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 <= -8.5e+102) {
		tmp = t_1;
	} else if (x <= 8.5e+159) {
		tmp = (y * (z * ((y * (-0.5 + (y * -0.3333333333333333))) + -1.0))) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -8.5e+102:
		tmp = t_1
	elif x <= 8.5e+159:
		tmp = (y * (z * ((y * (-0.5 + (y * -0.3333333333333333))) + -1.0))) - 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 <= -8.5e+102)
		tmp = t_1;
	elseif (x <= 8.5e+159)
		tmp = Float64(Float64(y * Float64(z * Float64(Float64(y * Float64(-0.5 + Float64(y * -0.3333333333333333))) + -1.0))) - 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 <= -8.5e+102)
		tmp = t_1;
	elseif (x <= 8.5e+159)
		tmp = (y * (z * ((y * (-0.5 + (y * -0.3333333333333333))) + -1.0))) - 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, -8.5e+102], t$95$1, If[LessEqual[x, 8.5e+159], N[(N[(y * N[(z * N[(N[(y * N[(-0.5 + N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.4999999999999996e102 or 8.50000000000000076e159 < x

   1. Initial program 95.9%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 80.2%

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

   5. Simplified 80.2%

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

   if -8.4999999999999996e102 < x < 8.50000000000000076e159

   1. Initial program 83.0%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

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

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

      5. *-commutative N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

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

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0

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

      1. --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 76.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+102}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+159}:\\ \;\;\;\;y \cdot \left(z \cdot \left(y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right) + -1\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.2% 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 87.4%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)}, t\right) \]
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 98.8%

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

5. Simplified 98.8%

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

6. Final simplification 98.8%

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

Alternative 9: 58.0% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

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

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

   5. *-commutative N/A

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

   6. mul-1-neg N/A

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

   7. +-commutative N/A

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

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

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

5. Simplified 99.2%

   \[\leadsto \color{blue}{\left(y \cdot \left(y \cdot \left(z \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) - z\right) + x \cdot \log y\right)} - t \]
6. Step-by-step derivation

   1. sub-neg N/A

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

   2. distribute-rgt-in N/A

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

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

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

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

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

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

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

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

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

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

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

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

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

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

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

   13. neg-sub0 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right), y\right), \left(0 - z \cdot y\right)\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), t\right) \]

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

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right), y\right), \mathsf{\_.f64}\left(0, \left(z \cdot y\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), t\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right), y\right), \mathsf{\_.f64}\left(0, \left(y \cdot z\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), t\right) \]

   16. *-lowering-*.f64 99.2%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \frac{-1}{3}\right)\right)\right)\right), y\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, z\right)\right)\right), \mathsf{*.f64}\left(x, \mathsf{log.f64}\left(y\right)\right)\right), t\right) \]

7. Applied egg-rr 99.2%

   \[\leadsto \left(\color{blue}{\left(\left(z \cdot \left(y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right)\right) \cdot y + \left(0 - y \cdot z\right)\right)} + x \cdot \log y\right) - t \]

8. Taylor expanded in x around 0

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

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

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

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

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

   3. unpow2 N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 57.5%

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

10. Simplified 57.5%

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

11. Final simplification 57.5%

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

Alternative 10: 48.9% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-73}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-56}:\\ \;\;\;\;0 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.6e-73) (- 0.0 t) (if (<= t 8e-56) (- 0.0 (* y z)) (- 0.0 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.6e-73) {
		tmp = 0.0 - t;
	} else if (t <= 8e-56) {
		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 <= (-1.6d-73)) then
        tmp = 0.0d0 - t
    else if (t <= 8d-56) 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 <= -1.6e-73) {
		tmp = 0.0 - t;
	} else if (t <= 8e-56) {
		tmp = 0.0 - (y * z);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.6e-73:
		tmp = 0.0 - t
	elif t <= 8e-56:
		tmp = 0.0 - (y * z)
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.6e-73)
		tmp = Float64(0.0 - t);
	elseif (t <= 8e-56)
		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 <= -1.6e-73)
		tmp = 0.0 - t;
	elseif (t <= 8e-56)
		tmp = 0.0 - (y * z);
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.6e-73], N[(0.0 - t), $MachinePrecision], If[LessEqual[t, 8e-56], N[(0.0 - N[(y * z), $MachinePrecision]), $MachinePrecision], N[(0.0 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.59999999999999993e-73 or 8.0000000000000003e-56 < t

   1. Initial program 94.9%

      \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
   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 64.0%

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

   5. Simplified 64.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 64.0%

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

   7. Applied egg-rr 64.0%

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

   if -1.59999999999999993e-73 < t < 8.0000000000000003e-56

   1. Initial program 74.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)}, t\right) \]
   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. Taylor expanded in y around inf

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

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

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

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

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

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

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

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

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

      12. log-lowering-log.f64 77.6%

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

   8. Simplified 77.6%

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

   9. Taylor expanded in y around inf

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

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(y \cdot z\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{y \cdot z} \]

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

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

       4. *-lowering-*.f64 27.9%

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

   11. Simplified 27.9%

       \[\leadsto \color{blue}{0 - y \cdot z} \]
   12. Step-by-step derivation

       1. sub0-neg N/A

          \[\leadsto \mathsf{neg}\left(y \cdot z\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(z \cdot y\right) \]

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

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

       4. mul-1-neg N/A

          \[\leadsto \left(-1 \cdot z\right) \cdot y \]

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

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

       6. mul-1-neg N/A

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

       7. neg-lowering-neg.f64 27.9%

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

   13. Applied egg-rr 27.9%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-73}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-56}:\\ \;\;\;\;0 - y \cdot z\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.0% accurate, 14.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

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

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

   5. *-commutative N/A

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

   6. mul-1-neg N/A

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

   7. +-commutative N/A

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

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

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

   1. --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 57.5%

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

9. Final simplification 57.5%

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

Alternative 12: 57.9% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

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

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

   5. *-commutative N/A

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

   6. mul-1-neg N/A

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

   7. +-commutative N/A

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

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

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

   1. --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 57.5%

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

9. Taylor expanded in y around 0

   \[\leadsto \color{blue}{y \cdot \left(-1 \cdot z + \frac{-1}{2} \cdot \left(y \cdot z\right)\right) - t} \]
10. 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. +-commutative N/A

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

    4. associate-*r* N/A

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

    5. distribute-rgt-out N/A

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

    6. metadata-eval N/A

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

    7. sub-neg N/A

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

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

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

    9. sub-neg N/A

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

    10. metadata-eval N/A

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

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

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

    12. *-commutative N/A

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

    13. *-lowering-*.f64 57.4%

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

11. Simplified 57.4%

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

Alternative 13: 57.6% accurate, 30.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

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

3. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot z\right) + x \cdot \log y\right)}, t\right) \]
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 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\left(t + y \cdot z\right)\right) \]

   2. distribute-neg-in N/A

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

   3. unsub-neg N/A

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

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

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

   5. neg-sub0 N/A

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 57.0%

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

8. Simplified 57.0%

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

9. Final simplification 57.0%

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

Alternative 14: 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 87.4%

   \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t \]
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 44.3%

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

5. Simplified 44.3%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 44.3%

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

7. Applied egg-rr 44.3%

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

8. Final simplification 44.3%

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

Developer Target 1: 99.7% 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 2024161 
(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))