Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2

Percentage Accurate: 88.9% → 99.8%

Time: 24.3s

Alternatives: 17

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 88.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{log1p}\left(0 - y\right) \cdot \left(z + -1\right) + \log y \cdot \left(x + -1\right)\right) - t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- (+ (* (log1p (- 0.0 y)) (+ z -1.0)) (* (log y) (+ x -1.0))) t))

C

double code(double x, double y, double z, double t) {
	return ((log1p((0.0 - y)) * (z + -1.0)) + (log(y) * (x + -1.0))) - t;
}

Java

public static double code(double x, double y, double z, double t) {
	return ((Math.log1p((0.0 - y)) * (z + -1.0)) + (Math.log(y) * (x + -1.0))) - t;
}

Python

def code(x, y, z, t):
	return ((math.log1p((0.0 - y)) * (z + -1.0)) + (math.log(y) * (x + -1.0))) - t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(log1p(Float64(0.0 - y)) * Float64(z + -1.0)) + Float64(log(y) * Float64(x + -1.0))) - t)
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[Log[1 + N[(0.0 - y), $MachinePrecision]], $MachinePrecision] * N[(z + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 91.6%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

   5. --lowering--.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

5. Final simplification 99.8%

   \[\leadsto \left(\mathsf{log1p}\left(0 - y\right) \cdot \left(z + -1\right) + \log y \cdot \left(x + -1\right)\right) - t \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in y around 0

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

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

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in y around 0

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

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

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

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

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

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

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 95.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot \left(x + -1\right) - t\\ \mathbf{if}\;x \leq -0.00017:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) (+ x -1.0)) t)))
   (if (<= x -0.00017)
     t_1
     (if (<= x 2.6e+18) (- (- (* y (- 1.0 z)) (log y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * (x + -1.0)) - t;
	double tmp;
	if (x <= -0.00017) {
		tmp = t_1;
	} else if (x <= 2.6e+18) {
		tmp = ((y * (1.0 - z)) - log(y)) - 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 = (log(y) * (x + (-1.0d0))) - t
    if (x <= (-0.00017d0)) then
        tmp = t_1
    else if (x <= 2.6d+18) then
        tmp = ((y * (1.0d0 - z)) - log(y)) - 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 = (Math.log(y) * (x + -1.0)) - t;
	double tmp;
	if (x <= -0.00017) {
		tmp = t_1;
	} else if (x <= 2.6e+18) {
		tmp = ((y * (1.0 - z)) - Math.log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * (x + -1.0)) - t
	tmp = 0
	if x <= -0.00017:
		tmp = t_1
	elif x <= 2.6e+18:
		tmp = ((y * (1.0 - z)) - math.log(y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * Float64(x + -1.0)) - t)
	tmp = 0.0
	if (x <= -0.00017)
		tmp = t_1;
	elseif (x <= 2.6e+18)
		tmp = Float64(Float64(Float64(y * Float64(1.0 - z)) - log(y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (log(y) * (x + -1.0)) - t;
	tmp = 0.0;
	if (x <= -0.00017)
		tmp = t_1;
	elseif (x <= 2.6e+18)
		tmp = ((y * (1.0 - z)) - log(y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -0.00017], t$95$1, If[LessEqual[x, 2.6e+18], N[(N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e-4 or 2.6e18 < x

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 95.1%

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

   5. Simplified 95.1%

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

   if -1.7e-4 < x < 2.6e18

   1. Initial program 87.1%

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

      1. sub-neg N/A

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

      2. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   7. Simplified 98.0%

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

   8. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

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

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

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

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

      10. mul-1-neg N/A

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

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

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-in N/A

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

      15. metadata-eval N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. sub-neg N/A

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

      19. --lowering--.f64 N/A

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

      20. log-lowering-log.f64 98.0%

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

   10. Simplified 98.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00017:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\left(y \cdot \left(1 - z\right) - \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot \left(x + -1\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

   5. --lowering--.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

5. Taylor expanded in y around 0

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

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

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.2%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 6: 86.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - t\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\left(0 - t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) t)))
   (if (<= x -3.9e-13) t_1 (if (<= x 2.6e+18) (- (- 0.0 t) (log y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - t;
	double tmp;
	if (x <= -3.9e-13) {
		tmp = t_1;
	} else if (x <= 2.6e+18) {
		tmp = (0.0 - t) - log(y);
	} 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 <= (-3.9d-13)) then
        tmp = t_1
    else if (x <= 2.6d+18) then
        tmp = (0.0d0 - t) - log(y)
    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 <= -3.9e-13) {
		tmp = t_1;
	} else if (x <= 2.6e+18) {
		tmp = (0.0 - t) - Math.log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - t
	tmp = 0
	if x <= -3.9e-13:
		tmp = t_1
	elif x <= 2.6e+18:
		tmp = (0.0 - t) - math.log(y)
	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 <= -3.9e-13)
		tmp = t_1;
	elseif (x <= 2.6e+18)
		tmp = Float64(Float64(0.0 - t) - log(y));
	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 <= -3.9e-13)
		tmp = t_1;
	elseif (x <= 2.6e+18)
		tmp = (0.0 - t) - log(y);
	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, -3.9e-13], t$95$1, If[LessEqual[x, 2.6e+18], N[(N[(0.0 - t), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.90000000000000004e-13 or 2.6e18 < x

   1. Initial program 95.2%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 93.8%

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

   5. Simplified 93.8%

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

   if -3.90000000000000004e-13 < x < 2.6e18

   1. Initial program 87.0%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 85.0%

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

   5. Simplified 85.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \log y - t} \]
   7. Step-by-step derivation

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

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

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

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

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

      5. log-lowering-log.f64 85.0%

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

   8. Simplified 85.0%

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

      1. sub0-neg N/A

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

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

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

      3. log-lowering-log.f64 85.0%

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

   10. Applied egg-rr 85.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \log y - t\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+18}:\\ \;\;\;\;\left(0 - t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+25}:\\ \;\;\;\;t \cdot \left(-1 + \frac{y \cdot \left(1 - z\right)}{t}\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -3.2e+25)
   (* t (+ -1.0 (/ (* y (- 1.0 z)) t)))
   (if (<= t 5e+77)
     (* (log y) (+ x -1.0))
     (- (* y (* z (+ -1.0 (* y -0.5)))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.2e+25) {
		tmp = t * (-1.0 + ((y * (1.0 - z)) / t));
	} else if (t <= 5e+77) {
		tmp = log(y) * (x + -1.0);
	} else {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-3.2d+25)) then
        tmp = t * ((-1.0d0) + ((y * (1.0d0 - z)) / t))
    else if (t <= 5d+77) then
        tmp = log(y) * (x + (-1.0d0))
    else
        tmp = (y * (z * ((-1.0d0) + (y * (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -3.2e+25) {
		tmp = t * (-1.0 + ((y * (1.0 - z)) / t));
	} else if (t <= 5e+77) {
		tmp = Math.log(y) * (x + -1.0);
	} else {
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -3.2e+25:
		tmp = t * (-1.0 + ((y * (1.0 - z)) / t))
	elif t <= 5e+77:
		tmp = math.log(y) * (x + -1.0)
	else:
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -3.2e+25)
		tmp = Float64(t * Float64(-1.0 + Float64(Float64(y * Float64(1.0 - z)) / t)));
	elseif (t <= 5e+77)
		tmp = Float64(log(y) * Float64(x + -1.0));
	else
		tmp = Float64(Float64(y * Float64(z * Float64(-1.0 + Float64(y * -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -3.2e+25)
		tmp = t * (-1.0 + ((y * (1.0 - z)) / t));
	elseif (t <= 5e+77)
		tmp = log(y) * (x + -1.0);
	else
		tmp = (y * (z * (-1.0 + (y * -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -3.2e+25], N[(t * N[(-1.0 + N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5e+77], N[(N[Log[y], $MachinePrecision] * N[(x + -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(z * N[(-1.0 + N[(y * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.1999999999999999e25

   1. Initial program 94.7%

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

   3. Taylor expanded in t around inf

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

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

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

      15. associate-/l* N/A

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

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

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

   5. Simplified 94.6%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

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

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around inf

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

       1. div-sub N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. metadata-eval N/A

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

       6. distribute-lft-in N/A

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

       7. metadata-eval N/A

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

       8. sub-neg N/A

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

       9. associate-*r/ N/A

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

       10. mul-1-neg N/A

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

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

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

       12. associate-/l* N/A

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

       13. distribute-frac-neg N/A

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

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

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

   11. Simplified 83.1%

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

   if -3.1999999999999999e25 < t < 5.00000000000000004e77

   1. Initial program 88.8%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 86.4%

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

   5. Simplified 86.4%

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

   6. Taylor expanded in t around 0

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-lowering-+.f64 82.5%

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

   8. Simplified 82.5%

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

   if 5.00000000000000004e77 < t

   1. Initial program 95.6%

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

      1. sub-neg N/A

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

      2. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

      5. --lowering--.f64 99.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

   5. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in z around inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 80.0%

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

   10. Simplified 80.0%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{+25}:\\ \;\;\;\;t \cdot \left(-1 + \frac{y \cdot \left(1 - z\right)}{t}\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+77}:\\ \;\;\;\;\log y \cdot \left(x + -1\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot \left(-1 + y \cdot -0.5\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+159}:\\ \;\;\;\;\left(0 - t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -3.4e+100) t_1 (if (<= x 1.65e+159) (- (- 0.0 t) (log y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -3.4e+100) {
		tmp = t_1;
	} else if (x <= 1.65e+159) {
		tmp = (0.0 - t) - log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-3.4d+100)) then
        tmp = t_1
    else if (x <= 1.65d+159) then
        tmp = (0.0d0 - t) - log(y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -3.4e+100) {
		tmp = t_1;
	} else if (x <= 1.65e+159) {
		tmp = (0.0 - t) - Math.log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -3.4e+100:
		tmp = t_1
	elif x <= 1.65e+159:
		tmp = (0.0 - t) - math.log(y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -3.4e+100)
		tmp = t_1;
	elseif (x <= 1.65e+159)
		tmp = Float64(Float64(0.0 - t) - log(y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -3.4e+100)
		tmp = t_1;
	elseif (x <= 1.65e+159)
		tmp = (0.0 - t) - log(y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e+100], t$95$1, If[LessEqual[x, 1.65e+159], N[(N[(0.0 - t), $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.39999999999999994e100 or 1.6499999999999999e159 < x

   1. Initial program 95.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 80.2%

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

   5. Simplified 80.2%

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

   if -3.39999999999999994e100 < x < 1.6499999999999999e159

   1. Initial program 89.4%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 87.7%

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

   5. Simplified 87.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \log y - t} \]
   7. Step-by-step derivation

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

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

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

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

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

      5. log-lowering-log.f64 76.4%

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

   8. Simplified 76.4%

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

      1. sub0-neg N/A

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

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

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

      3. log-lowering-log.f64 76.4%

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

   10. Applied egg-rr 76.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{+100}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+159}:\\ \;\;\;\;\left(0 - t\right) - \log y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+159}:\\ \;\;\;\;\left(-1 + y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) \cdot \left(y \cdot z\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -5.6e+99)
     t_1
     (if (<= x 3.9e+159)
       (- (* (+ -1.0 (* y (+ -0.5 (* y -0.3333333333333333)))) (* y z)) t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -5.6e+99) {
		tmp = t_1;
	} else if (x <= 3.9e+159) {
		tmp = ((-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))) * (y * z)) - 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 <= (-5.6d+99)) then
        tmp = t_1
    else if (x <= 3.9d+159) then
        tmp = (((-1.0d0) + (y * ((-0.5d0) + (y * (-0.3333333333333333d0))))) * (y * z)) - 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 <= -5.6e+99) {
		tmp = t_1;
	} else if (x <= 3.9e+159) {
		tmp = ((-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))) * (y * z)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -5.6e+99:
		tmp = t_1
	elif x <= 3.9e+159:
		tmp = ((-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))) * (y * z)) - 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 <= -5.6e+99)
		tmp = t_1;
	elseif (x <= 3.9e+159)
		tmp = Float64(Float64(Float64(-1.0 + Float64(y * Float64(-0.5 + Float64(y * -0.3333333333333333)))) * Float64(y * z)) - 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 <= -5.6e+99)
		tmp = t_1;
	elseif (x <= 3.9e+159)
		tmp = ((-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))) * (y * z)) - 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, -5.6e+99], t$95$1, If[LessEqual[x, 3.9e+159], N[(N[(N[(-1.0 + N[(y * N[(-0.5 + N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.6e99 or 3.9000000000000001e159 < x

   1. Initial program 95.9%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

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

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

      3. log-lowering-log.f64 80.2%

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

   5. Simplified 80.2%

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

   if -5.6e99 < x < 3.9000000000000001e159

   1. Initial program 89.4%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

   5. Simplified 99.1%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-lowering-*.f64 58.9%

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

   8. Simplified 58.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+99}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+159}:\\ \;\;\;\;\left(-1 + y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) \cdot \left(y \cdot z\right) - t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-7}:\\ \;\;\;\;t \cdot \left(-1 + \frac{y \cdot \left(1 - z\right)}{t}\right)\\ \mathbf{elif}\;t \leq 25:\\ \;\;\;\;0 - \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) \cdot \left(y \cdot z\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.05e-7)
   (* t (+ -1.0 (/ (* y (- 1.0 z)) t)))
   (if (<= t 25.0)
     (- 0.0 (log y))
     (- (* (+ -1.0 (* y (+ -0.5 (* y -0.3333333333333333)))) (* y z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.05e-7) {
		tmp = t * (-1.0 + ((y * (1.0 - z)) / t));
	} else if (t <= 25.0) {
		tmp = 0.0 - log(y);
	} else {
		tmp = ((-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))) * (y * z)) - 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 <= (-1.05d-7)) then
        tmp = t * ((-1.0d0) + ((y * (1.0d0 - z)) / t))
    else if (t <= 25.0d0) then
        tmp = 0.0d0 - log(y)
    else
        tmp = (((-1.0d0) + (y * ((-0.5d0) + (y * (-0.3333333333333333d0))))) * (y * z)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.05e-7) {
		tmp = t * (-1.0 + ((y * (1.0 - z)) / t));
	} else if (t <= 25.0) {
		tmp = 0.0 - Math.log(y);
	} else {
		tmp = ((-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))) * (y * z)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.05e-7:
		tmp = t * (-1.0 + ((y * (1.0 - z)) / t))
	elif t <= 25.0:
		tmp = 0.0 - math.log(y)
	else:
		tmp = ((-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))) * (y * z)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.05e-7)
		tmp = Float64(t * Float64(-1.0 + Float64(Float64(y * Float64(1.0 - z)) / t)));
	elseif (t <= 25.0)
		tmp = Float64(0.0 - log(y));
	else
		tmp = Float64(Float64(Float64(-1.0 + Float64(y * Float64(-0.5 + Float64(y * -0.3333333333333333)))) * Float64(y * z)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.05e-7)
		tmp = t * (-1.0 + ((y * (1.0 - z)) / t));
	elseif (t <= 25.0)
		tmp = 0.0 - log(y);
	else
		tmp = ((-1.0 + (y * (-0.5 + (y * -0.3333333333333333)))) * (y * z)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.05e-7], N[(t * N[(-1.0 + N[(N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 25.0], N[(0.0 - N[Log[y], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 + N[(y * N[(-0.5 + N[(y * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.05e-7

   1. Initial program 93.7%

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

   3. Taylor expanded in t around inf

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

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

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

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

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

      7. associate-/l* N/A

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

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

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

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

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. +-commutative N/A

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

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

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

      15. associate-/l* N/A

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

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

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

   5. Simplified 93.6%

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

   6. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

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

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around inf

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

       1. div-sub N/A

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

       2. sub-neg N/A

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

       3. mul-1-neg N/A

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

       4. +-commutative N/A

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

       5. metadata-eval N/A

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

       6. distribute-lft-in N/A

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

       7. metadata-eval N/A

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

       8. sub-neg N/A

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

       9. associate-*r/ N/A

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

       10. mul-1-neg N/A

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

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

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

       12. associate-/l* N/A

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

       13. distribute-frac-neg N/A

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

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

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

   11. Simplified 80.1%

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

   if -1.05e-7 < t < 25

   1. Initial program 88.7%

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

   3. Taylor expanded in y around 0

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

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

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

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

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

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

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. +-lowering-+.f64 86.8%

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

   5. Simplified 86.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \log y - t} \]
   7. Step-by-step derivation

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

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

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

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

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

      5. log-lowering-log.f64 41.3%

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

   8. Simplified 41.3%

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

   9. Taylor expanded in t around 0

      \[\leadsto \color{blue}{-1 \cdot \log y} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

       2. log-rec N/A

          \[\leadsto \log \left(\frac{1}{y}\right) \]

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

          \[\leadsto \mathsf{log.f64}\left(\left(\frac{1}{y}\right)\right) \]

       4. /-lowering-/.f64 40.4%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{/.f64}\left(1, y\right)\right) \]

   11. Simplified 40.4%

       \[\leadsto \color{blue}{\log \left(\frac{1}{y}\right)} \]
   12. Step-by-step derivation

       1. log-rec N/A

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

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

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

       3. log-lowering-log.f64 40.4%

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

   13. Applied egg-rr 40.4%

       \[\leadsto \color{blue}{-\log y} \]

   if 25 < t

   1. Initial program 95.1%

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

   3. Taylor expanded in y around 0

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

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

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

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

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

   5. Simplified 98.9%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

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

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

      13. *-commutative N/A

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

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

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

      15. *-lowering-*.f64 69.3%

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

   8. Simplified 69.3%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-7}:\\ \;\;\;\;t \cdot \left(-1 + \frac{y \cdot \left(1 - z\right)}{t}\right)\\ \mathbf{elif}\;t \leq 25:\\ \;\;\;\;0 - \log y\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + y \cdot \left(-0.5 + y \cdot -0.3333333333333333\right)\right) \cdot \left(y \cdot z\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

   5. --lowering--.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

7. Simplified 98.8%

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

8. Taylor expanded in z around inf

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

   1. *-lowering-*.f64 98.8%

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

10. Simplified 98.8%

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

Alternative 12: 87.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in y around 0

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

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

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

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

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

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

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. +-commutative N/A

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

   7. +-lowering-+.f64 90.3%

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

5. Simplified 90.3%

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

6. Final simplification 90.3%

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

Alternative 13: 43.1% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.32 \cdot 10^{+29}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.32e+29) (- 0.0 t) (if (<= t 3e+24) (* y (- 1.0 z)) (- 0.0 t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.32e+29) {
		tmp = 0.0 - t;
	} else if (t <= 3e+24) {
		tmp = y * (1.0 - 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 <= (-1.32d+29)) then
        tmp = 0.0d0 - t
    else if (t <= 3d+24) then
        tmp = y * (1.0d0 - 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 <= -1.32e+29) {
		tmp = 0.0 - t;
	} else if (t <= 3e+24) {
		tmp = y * (1.0 - z);
	} else {
		tmp = 0.0 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.32e+29:
		tmp = 0.0 - t
	elif t <= 3e+24:
		tmp = y * (1.0 - z)
	else:
		tmp = 0.0 - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.32e+29)
		tmp = Float64(0.0 - t);
	elseif (t <= 3e+24)
		tmp = Float64(y * Float64(1.0 - 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 <= -1.32e+29)
		tmp = 0.0 - t;
	elseif (t <= 3e+24)
		tmp = y * (1.0 - z);
	else
		tmp = 0.0 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.32e+29], N[(0.0 - t), $MachinePrecision], If[LessEqual[t, 3e+24], N[(y * N[(1.0 - z), $MachinePrecision]), $MachinePrecision], N[(0.0 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.32e29 or 2.99999999999999995e24 < t

   1. Initial program 96.2%

      \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
   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 74.0%

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

   5. Simplified 74.0%

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

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 74.0%

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

   7. Applied egg-rr 74.0%

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

   if -1.32e29 < t < 2.99999999999999995e24

   1. Initial program 87.3%

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

      1. sub-neg N/A

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

      2. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

      5. --lowering--.f64 99.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

   4. Applied egg-rr 99.7%

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

   7. Simplified 97.8%

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

   8. Taylor expanded in y around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

         \[\leadsto y \cdot \left(\left(\mathsf{neg}\left(z\right)\right) + 1\right) \]

      5. metadata-eval N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(z + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(z - 1\right)\right)\right) \]

      9. mul-1-neg N/A

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

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

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. distribute-lft-in N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

      16. mul-1-neg N/A

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

      17. sub-neg N/A

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

      18. --lowering--.f64 15.1%

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

   10. Simplified 15.1%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.32 \cdot 10^{+29}:\\ \;\;\;\;0 - t\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \left(1 - z\right)\\ \mathbf{else}:\\ \;\;\;\;0 - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.7% accurate, 14.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

3. Taylor expanded in y around 0

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

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

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

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

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

5. Simplified 99.3%

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

6. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

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

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

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

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

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

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

   13. *-commutative N/A

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

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

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

   15. *-lowering-*.f64 45.9%

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

8. Simplified 45.9%

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

9. Final simplification 45.9%

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

Alternative 15: 46.7% accurate, 19.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

   5. --lowering--.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

5. Taylor expanded in y around 0

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

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

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. distribute-rgt-out N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

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

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

   8. sub-neg N/A

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

   9. metadata-eval N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

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

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

   14. *-commutative N/A

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

   15. *-lowering-*.f64 99.2%

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

7. Simplified 99.2%

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

8. Taylor expanded in z around inf

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

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

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

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

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 45.8%

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

10. Simplified 45.8%

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

Alternative 16: 46.6% accurate, 30.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.6%

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

   1. sub-neg N/A

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

   2. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \left(\mathsf{log1p}\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(y\right)\right)\right)\right)\right), t\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\left(0 - y\right)\right)\right)\right), t\right) \]

   5. --lowering--.f64 99.8%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(x, 1\right), \mathsf{log.f64}\left(y\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(z, 1\right), \mathsf{log1p.f64}\left(\mathsf{\_.f64}\left(0, y\right)\right)\right)\right), t\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\mathsf{log1p}\left(0 - y\right)}\right) - t \]

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

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

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

7. Simplified 98.8%

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

8. Taylor expanded in y around inf

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

   1. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. metadata-eval N/A

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

   6. distribute-neg-in N/A

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

   7. metadata-eval N/A

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

   8. sub-neg N/A

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

   9. mul-1-neg N/A

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

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

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. distribute-lft-in N/A

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

   14. metadata-eval N/A

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

   15. +-commutative N/A

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

   16. mul-1-neg N/A

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

   17. sub-neg N/A

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

   18. --lowering--.f64 45.7%

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

10. Simplified 45.7%

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

Alternative 17: 35.6% accurate, 71.7× 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 91.6%

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t \]
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 37.6%

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

5. Simplified 37.6%

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

   1. sub0-neg N/A

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

   2. neg-lowering-neg.f64 37.6%

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

7. Applied egg-rr 37.6%

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

8. Final simplification 37.6%

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

Reproduce

herbie shell --seed 2024161 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))