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

Percentage Accurate: 89.4% → 99.8%

Time: 12.3s

Alternatives: 16

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: 89.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 N/A

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

   9. *-lft-identity N/A

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

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

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

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

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

   12. *-lft-identity N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower--.f64 99.8

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z - 1, \mathsf{log1p}\left(-y\right), \log y \cdot \left(x - 1\right) - t\right)} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z, t):
	return (((x - 1.0) * math.log(y)) + ((z - 1.0) * (((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 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) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y))) - t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x - 1.0) * log(y)) + ((z - 1.0) * (((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 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[(N[(N[(N[(N[(N[(N[(-0.25 * y), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 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

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 3: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift--.f64 N/A

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

   9. *-lft-identity N/A

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

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

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

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

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

   12. *-lft-identity N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower--.f64 99.8

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

   4. mul-1-neg N/A

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. mul-1-neg N/A

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

   8. log-rec N/A

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

   9. lower-fma.f64 N/A

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

   10. log-rec N/A

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

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

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

   12. mul-1-neg N/A

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

   13. remove-double-neg N/A

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

   14. lower-log.f64 N/A

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

   15. lower--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

8. Applied rewrites 99.3%

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

Alternative 5: 95.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0006 \lor \neg \left(x \leq 8.5 \cdot 10^{+19}\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\mathsf{fma}\left(z - 1, y, \log y\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -0.0006) (not (<= x 8.5e+19)))
   (- (* (+ -1.0 x) (log y)) t)
   (- (- (fma (- z 1.0) y (log y))) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -0.0006) || !(x <= 8.5e+19)) {
		tmp = ((-1.0 + x) * log(y)) - t;
	} else {
		tmp = -fma((z - 1.0), y, log(y)) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -0.0006) || !(x <= 8.5e+19))
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	else
		tmp = Float64(Float64(-fma(Float64(z - 1.0), y, log(y))) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -0.0006], N[Not[LessEqual[x, 8.5e+19]], $MachinePrecision]], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[((-N[(N[(z - 1.0), $MachinePrecision] * y + N[Log[y], $MachinePrecision]), $MachinePrecision]) - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.99999999999999947e-4 or 8.5e19 < x

   1. Initial program 96.6%

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

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

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

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. remove-double-neg N/A

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

      16. log-rec N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. log-rec N/A

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

   5. Applied rewrites 95.8%

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

   if -5.99999999999999947e-4 < x < 8.5e19

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0006 \lor \neg \left(x \leq 8.5 \cdot 10^{+19}\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(-\mathsf{fma}\left(z - 1, y, \log y\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-70}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+19}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t)))
   (if (<= x -1.0)
     t_1
     (if (<= x 3e-70)
       (- (- y (log y)) t)
       (if (<= x 8.5e+19) (- (* (- 1.0 z) y) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double tmp;
	if (x <= -1.0) {
		tmp = t_1;
	} else if (x <= 3e-70) {
		tmp = (y - log(y)) - t;
	} else if (x <= 8.5e+19) {
		tmp = ((1.0 - z) * 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) - t
    if (x <= (-1.0d0)) then
        tmp = t_1
    else if (x <= 3d-70) then
        tmp = (y - log(y)) - t
    else if (x <= 8.5d+19) then
        tmp = ((1.0d0 - z) * 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) - t;
	double tmp;
	if (x <= -1.0) {
		tmp = t_1;
	} else if (x <= 3e-70) {
		tmp = (y - Math.log(y)) - t;
	} else if (x <= 8.5e+19) {
		tmp = ((1.0 - z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.log(y) * x) - t
	tmp = 0
	if x <= -1.0:
		tmp = t_1
	elif x <= 3e-70:
		tmp = (y - math.log(y)) - t
	elif x <= 8.5e+19:
		tmp = ((1.0 - z) * y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	tmp = 0.0
	if (x <= -1.0)
		tmp = t_1;
	elseif (x <= 3e-70)
		tmp = Float64(Float64(y - log(y)) - t);
	elseif (x <= 8.5e+19)
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (log(y) * x) - t;
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_1;
	elseif (x <= 3e-70)
		tmp = (y - log(y)) - t;
	elseif (x <= 8.5e+19)
		tmp = ((1.0 - z) * 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] * x), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$1, If[LessEqual[x, 3e-70], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 8.5e+19], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1 or 8.5e19 < x

   1. Initial program 96.5%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if -1 < x < 3.0000000000000001e-70

   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. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 85.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 84.8%

       \[\leadsto \left(y - \log y\right) - t \]

   if 3.0000000000000001e-70 < x < 8.5e19

   1. Initial program 63.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in y around inf

      \[\leadsto \left(1 - z\right) \cdot y - t \]
   10. Step-by-step derivation

   11. Applied rewrites 78.3%

       \[\leadsto \left(1 - z\right) \cdot y - t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-70}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+39}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -4.4e+51)
     t_1
     (if (<= x 3e-70)
       (- (- y (log y)) t)
       (if (<= x 4.3e+39) (- (* (- 1.0 z) y) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -4.4e+51) {
		tmp = t_1;
	} else if (x <= 3e-70) {
		tmp = (y - log(y)) - t;
	} else if (x <= 4.3e+39) {
		tmp = ((1.0 - z) * 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
    if (x <= (-4.4d+51)) then
        tmp = t_1
    else if (x <= 3d-70) then
        tmp = (y - log(y)) - t
    else if (x <= 4.3d+39) then
        tmp = ((1.0d0 - z) * 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;
	double tmp;
	if (x <= -4.4e+51) {
		tmp = t_1;
	} else if (x <= 3e-70) {
		tmp = (y - Math.log(y)) - t;
	} else if (x <= 4.3e+39) {
		tmp = ((1.0 - z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -4.4e+51:
		tmp = t_1
	elif x <= 3e-70:
		tmp = (y - math.log(y)) - t
	elif x <= 4.3e+39:
		tmp = ((1.0 - z) * y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -4.4e+51)
		tmp = t_1;
	elseif (x <= 3e-70)
		tmp = Float64(Float64(y - log(y)) - t);
	elseif (x <= 4.3e+39)
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -4.4e+51)
		tmp = t_1;
	elseif (x <= 3e-70)
		tmp = (y - log(y)) - t;
	elseif (x <= 4.3e+39)
		tmp = ((1.0 - z) * y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.4e+51], t$95$1, If[LessEqual[x, 3e-70], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 4.3e+39], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.39999999999999984e51 or 4.3e39 < x

   1. Initial program 96.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 99.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. log-rec N/A

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

      6. lower-*.f64 N/A

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

      7. log-rec N/A

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

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

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-log.f64 78.2

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

   7. Applied rewrites 78.2%

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

   if -4.39999999999999984e51 < x < 3.0000000000000001e-70

   1. Initial program 88.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 86.5%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 83.1%

       \[\leadsto \left(y - \log y\right) - t \]

   if 3.0000000000000001e-70 < x < 4.3e39

   1. Initial program 65.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in y around inf

      \[\leadsto \left(1 - z\right) \cdot y - t \]
   10. Step-by-step derivation

   11. Applied rewrites 76.9%

       \[\leadsto \left(1 - z\right) \cdot y - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+51}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-70}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+39}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -4.4 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-70}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+39}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -4.4e+51)
     t_1
     (if (<= x 3e-70)
       (- (- (log y)) t)
       (if (<= x 4.3e+39) (- (* (- 1.0 z) y) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -4.4e+51) {
		tmp = t_1;
	} else if (x <= 3e-70) {
		tmp = -log(y) - t;
	} else if (x <= 4.3e+39) {
		tmp = ((1.0 - z) * 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
    if (x <= (-4.4d+51)) then
        tmp = t_1
    else if (x <= 3d-70) then
        tmp = -log(y) - t
    else if (x <= 4.3d+39) then
        tmp = ((1.0d0 - z) * 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;
	double tmp;
	if (x <= -4.4e+51) {
		tmp = t_1;
	} else if (x <= 3e-70) {
		tmp = -Math.log(y) - t;
	} else if (x <= 4.3e+39) {
		tmp = ((1.0 - z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -4.4e+51:
		tmp = t_1
	elif x <= 3e-70:
		tmp = -math.log(y) - t
	elif x <= 4.3e+39:
		tmp = ((1.0 - z) * y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -4.4e+51)
		tmp = t_1;
	elseif (x <= 3e-70)
		tmp = Float64(Float64(-log(y)) - t);
	elseif (x <= 4.3e+39)
		tmp = Float64(Float64(Float64(1.0 - z) * y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -4.4e+51)
		tmp = t_1;
	elseif (x <= 3e-70)
		tmp = -log(y) - t;
	elseif (x <= 4.3e+39)
		tmp = ((1.0 - z) * y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -4.4e+51], t$95$1, If[LessEqual[x, 3e-70], N[((-N[Log[y], $MachinePrecision]) - t), $MachinePrecision], If[LessEqual[x, 4.3e+39], N[(N[(N[(1.0 - z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.39999999999999984e51 or 4.3e39 < x

   1. Initial program 96.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 99.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. log-rec N/A

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

      6. lower-*.f64 N/A

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

      7. log-rec N/A

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

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

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-log.f64 78.2

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

   7. Applied rewrites 78.2%

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

   if -4.39999999999999984e51 < x < 3.0000000000000001e-70

   1. Initial program 88.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.3%

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

   9. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \log y - t \]
   10. Step-by-step derivation

   11. Applied rewrites 83.0%

       \[\leadsto \left(-\log y\right) - t \]

   if 3.0000000000000001e-70 < x < 4.3e39

   1. Initial program 65.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in y around inf

      \[\leadsto \left(1 - z\right) \cdot y - t \]
   10. Step-by-step derivation

   11. Applied rewrites 76.9%

       \[\leadsto \left(1 - z\right) \cdot y - t \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+51}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-70}:\\ \;\;\;\;\left(-\log y\right) - t\\ \mathbf{elif}\;x \leq 4.3 \cdot 10^{+39}:\\ \;\;\;\;\left(1 - z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -4.2 \cdot 10^{+51} \lor \neg \left(x - 1 \leq 4 \cdot 10^{+36}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (- x 1.0) -4.2e+51) (not (<= (- x 1.0) 4e+36)))
   (* (log y) x)
   (- (* (* (- (* -0.5 y) 1.0) z) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -4.2e+51) || !((x - 1.0) <= 4e+36)) {
		tmp = log(y) * x;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - 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 (((x - 1.0d0) <= (-4.2d+51)) .or. (.not. ((x - 1.0d0) <= 4d+36))) then
        tmp = log(y) * x
    else
        tmp = (((((-0.5d0) * y) - 1.0d0) * z) * y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -4.2e+51) || !((x - 1.0) <= 4e+36)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x - 1.0) <= -4.2e+51) or not ((x - 1.0) <= 4e+36):
		tmp = math.log(y) * x
	else:
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x - 1.0) <= -4.2e+51) || !(Float64(x - 1.0) <= 4e+36))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * z) * y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x - 1.0) <= -4.2e+51) || ~(((x - 1.0) <= 4e+36)))
		tmp = log(y) * x;
	else
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x - 1.0), $MachinePrecision], -4.2e+51], N[Not[LessEqual[N[(x - 1.0), $MachinePrecision], 4e+36]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -4.2000000000000002e51 or 4.00000000000000017e36 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 96.7%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-log.f64 N/A

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

      8. lift--.f64 N/A

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

      9. *-lft-identity N/A

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

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

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

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

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

      12. *-lft-identity N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower--.f64 99.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. log-rec N/A

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

      6. lower-*.f64 N/A

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

      7. log-rec N/A

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

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

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-log.f64 78.2

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

   7. Applied rewrites 78.2%

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

   if -4.2000000000000002e51 < (-.f64 x #s(literal 1 binary64)) < 4.00000000000000017e36

   1. Initial program 84.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. log-rec N/A

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

      9. lower-fma.f64 N/A

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

      10. log-rec N/A

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

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

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

      12. mul-1-neg N/A

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

      13. remove-double-neg N/A

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

      14. lower-log.f64 N/A

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

      15. lower--.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

   8. Applied rewrites 99.3%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 63.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -4.2 \cdot 10^{+51} \lor \neg \left(x - 1 \leq 4 \cdot 10^{+36}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

   4. mul-1-neg N/A

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 99.0%

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

Alternative 11: 89.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+236}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8.5e+236) (- (* (- y) z) t) (- (fma (log y) (- x 1.0) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.5e+236) {
		tmp = (-y * z) - t;
	} else {
		tmp = fma(log(y), (x - 1.0), y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8.5e+236)
		tmp = Float64(Float64(Float64(-y) * z) - t);
	else
		tmp = Float64(fma(log(y), Float64(x - 1.0), y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8.5e+236], N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5000000000000008e236

   1. Initial program 41.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 82.2%

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

   if -8.5000000000000008e236 < z

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 92.6%

      \[\leadsto \mathsf{fma}\left(\log y, \color{blue}{x - 1}, y\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 89.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+236}:\\ \;\;\;\;\left(-y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8.5e+236) (- (* (- y) z) t) (- (* (+ -1.0 x) (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.5e+236) {
		tmp = (-y * z) - t;
	} else {
		tmp = ((-1.0 + x) * log(y)) - 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 (z <= (-8.5d+236)) then
        tmp = (-y * z) - t
    else
        tmp = (((-1.0d0) + x) * log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.5e+236) {
		tmp = (-y * z) - t;
	} else {
		tmp = ((-1.0 + x) * Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -8.5e+236:
		tmp = (-y * z) - t
	else:
		tmp = ((-1.0 + x) * math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8.5e+236)
		tmp = Float64(Float64(Float64(-y) * z) - t);
	else
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -8.5e+236)
		tmp = (-y * z) - t;
	else
		tmp = ((-1.0 + x) * log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8.5e+236], N[(N[((-y) * z), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5000000000000008e236

   1. Initial program 41.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 82.2%

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

   if -8.5000000000000008e236 < z

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

      4. distribute-rgt-out-- N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

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

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. remove-double-neg N/A

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

      16. log-rec N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. log-rec N/A

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

   5. Applied rewrites 92.5%

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

Alternative 13: 46.6% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

   4. mul-1-neg N/A

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

   6. mul-1-neg N/A

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

   7. mul-1-neg N/A

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

   8. log-rec N/A

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

   9. lower-fma.f64 N/A

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

   10. log-rec N/A

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

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

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

   12. mul-1-neg N/A

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

   13. remove-double-neg N/A

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

   14. lower-log.f64 N/A

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

   15. lower--.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

8. Applied rewrites 99.3%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 47.8%

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

Alternative 14: 46.5% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

   4. mul-1-neg N/A

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 81.8%

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

9. Taylor expanded in y around inf

   \[\leadsto \left(1 - z\right) \cdot y - t \]
10. Step-by-step derivation

11. Applied rewrites 47.7%

    \[\leadsto \left(1 - z\right) \cdot y - t \]
12. Add Preprocessing

Alternative 15: 46.3% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. log-rec N/A

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

   4. mul-1-neg N/A

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 47.5%

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

Alternative 16: 36.2% accurate, 75.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := (-t)

Derivation

1. Initial program 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 t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 37.2

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

5. Applied rewrites 37.2%

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

Reproduce

herbie shell --seed 2024337 
(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))