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

Percentage Accurate: 85.7% → 99.8%

Time: 14.3s

Alternatives: 16

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. *-lft-identity N/A

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

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

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

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

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

   13. *-lft-identity N/A

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

   14. lower-log1p.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower--.f64 99.8

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate--l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-*r* N/A

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

   10. distribute-rgt-out N/A

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

   11. lower-*.f64 N/A

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

   12. lower-fma.f64 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f64 N/A

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

   15. *-lft-identity N/A

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

   16. metadata-eval N/A

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

10. Applied rewrites 99.4%

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

Alternative 4: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-51} \lor \neg \left(t \leq 3.7 \cdot 10^{-172}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, \log y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.8e-51) (not (<= t 3.7e-172)))
   (fma (log y) x (- t))
   (fma (- y) z (* (log y) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.8e-51) || !(t <= 3.7e-172)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = fma(-y, z, (log(y) * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.8e-51) || !(t <= 3.7e-172))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = fma(Float64(-y), z, Float64(log(y) * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.8e-51], N[Not[LessEqual[t, 3.7e-172]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[((-y) * z + N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.8e-51 or 3.70000000000000001e-172 < t

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. log-rec N/A

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

      15. mul-1-neg N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 92.1%

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

   if -4.8e-51 < t < 3.70000000000000001e-172

   1. Initial program 73.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 92.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-51} \lor \neg \left(t \leq 3.7 \cdot 10^{-172}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, \log y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-51} \lor \neg \left(t \leq 3.7 \cdot 10^{-172}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \left(-z\right) \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.8e-51) (not (<= t 3.7e-172)))
   (fma (log y) x (- t))
   (fma (log y) x (* (- z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.8e-51) || !(t <= 3.7e-172)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = fma(log(y), x, (-z * y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.8e-51) || !(t <= 3.7e-172))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = fma(log(y), x, Float64(Float64(-z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.8e-51], N[Not[LessEqual[t, 3.7e-172]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[((-z) * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.8e-51 or 3.70000000000000001e-172 < t

   1. Initial program 92.9%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. log-rec N/A

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

      15. mul-1-neg N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 92.1%

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

   if -4.8e-51 < t < 3.70000000000000001e-172

   1. Initial program 73.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 92.5%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-51} \lor \neg \left(t \leq 3.7 \cdot 10^{-172}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \left(-z\right) \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 99.5

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-out N/A

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

   6. lower-*.f64 N/A

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

   7. lower-fma.f64 99.2

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

10. Applied rewrites 99.2%

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

Alternative 7: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in y around 0

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

   1. remove-double-neg N/A

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

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

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

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

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

   4. *-commutative N/A

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

   5. remove-double-neg N/A

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

   6. mul-1-neg N/A

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

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

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

   8. log-rec N/A

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

   9. lower-fma.f64 N/A

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

   10. log-rec N/A

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

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

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

   12. mul-1-neg N/A

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

   13. remove-double-neg N/A

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

   14. lower-log.f64 N/A

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

   15. associate-*r* N/A

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

   16. distribute-rgt-out N/A

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

   17. associate-*r* N/A

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

5. Applied rewrites 99.2%

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

Alternative 8: 89.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-137} \lor \neg \left(x \leq 6.4 \cdot 10^{-233}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.7e-137) (not (<= x 6.4e-233)))
   (fma (log y) x (- t))
   (fma (log1p (- y)) z (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.7e-137) || !(x <= 6.4e-233)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = fma(log1p(-y), z, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.7e-137) || !(x <= 6.4e-233))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = fma(log1p(Float64(-y)), z, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.7e-137], N[Not[LessEqual[x, 6.4e-233]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[(N[Log[1 + (-y)], $MachinePrecision] * z + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7e-137 or 6.3999999999999997e-233 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. log-rec N/A

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

      15. mul-1-neg N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 88.8%

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

   if -3.7e-137 < x < 6.3999999999999997e-233

   1. Initial program 73.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. *-lft-identity N/A

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

      14. lower-log1p.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower--.f64 100.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.3

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

   7. Applied rewrites 98.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-137} \lor \neg \left(x \leq 6.4 \cdot 10^{-233}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{log1p}\left(-y\right), z, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-137} \lor \neg \left(x \leq 6.4 \cdot 10^{-233}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y, z, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -3.7e-137) (not (<= x 6.4e-233)))
   (fma (log y) x (- t))
   (fma
    (* (- (* (- (* (- (* -0.25 y) 0.3333333333333333) y) 0.5) y) 1.0) y)
    z
    (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -3.7e-137) || !(x <= 6.4e-233)) {
		tmp = fma(log(y), x, -t);
	} else {
		tmp = fma((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y), z, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -3.7e-137) || !(x <= 6.4e-233))
		tmp = fma(log(y), x, Float64(-t));
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y), z, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -3.7e-137], N[Not[LessEqual[x, 6.4e-233]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x + (-t)), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(-0.25 * y), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.7e-137 or 6.3999999999999997e-233 < x

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. log-rec N/A

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

      6. mul-1-neg N/A

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

      7. *-rgt-identity N/A

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

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

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

      9. log-rec N/A

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

      10. mul-1-neg N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. mul-1-neg N/A

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

      14. log-rec N/A

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

      15. mul-1-neg N/A

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

      16. *-rgt-identity N/A

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

      17. lower-fma.f64 N/A

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

   5. Applied rewrites 88.8%

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

   if -3.7e-137 < x < 6.3999999999999997e-233

   1. Initial program 73.3%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. *-lft-identity N/A

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

      14. lower-log1p.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower--.f64 100.0

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.3

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 97.0

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

   10. Applied rewrites 97.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-137} \lor \neg \left(x \leq 6.4 \cdot 10^{-233}\right):\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y, z, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+51} \lor \neg \left(x \leq 7 \cdot 10^{+38}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y, z, -t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4.4e+51) (not (<= x 7e+38)))
   (* (log y) x)
   (fma
    (* (- (* (- (* (- (* -0.25 y) 0.3333333333333333) y) 0.5) y) 1.0) y)
    z
    (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4.4e+51) || !(x <= 7e+38)) {
		tmp = log(y) * x;
	} else {
		tmp = fma((((((((-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y), z, -t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4.4e+51) || !(x <= 7e+38))
		tmp = Float64(log(y) * x);
	else
		tmp = fma(Float64(Float64(Float64(Float64(Float64(Float64(Float64(-0.25 * y) - 0.3333333333333333) * y) - 0.5) * y) - 1.0) * y), z, Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4.4e+51], N[Not[LessEqual[x, 7e+38]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(N[(N[(N[(-0.25 * y), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * y), $MachinePrecision] - 0.5), $MachinePrecision] * y), $MachinePrecision] - 1.0), $MachinePrecision] * y), $MachinePrecision] * z + (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.39999999999999984e51 or 7.00000000000000003e38 < x

   1. Initial program 96.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 99.6

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r* N/A

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

      10. distribute-rgt-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 N/A

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

      15. *-lft-identity N/A

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

      16. metadata-eval N/A

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in x around inf

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

       1. remove-double-neg N/A

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

       2. log-rec N/A

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

       3. mul-1-neg N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. mul-1-neg N/A

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

       7. log-rec N/A

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

       8. remove-double-neg N/A

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

       9. lower-log.f64 78.2

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

   13. Applied rewrites 78.2%

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

   if -4.39999999999999984e51 < x < 7.00000000000000003e38

   1. Initial program 79.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-log.f64 N/A

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

      9. lift--.f64 N/A

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

      10. *-lft-identity N/A

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

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

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

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

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

      13. *-lft-identity N/A

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

      14. lower-log1p.f64 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower--.f64 99.9

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

      17. lift-*.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 84.1

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

   7. Applied rewrites 84.1%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 83.6

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

   10. Applied rewrites 83.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+51} \lor \neg \left(x \leq 7 \cdot 10^{+38}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(\left(-0.25 \cdot y - 0.3333333333333333\right) \cdot y - 0.5\right) \cdot y - 1\right) \cdot y, z, -t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. *-commutative N/A

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

   9. cancel-sign-sub N/A

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

   10. mul-1-neg N/A

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

   11. associate-*r* N/A

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

   12. mul-1-neg N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. *-commutative N/A

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.8%

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

9. Final simplification 98.8%

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

Alternative 12: 58.0% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-log.f64 N/A

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

   9. lift--.f64 N/A

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

   10. *-lft-identity N/A

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

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

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

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

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

   13. *-lft-identity N/A

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

   14. lower-log1p.f64 N/A

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

   15. lower-neg.f64 N/A

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

   16. lower--.f64 99.8

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 60.1

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

7. Applied rewrites 60.1%

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

8. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower--.f64 N/A

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

   10. lower-*.f64 59.9

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

10. Applied rewrites 59.9%

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

Alternative 13: 48.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-51} \lor \neg \left(t \leq 8 \cdot 10^{-205}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.8e-51) (not (<= t 8e-205))) (- t) (* (- z) y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.8e-51) || !(t <= 8e-205)) {
		tmp = -t;
	} else {
		tmp = -z * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-4.8d-51)) .or. (.not. (t <= 8d-205))) then
        tmp = -t
    else
        tmp = -z * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.8e-51) || !(t <= 8e-205)) {
		tmp = -t;
	} else {
		tmp = -z * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -4.8e-51) or not (t <= 8e-205):
		tmp = -t
	else:
		tmp = -z * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.8e-51) || !(t <= 8e-205))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(-z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4.8e-51) || ~((t <= 8e-205)))
		tmp = -t;
	else
		tmp = -z * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.8e-51], N[Not[LessEqual[t, 8e-205]], $MachinePrecision]], (-t), N[((-z) * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.8e-51 or 8e-205 < t

   1. Initial program 92.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 63.2

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

   5. Applied rewrites 63.2%

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

   if -4.8e-51 < t < 8e-205

   1. Initial program 72.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. mul-1-neg N/A

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

      8. *-commutative N/A

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

      9. cancel-sign-sub N/A

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

      10. mul-1-neg N/A

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

      11. associate-*r* N/A

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

      12. mul-1-neg N/A

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

      13. associate-*r* N/A

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

      14. *-commutative N/A

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

      15. *-commutative N/A

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 35.5%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 30.9%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-51} \lor \neg \left(t \leq 8 \cdot 10^{-205}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.8% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

   1. lift-+.f64 N/A

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

   2. flip-+ N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. div-add N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. swap-sqr N/A

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

   8. associate-/l* N/A

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

4. Applied rewrites 67.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{{\log y}^{2}}{\mathsf{fma}\left(-z, \mathsf{log1p}\left(-y\right), \log y \cdot x\right)}, \frac{-{\left(\mathsf{log1p}\left(-y\right) \cdot z\right)}^{2}}{\mathsf{fma}\left(-z, \mathsf{log1p}\left(-y\right), \log y \cdot x\right)}\right)} - t \]

5. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-log.f64 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 59.7%

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

Alternative 15: 57.4% accurate, 24.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. mul-1-neg N/A

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

   8. *-commutative N/A

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

   9. cancel-sign-sub N/A

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

   10. mul-1-neg N/A

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

   11. associate-*r* N/A

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

   12. mul-1-neg N/A

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

   13. associate-*r* N/A

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

   14. *-commutative N/A

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

   15. *-commutative N/A

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 59.4%

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

9. Final simplification 59.4%

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

Alternative 16: 43.4% accurate, 73.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.3%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 45.4

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

5. Applied rewrites 45.4%

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

6. Final simplification 45.4%

   \[\leadsto -t \]
7. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* (- z) (+ (+ (* 1/2 (* y y)) y) (* (/ 1/3 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y)))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))