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

Percentage Accurate: 89.2% → 99.8%

Time: 14.3s

Alternatives: 19

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

Alternative 2: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 99.7

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

7. Applied rewrites 99.7%

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

Alternative 3: 99.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 99.6

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

5. Applied rewrites 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 95.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{elif}\;x - 1 \leq -1:\\ \;\;\;\;\mathsf{fma}\left(1 - z, y, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- x 1.0) -1e+21)
   (fma (- x 1.0) (log y) (- t))
   (if (<= (- x 1.0) -1.0)
     (- (fma (- 1.0 z) y (- (log y))) t)
     (- (fma (- x 1.0) (log y) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x - 1.0) <= -1e+21) {
		tmp = fma((x - 1.0), log(y), -t);
	} else if ((x - 1.0) <= -1.0) {
		tmp = fma((1.0 - z), y, -log(y)) - t;
	} else {
		tmp = fma((x - 1.0), log(y), y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -1e+21)
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	elseif (Float64(x - 1.0) <= -1.0)
		tmp = Float64(fma(Float64(1.0 - z), y, Float64(-log(y))) - t);
	else
		tmp = Float64(fma(Float64(x - 1.0), log(y), y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x - 1.0), $MachinePrecision], -1e+21], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.0], N[(N[(N[(1.0 - z), $MachinePrecision] * y + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e21

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 91.9

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

   5. Applied rewrites 91.9%

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

   if -1e21 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 78.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-lft-in N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

       9. mul-1-neg N/A

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

       10. sub-neg N/A

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

       11. lower--.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower--.f64 N/A

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

       15. lower-log.f64 99.2

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

   11. Applied rewrites 99.2%

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

   12. Taylor expanded in x around 0

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

   14. Applied rewrites 98.6%

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

   if -1 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 90.8%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 33.5

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

   5. Applied rewrites 33.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 33.5%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-lft-in N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

       9. mul-1-neg N/A

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

       10. sub-neg N/A

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

       11. lower--.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower--.f64 N/A

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

       15. lower-log.f64 99.8

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

   11. Applied rewrites 99.8%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 90.8%

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

Alternative 5: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--l+ N/A

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

   4. lift-*.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. sub-neg N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lift-log.f64 N/A

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

   11. lift--.f64 N/A

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

   12. sub-neg N/A

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

   13. lower-log1p.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lower-neg.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. distribute-rgt-out N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. sub-neg N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-fma.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 7: 95.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{if}\;t \leq -24:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 350000:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (- x 1.0) (log y) (- t))))
   (if (<= t -24.0)
     t_1
     (if (<= t 350000.0) (fma (- x 1.0) (log y) (- (* z y))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((x - 1.0), log(y), -t);
	double tmp;
	if (t <= -24.0) {
		tmp = t_1;
	} else if (t <= 350000.0) {
		tmp = fma((x - 1.0), log(y), -(z * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(x - 1.0), log(y), Float64(-t))
	tmp = 0.0
	if (t <= -24.0)
		tmp = t_1;
	elseif (t <= 350000.0)
		tmp = fma(Float64(x - 1.0), log(y), Float64(-Float64(z * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision]}, If[LessEqual[t, -24.0], t$95$1, If[LessEqual[t, 350000.0], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-N[(z * y), $MachinePrecision])), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -24 or 3.5e5 < t

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 95.1

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

   5. Applied rewrites 95.1%

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

   if -24 < t < 3.5e5

   1. Initial program 73.4%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--l+ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. sub-neg N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-log.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. lower-log1p.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-neg.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-out N/A

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

      4. lower-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 98.1%

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

Alternative 8: 76.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x - t\\ \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq -1:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* (log y) x) t)))
   (if (<= (- x 1.0) -1e+21)
     t_1
     (if (<= (- x 1.0) -1.0) (- (fma (- y) z y) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (log(y) * x) - t;
	double tmp;
	if ((x - 1.0) <= -1e+21) {
		tmp = t_1;
	} else if ((x - 1.0) <= -1.0) {
		tmp = fma(-y, z, y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(log(y) * x) - t)
	tmp = 0.0
	if (Float64(x - 1.0) <= -1e+21)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= -1.0)
		tmp = Float64(fma(Float64(-y), z, y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -1e+21], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], -1.0], N[(N[((-y) * z + y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -1e21 or -1 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 91.9%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 91.1

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

   5. Applied rewrites 91.1%

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

   if -1e21 < (-.f64 x #s(literal 1 binary64)) < -1

   1. Initial program 78.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.1%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-lft-in N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

       9. mul-1-neg N/A

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

       10. sub-neg N/A

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

       11. lower--.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower--.f64 N/A

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

       15. lower-log.f64 99.2

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

   11. Applied rewrites 99.2%

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

   12. Taylor expanded in y around inf

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

   14. Applied rewrites 67.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -1 \cdot 10^{+21}:\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{elif}\;x - 1 \leq -1:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x - 1 \leq 8.1 \cdot 10^{+18}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= (- x 1.0) -2e+138)
     t_1
     (if (<= (- x 1.0) 8.1e+18)
       (-
        (* (* (fma (fma (fma -0.25 y -0.3333333333333333) y -0.5) y -1.0) y) z)
        t)
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if ((x - 1.0) <= -2e+138) {
		tmp = t_1;
	} else if ((x - 1.0) <= 8.1e+18) {
		tmp = ((fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (Float64(x - 1.0) <= -2e+138)
		tmp = t_1;
	elseif (Float64(x - 1.0) <= 8.1e+18)
		tmp = Float64(Float64(Float64(fma(fma(fma(-0.25, y, -0.3333333333333333), y, -0.5), y, -1.0) * y) * z) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(x - 1.0), $MachinePrecision], -2e+138], t$95$1, If[LessEqual[N[(x - 1.0), $MachinePrecision], 8.1e+18], N[(N[(N[(N[(N[(N[(-0.25 * y + -0.3333333333333333), $MachinePrecision] * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -2.0000000000000001e138 or 8.1e18 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 93.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if -2.0000000000000001e138 < (-.f64 x #s(literal 1 binary64)) < 8.1e18

   1. Initial program 79.6%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.1%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -2 \cdot 10^{+138}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x - 1 \leq 8.1 \cdot 10^{+18}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, y, -0.3333333333333333\right), y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 2 \cdot 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 2e+222)
   (- (fma (- x 1.0) (log y) y) t)
   (- (* (* (fma (fma -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 ((z - 1.0) <= 2e+222) {
		tmp = fma((x - 1.0), log(y), y) - t;
	} else {
		tmp = ((fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 2e+222)
		tmp = Float64(fma(Float64(x - 1.0), log(y), y) - t);
	else
		tmp = Float64(Float64(Float64(fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 2e+222], N[(N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(N[(-0.3333333333333333 * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 2.0000000000000001e222

   1. Initial program 88.9%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 47.6

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

   5. Applied rewrites 47.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 47.5%

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

   9. Taylor expanded in y around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-lft-in N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

       9. mul-1-neg N/A

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

       10. sub-neg N/A

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

       11. lower--.f64 N/A

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

       12. *-commutative N/A

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

       13. lower-*.f64 N/A

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

       14. lower--.f64 N/A

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

       15. lower-log.f64 99.4

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

   11. Applied rewrites 99.4%

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

   12. Taylor expanded in z around 0

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

   14. Applied rewrites 88.4%

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

   if 2.0000000000000001e222 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 55.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - 1 \leq 2 \cdot 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 89.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 2 \cdot 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 2e+222)
   (fma (- x 1.0) (log y) (- t))
   (- (* (* (fma (fma -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 ((z - 1.0) <= 2e+222) {
		tmp = fma((x - 1.0), log(y), -t);
	} else {
		tmp = ((fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y) * z) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 2e+222)
		tmp = fma(Float64(x - 1.0), log(y), Float64(-t));
	else
		tmp = Float64(Float64(Float64(fma(fma(-0.3333333333333333, y, -0.5), y, -1.0) * y) * z) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 2e+222], N[(N[(x - 1.0), $MachinePrecision] * N[Log[y], $MachinePrecision] + (-t)), $MachinePrecision], N[(N[(N[(N[(N[(-0.3333333333333333 * y + -0.5), $MachinePrecision] * y + -1.0), $MachinePrecision] * y), $MachinePrecision] * z), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 2.0000000000000001e222

   1. Initial program 88.9%

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

   3. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-neg.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if 2.0000000000000001e222 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 55.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. sub-neg N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-neg.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.8%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - 1 \leq 2 \cdot 10^{+222}:\\ \;\;\;\;\mathsf{fma}\left(x - 1, \log y, -t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, y, -0.5\right), y, -1\right) \cdot y\right) \cdot z - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in y around 0

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

   1. associate--l+ N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. mul-1-neg N/A

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

   6. neg-sub0 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. +-commutative N/A

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

   10. associate--r+ N/A

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

   11. metadata-eval N/A

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

   12. lower--.f64 N/A

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

   13. sub-neg N/A

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

   14. *-commutative N/A

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

   15. lower-fma.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lower-log.f64 N/A

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

   18. lower-neg.f64 99.3

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

5. Applied rewrites 99.3%

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

Alternative 13: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 50.0

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

5. Applied rewrites 50.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 49.7%

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

9. Taylor expanded in y around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. lower-fma.f64 N/A

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

    4. sub-neg N/A

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

    5. metadata-eval N/A

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

    6. distribute-lft-in N/A

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

    7. metadata-eval N/A

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

    8. +-commutative N/A

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

    9. mul-1-neg N/A

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

    10. sub-neg N/A

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

    11. lower--.f64 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 N/A

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

    14. lower--.f64 N/A

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

    15. lower-log.f64 99.3

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

11. Applied rewrites 99.3%

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

12. Taylor expanded in z around inf

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

14. Applied rewrites 99.3%

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

15. Final simplification 99.3%

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

Alternative 14: 46.4% accurate, 7.1× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 50.0

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

5. Applied rewrites 50.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 50.0%

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

9. Final simplification 50.0%

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

Alternative 15: 46.4% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 50.0

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

5. Applied rewrites 50.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 50.0%

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

9. Final simplification 50.0%

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

Alternative 16: 46.3% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 50.0

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

5. Applied rewrites 50.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 49.9%

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

9. Final simplification 49.9%

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

Alternative 17: 46.2% accurate, 18.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 50.0

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

5. Applied rewrites 50.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 49.7%

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

9. Taylor expanded in y around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. lower-fma.f64 N/A

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

    4. sub-neg N/A

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

    5. metadata-eval N/A

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

    6. distribute-lft-in N/A

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

    7. metadata-eval N/A

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

    8. +-commutative N/A

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

    9. mul-1-neg N/A

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

    10. sub-neg N/A

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

    11. lower--.f64 N/A

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

    12. *-commutative N/A

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

    13. lower-*.f64 N/A

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

    14. lower--.f64 N/A

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

    15. lower-log.f64 99.3

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

11. Applied rewrites 99.3%

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

12. Taylor expanded in y around inf

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

14. Applied rewrites 49.9%

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

Alternative 18: 46.0% accurate, 20.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-neg.f64 50.0

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

5. Applied rewrites 50.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 49.7%

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

9. Final simplification 49.7%

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

Alternative 19: 35.6% accurate, 75.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.1%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 37.1

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

5. Applied rewrites 37.1%

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

Reproduce

herbie shell --seed 2024255 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))