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

Percentage Accurate: 89.7% → 99.5%

Time: 13.3s

Alternatives: 15

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: 89.7% 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: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in y around 0

   \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
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. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

5. Applied rewrites 99.5%

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

Alternative 3: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

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

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

5. Applied rewrites 99.3%

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

Alternative 4: 96.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-10} \lor \neg \left(x \leq 9.5 \cdot 10^{-6}\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\left(\mathsf{fma}\left(z - 1, y, \log y\right) + t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -4e-10) (not (<= x 9.5e-6)))
   (- (* (+ -1.0 x) (log y)) t)
   (- (+ (fma (- z 1.0) y (log y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -4e-10) || !(x <= 9.5e-6)) {
		tmp = ((-1.0 + x) * log(y)) - t;
	} else {
		tmp = -(fma((z - 1.0), y, log(y)) + t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -4e-10) || !(x <= 9.5e-6))
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	else
		tmp = Float64(-Float64(fma(Float64(z - 1.0), y, log(y)) + t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -4e-10], N[Not[LessEqual[x, 9.5e-6]], $MachinePrecision]], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], (-N[(N[(N[(z - 1.0), $MachinePrecision] * y + N[Log[y], $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < -4.00000000000000015e-10 or 9.5000000000000005e-6 < x

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

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

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. remove-double-neg N/A

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

      16. log-rec N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. log-rec N/A

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

   5. Applied rewrites 95.4%

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

   if -4.00000000000000015e-10 < x < 9.5000000000000005e-6

   1. Initial program 81.9%

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

   3. Taylor expanded in y around 0

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

      1. 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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-10} \lor \neg \left(x \leq 9.5 \cdot 10^{-6}\right):\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;-\left(\mathsf{fma}\left(z - 1, y, \log y\right) + t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x - 1 \leq -40000000 \lor \neg \left(x - 1 \leq -0.5\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (- x 1.0) -40000000.0) (not (<= (- x 1.0) -0.5)))
   (- (* (log y) x) t)
   (- (- y (log y)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -40000000.0) || !((x - 1.0) <= -0.5)) {
		tmp = (log(y) * x) - t;
	} else {
		tmp = (y - log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x - 1.0d0) <= (-40000000.0d0)) .or. (.not. ((x - 1.0d0) <= (-0.5d0)))) then
        tmp = (log(y) * x) - t
    else
        tmp = (y - log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x - 1.0) <= -40000000.0) || !((x - 1.0) <= -0.5)) {
		tmp = (Math.log(y) * x) - t;
	} else {
		tmp = (y - Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x - 1.0) <= -40000000.0) or not ((x - 1.0) <= -0.5):
		tmp = (math.log(y) * x) - t
	else:
		tmp = (y - math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x - 1.0) <= -40000000.0) || !(Float64(x - 1.0) <= -0.5))
		tmp = Float64(Float64(log(y) * x) - t);
	else
		tmp = Float64(Float64(y - log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x - 1.0) <= -40000000.0) || ~(((x - 1.0) <= -0.5)))
		tmp = (log(y) * x) - t;
	else
		tmp = (y - log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x - 1.0), $MachinePrecision], -40000000.0], N[Not[LessEqual[N[(x - 1.0), $MachinePrecision], -0.5]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision] - t), $MachinePrecision], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -4e7 or -0.5 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 95.6%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. log-rec N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. log-rec N/A

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

      8. remove-double-neg N/A

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

      9. lower-log.f64 95.3

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

   5. Applied rewrites 95.3%

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

   if -4e7 < (-.f64 x #s(literal 1 binary64)) < -0.5

   1. Initial program 82.8%

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

   3. Taylor expanded in y around 0

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

      1. 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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 80.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 78.6%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - 1 \leq -40000000 \lor \neg \left(x - 1 \leq -0.5\right):\\ \;\;\;\;\log y \cdot x - t\\ \mathbf{else}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-37}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -1.5e+22)
     t_1
     (if (<= x 2.4e-37)
       (- (- y (log y)) t)
       (if (<= x 1.5e+58) (- (* (* (- (* -0.5 y) 1.0) z) y) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -1.5e+22) {
		tmp = t_1;
	} else if (x <= 2.4e-37) {
		tmp = (y - log(y)) - t;
	} else if (x <= 1.5e+58) {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-1.5d+22)) then
        tmp = t_1
    else if (x <= 2.4d-37) then
        tmp = (y - log(y)) - t
    else if (x <= 1.5d+58) then
        tmp = (((((-0.5d0) * y) - 1.0d0) * z) * y) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -1.5e+22) {
		tmp = t_1;
	} else if (x <= 2.4e-37) {
		tmp = (y - Math.log(y)) - t;
	} else if (x <= 1.5e+58) {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -1.5e+22:
		tmp = t_1
	elif x <= 2.4e-37:
		tmp = (y - math.log(y)) - t
	elif x <= 1.5e+58:
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -1.5e+22)
		tmp = t_1;
	elseif (x <= 2.4e-37)
		tmp = Float64(Float64(y - log(y)) - t);
	elseif (x <= 1.5e+58)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * z) * y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -1.5e+22)
		tmp = t_1;
	elseif (x <= 2.4e-37)
		tmp = (y - log(y)) - t;
	elseif (x <= 1.5e+58)
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.5e+22], t$95$1, If[LessEqual[x, 2.4e-37], N[(N[(y - N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[x, 1.5e+58], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e22 or 1.5000000000000001e58 < x

   1. Initial program 96.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 77.4

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

   10. Applied rewrites 77.4%

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

   if -1.5e22 < x < 2.39999999999999991e-37

   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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 83.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 80.6%

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

   if 2.39999999999999991e-37 < x < 1.5000000000000001e58

   1. Initial program 73.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 71.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-37}:\\ \;\;\;\;\left(y - \log y\right) - t\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log y \cdot x\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-37}:\\ \;\;\;\;-\left(\log y + t\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (log y) x)))
   (if (<= x -1.5e+22)
     t_1
     (if (<= x 2.4e-37)
       (- (+ (log y) t))
       (if (<= x 1.5e+58) (- (* (* (- (* -0.5 y) 1.0) z) y) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log(y) * x;
	double tmp;
	if (x <= -1.5e+22) {
		tmp = t_1;
	} else if (x <= 2.4e-37) {
		tmp = -(log(y) + t);
	} else if (x <= 1.5e+58) {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log(y) * x
    if (x <= (-1.5d+22)) then
        tmp = t_1
    else if (x <= 2.4d-37) then
        tmp = -(log(y) + t)
    else if (x <= 1.5d+58) then
        tmp = (((((-0.5d0) * y) - 1.0d0) * z) * y) - t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log(y) * x;
	double tmp;
	if (x <= -1.5e+22) {
		tmp = t_1;
	} else if (x <= 2.4e-37) {
		tmp = -(Math.log(y) + t);
	} else if (x <= 1.5e+58) {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log(y) * x
	tmp = 0
	if x <= -1.5e+22:
		tmp = t_1
	elif x <= 2.4e-37:
		tmp = -(math.log(y) + t)
	elif x <= 1.5e+58:
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log(y) * x)
	tmp = 0.0
	if (x <= -1.5e+22)
		tmp = t_1;
	elseif (x <= 2.4e-37)
		tmp = Float64(-Float64(log(y) + t));
	elseif (x <= 1.5e+58)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * z) * y) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = log(y) * x;
	tmp = 0.0;
	if (x <= -1.5e+22)
		tmp = t_1;
	elseif (x <= 2.4e-37)
		tmp = -(log(y) + t);
	elseif (x <= 1.5e+58)
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.5e+22], t$95$1, If[LessEqual[x, 2.4e-37], (-N[(N[Log[y], $MachinePrecision] + t), $MachinePrecision]), If[LessEqual[x, 1.5e+58], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e22 or 1.5000000000000001e58 < x

   1. Initial program 96.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 77.4

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

   10. Applied rewrites 77.4%

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

   if -1.5e22 < x < 2.39999999999999991e-37

   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

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

      2. lift--.f64 N/A

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

      3. flip-- N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. difference-of-squares-rev N/A

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

      8. difference-of-sqr--1-rev N/A

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

      9. metadata-eval N/A

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

      10. lower-fma.f64 N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 85.0

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

   4. Applied rewrites 85.0%

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

   5. Taylor expanded in y around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-+.f64 82.6

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

   7. Applied rewrites 82.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 80.2%

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

   if 2.39999999999999991e-37 < x < 1.5000000000000001e58

   1. Initial program 73.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 71.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+22}:\\ \;\;\;\;\log y \cdot x\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-37}:\\ \;\;\;\;-\left(\log y + t\right)\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \mathbf{else}:\\ \;\;\;\;\log y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 10^{+269}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x - 1, y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 1e+269)
   (- (fma (log y) (- x 1.0) y) t)
   (- (* (* (- (* -0.5 y) 1.0) z) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 1e+269) {
		tmp = fma(log(y), (x - 1.0), y) - t;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 1e+269)
		tmp = Float64(fma(log(y), Float64(x - 1.0), y) - t);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * z) * y) - t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 1e+269], N[(N[(N[Log[y], $MachinePrecision] * N[(x - 1.0), $MachinePrecision] + y), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 1e269

   1. Initial program 91.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right) - t} \]
   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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 90.9%

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

   if 1e269 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 42.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 78.2%

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

Alternative 9: 89.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z - 1 \leq 10^{+269}:\\ \;\;\;\;\left(-1 + x\right) \cdot \log y - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (- z 1.0) 1e+269)
   (- (* (+ -1.0 x) (log y)) t)
   (- (* (* (- (* -0.5 y) 1.0) z) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 1e+269) {
		tmp = ((-1.0 + x) * log(y)) - t;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z - 1.0d0) <= 1d+269) then
        tmp = (((-1.0d0) + x) * log(y)) - t
    else
        tmp = (((((-0.5d0) * y) - 1.0d0) * z) * y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z - 1.0) <= 1e+269) {
		tmp = ((-1.0 + x) * Math.log(y)) - t;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z - 1.0) <= 1e+269:
		tmp = ((-1.0 + x) * math.log(y)) - t
	else:
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z - 1.0) <= 1e+269)
		tmp = Float64(Float64(Float64(-1.0 + x) * log(y)) - t);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * z) * y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z - 1.0) <= 1e+269)
		tmp = ((-1.0 + x) * log(y)) - t;
	else
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z - 1.0), $MachinePrecision], 1e+269], N[(N[(N[(-1.0 + x), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 z #s(literal 1 binary64)) < 1e269

   1. Initial program 91.6%

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

   3. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

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

      2. log-rec N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

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

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

      9. metadata-eval N/A

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

      10. mul-1-neg N/A

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

      11. log-rec N/A

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

      12. remove-double-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-rgt-out N/A

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

      15. remove-double-neg N/A

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

      16. log-rec N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 N/A

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

      20. lower-+.f64 N/A

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

      21. mul-1-neg N/A

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

      22. log-rec N/A

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

   5. Applied rewrites 90.7%

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

   if 1e269 < (-.f64 z #s(literal 1 binary64))

   1. Initial program 42.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 78.2%

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

Alternative 10: 99.1% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\left(-1 \cdot \left(y \cdot \left(z - 1\right)\right) + \log y \cdot \left(x - 1\right)\right) - t} \]
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. associate-*r* N/A

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

   3. mul-1-neg N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower--.f64 N/A

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

   7. *-rgt-identity N/A

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

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

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 99.1%

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

Alternative 11: 67.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+23} \lor \neg \left(x \leq 1.5 \cdot 10^{+58}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -2.4e+23) (not (<= x 1.5e+58)))
   (* (log y) x)
   (- (* (* (- (* -0.5 y) 1.0) z) y) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e+23) || !(x <= 1.5e+58)) {
		tmp = log(y) * x;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((x <= (-2.4d+23)) .or. (.not. (x <= 1.5d+58))) then
        tmp = log(y) * x
    else
        tmp = (((((-0.5d0) * y) - 1.0d0) * z) * y) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -2.4e+23) || !(x <= 1.5e+58)) {
		tmp = Math.log(y) * x;
	} else {
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -2.4e+23) or not (x <= 1.5e+58):
		tmp = math.log(y) * x
	else:
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -2.4e+23) || !(x <= 1.5e+58))
		tmp = Float64(log(y) * x);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(-0.5 * y) - 1.0) * z) * y) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -2.4e+23) || ~((x <= 1.5e+58)))
		tmp = log(y) * x;
	else
		tmp = ((((-0.5 * y) - 1.0) * z) * y) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -2.4e+23], N[Not[LessEqual[x, 1.5e+58]], $MachinePrecision]], N[(N[Log[y], $MachinePrecision] * x), $MachinePrecision], N[(N[(N[(N[(N[(-0.5 * y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e23 or 1.5000000000000001e58 < x

   1. Initial program 96.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\frac{-1}{3} \cdot y - \frac{1}{2}\right) - 1\right)\right)}\right) - t \]
   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. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 77.4

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

   10. Applied rewrites 77.4%

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

   if -2.4e23 < x < 1.5000000000000001e58

   1. Initial program 83.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. remove-double-neg N/A

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

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

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

      4. log-rec N/A

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

      5. mul-1-neg N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 59.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+23} \lor \neg \left(x \leq 1.5 \cdot 10^{+58}\right):\\ \;\;\;\;\log y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-0.5 \cdot y - 1\right) \cdot z\right) \cdot y - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.3% accurate, 10.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. remove-double-neg N/A

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

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

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

   4. log-rec N/A

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

   5. mul-1-neg N/A

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 41.6%

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

Alternative 13: 42.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+44} \lor \neg \left(t \leq 1.52 \cdot 10^{+29}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1e+44) (not (<= t 1.52e+29))) (- t) (fma (- y) z y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1e+44) || !(t <= 1.52e+29)) {
		tmp = -t;
	} else {
		tmp = fma(-y, z, y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1e+44) || !(t <= 1.52e+29))
		tmp = Float64(-t);
	else
		tmp = fma(Float64(-y), z, y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1e+44], N[Not[LessEqual[t, 1.52e+29]], $MachinePrecision]], (-t), N[((-y) * z + y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.0000000000000001e44 or 1.52e29 < t

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -1.0000000000000001e44 < t < 1.52e29

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. 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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 18.1%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 18.1%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+44} \lor \neg \left(t \leq 1.52 \cdot 10^{+29}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, z, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.4% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+44} \lor \neg \left(t \leq 1.52 \cdot 10^{+29}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1e+44) (not (<= t 1.52e+29))) (- t) (* (- y) z)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1e+44) || !(t <= 1.52e+29)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-1d+44)) .or. (.not. (t <= 1.52d+29))) then
        tmp = -t
    else
        tmp = -y * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1e+44) || !(t <= 1.52e+29)) {
		tmp = -t;
	} else {
		tmp = -y * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1e+44) or not (t <= 1.52e+29):
		tmp = -t
	else:
		tmp = -y * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1e+44) || !(t <= 1.52e+29))
		tmp = Float64(-t);
	else
		tmp = Float64(Float64(-y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1e+44) || ~((t <= 1.52e+29)))
		tmp = -t;
	else
		tmp = -y * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1e+44], N[Not[LessEqual[t, 1.52e+29]], $MachinePrecision]], (-t), N[((-y) * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.0000000000000001e44 or 1.52e29 < t

   1. Initial program 97.7%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -1.0000000000000001e44 < t < 1.52e29

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

      1. 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. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-rgt-identity N/A

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

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

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 17.6%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+44} \lor \neg \left(t \leq 1.52 \cdot 10^{+29}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.1% accurate, 75.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.7%

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

3. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 32.0

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

5. Applied rewrites 32.0%

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

Reproduce

herbie shell --seed 2024326 
(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))