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

Percentage Accurate: 89.3% → 99.4%

Time: 16.3s

Alternatives: 11

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: 89.3% 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: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x + -1.0) * log(y)) + Float64(Float64(z + -1.0) * Float64(y * fma(y, -0.5, -1.0)))) - 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[(y * N[(y * -0.5 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in y around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. metadata-eval N/A

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

   5. lower-fma.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 98.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, -z, x \cdot \log y\right) - t\\ \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x + -1 \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (fma y (- z) (* x (log y))) t)))
   (if (<= (+ x -1.0) -5e+15)
     t_1
     (if (<= (+ x -1.0) -0.2) (- (fma y (- 1.0 z) (- (log y))) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, -z, (x * log(y))) - t;
	double tmp;
	if ((x + -1.0) <= -5e+15) {
		tmp = t_1;
	} else if ((x + -1.0) <= -0.2) {
		tmp = fma(y, (1.0 - z), -log(y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(y, Float64(-z), Float64(x * log(y))) - t)
	tmp = 0.0
	if (Float64(x + -1.0) <= -5e+15)
		tmp = t_1;
	elseif (Float64(x + -1.0) <= -0.2)
		tmp = Float64(fma(y, Float64(1.0 - z), Float64(-log(y))) - t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * (-z) + N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[(x + -1.0), $MachinePrecision], -5e+15], t$95$1, If[LessEqual[N[(x + -1.0), $MachinePrecision], -0.2], N[(N[(y * N[(1.0 - z), $MachinePrecision] + (-N[Log[y], $MachinePrecision])), $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 x #s(literal 1 binary64)) < -5e15 or -0.20000000000000001 < (-.f64 x #s(literal 1 binary64))

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 98.8%

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

   if -5e15 < (-.f64 x #s(literal 1 binary64)) < -0.20000000000000001

   1. Initial program 84.5%

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

   3. Taylor expanded in 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. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-sub0 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. associate--r+ N/A

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

      11. metadata-eval N/A

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

      12. lower--.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. sub-neg N/A

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 97.3%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + -1 \leq -5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot \log y\right) - t\\ \mathbf{elif}\;x + -1 \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - z, -\log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -z, x \cdot \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(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))