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

Percentage Accurate: 86.1% → 99.3%

Time: 16.8s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{log1p}\left(-y\right)\\ t_2 := \mathsf{fma}\left(z, t\_1, \mathsf{fma}\left(x, \log y, t\right)\right)\\ t\_2 \cdot \frac{\mathsf{fma}\left(x, \log y, \mathsf{fma}\left(z, t\_1, -t\right)\right)}{t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (log1p (- y))) (t_2 (fma z t_1 (fma x (log y) t))))
   (* t_2 (/ (fma x (log y) (fma z t_1 (- t))) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log1p(-y);
	double t_2 = fma(z, t_1, fma(x, log(y), t));
	return t_2 * (fma(x, log(y), fma(z, t_1, -t)) / t_2);
}

Julia

function code(x, y, z, t)
	t_1 = log1p(Float64(-y))
	t_2 = fma(z, t_1, fma(x, log(y), t))
	return Float64(t_2 * Float64(fma(x, log(y), fma(z, t_1, Float64(-t))) / t_2))
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Log[1 + (-y)], $MachinePrecision]}, Block[{t$95$2 = N[(z * t$95$1 + N[(x * N[Log[y], $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]}, N[(t$95$2 * N[(N[(x * N[Log[y], $MachinePrecision] + N[(z * t$95$1 + (-t)), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 83.9%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. difference-of-squares N/A

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

   4. lift--.f64 N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.1%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip-- N/A

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

   7. lift--.f64 N/A

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

   8. lower-/.f64 85.9

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

4. Applied rewrites 99.6%

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

Developer Target 1: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024228 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

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

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