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

Percentage Accurate: 85.7% → 84.7%

Time: 3.3s

Alternatives: 5

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 84.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \log y - t \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * log(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)) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 86.4%

   \[\leadsto \color{blue}{x \cdot \log y} - t \]
6. Add Preprocessing

Alternative 2: 63.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -4.6 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-8}:\\ \;\;\;\;\log \left(1 - y\right) - t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -4.6e+140) t_1 (if (<= x 6.5e-8) (- (log (- 1.0 y)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -4.6e+140) {
		tmp = t_1;
	} else if (x <= 6.5e-8) {
		tmp = log((1.0 - 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 = x * log(y)
    if (x <= (-4.6d+140)) then
        tmp = t_1
    else if (x <= 6.5d-8) then
        tmp = log((1.0d0 - 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 = x * Math.log(y);
	double tmp;
	if (x <= -4.6e+140) {
		tmp = t_1;
	} else if (x <= 6.5e-8) {
		tmp = Math.log((1.0 - y)) - t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -4.6e+140:
		tmp = t_1
	elif x <= 6.5e-8:
		tmp = math.log((1.0 - y)) - t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -4.6e+140)
		tmp = t_1;
	elseif (x <= 6.5e-8)
		tmp = Float64(log(Float64(1.0 - y)) - t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -4.6e+140)
		tmp = t_1;
	elseif (x <= 6.5e-8)
		tmp = log((1.0 - y)) - t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.6e+140], t$95$1, If[LessEqual[x, 6.5e-8], N[(N[Log[N[(1.0 - y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.59999999999999981e140 or 6.49999999999999997e-8 < x

   1. Initial program 95.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 76.2%

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

   if -4.59999999999999981e140 < x < 6.49999999999999997e-8

   1. Initial program 80.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 65.5%

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

Alternative 3: 65.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - y\right) - t\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (- 1.0 y) t)))
   (if (<= t -2.3e+88) t_1 (if (<= t 5e+86) (* x (log y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - y) - t;
	double tmp;
	if (t <= -2.3e+88) {
		tmp = t_1;
	} else if (t <= 5e+86) {
		tmp = x * log(y);
	} 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 = (1.0d0 - y) - t
    if (t <= (-2.3d+88)) then
        tmp = t_1
    else if (t <= 5d+86) then
        tmp = x * log(y)
    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 = (1.0 - y) - t;
	double tmp;
	if (t <= -2.3e+88) {
		tmp = t_1;
	} else if (t <= 5e+86) {
		tmp = x * Math.log(y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (1.0 - y) - t
	tmp = 0
	if t <= -2.3e+88:
		tmp = t_1
	elif t <= 5e+86:
		tmp = x * math.log(y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(1.0 - y) - t)
	tmp = 0.0
	if (t <= -2.3e+88)
		tmp = t_1;
	elseif (t <= 5e+86)
		tmp = Float64(x * log(y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (1.0 - y) - t;
	tmp = 0.0;
	if (t <= -2.3e+88)
		tmp = t_1;
	elseif (t <= 5e+86)
		tmp = x * log(y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(1.0 - y), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[t, -2.3e+88], t$95$1, If[LessEqual[t, 5e+86], N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.3000000000000002e88 or 4.9999999999999998e86 < t

   1. Initial program 98.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 80.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 80.3%

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

   if -2.3000000000000002e88 < t < 4.9999999999999998e86

   1. Initial program 76.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 59.7%

      \[\leadsto \color{blue}{x \cdot \log y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 35.9% accurate, 31.4× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (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 = (1.0d0 - y) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 86.4%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 47.2%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 41.3%

    \[\leadsto \color{blue}{\left(1 - y\right)} - t \]
12. Add Preprocessing

Alternative 5: 3.3% accurate, 55.0× speedup

Math

\[\begin{array}{l} \\ 1 - y \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 1.0 - 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 = 1.0d0 - y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 86.4%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 2.9%

   \[\leadsto \color{blue}{1 - y} \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024313 
(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))