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

Percentage Accurate: 96.3% → 96.3%

Time: 11.2s

Alternatives: 7

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 1.1055524551340896e-99
  2. y = 2.720435503382072e-77
  3. z = 3.785678035480415e-12
  4. t = 7.531594691962872e+39
  5. a = 9.59777746610844e-256
  6. b = 4.851798829716403e-197

• 41.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Final simplification 96.9%

   \[\leadsto e^{\left(\log \left(1 - z\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y} \cdot x \]
4. Add Preprocessing

Alternative 2: 87.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{z}{e^{t}}\right)}^{y} \cdot x\\ \mathbf{if}\;y \leq -480:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-69}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (pow (/ z (exp t)) y) x)))
   (if (<= y -480.0)
     t_1
     (if (<= y 9.5e-69) (* (exp (* (- (- z) b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow((z / exp(t)), y) * x;
	double tmp;
	if (y <= -480.0) {
		tmp = t_1;
	} else if (y <= 9.5e-69) {
		tmp = exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((z / exp(t)) ** y) * x
    if (y <= (-480.0d0)) then
        tmp = t_1
    else if (y <= 9.5d-69) then
        tmp = exp(((-z - b) * a)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow((z / Math.exp(t)), y) * x;
	double tmp;
	if (y <= -480.0) {
		tmp = t_1;
	} else if (y <= 9.5e-69) {
		tmp = Math.exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow((z / math.exp(t)), y) * x
	tmp = 0
	if y <= -480.0:
		tmp = t_1
	elif y <= 9.5e-69:
		tmp = math.exp(((-z - b) * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64((Float64(z / exp(t)) ^ y) * x)
	tmp = 0.0
	if (y <= -480.0)
		tmp = t_1;
	elseif (y <= 9.5e-69)
		tmp = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = ((z / exp(t)) ^ y) * x;
	tmp = 0.0;
	if (y <= -480.0)
		tmp = t_1;
	elseif (y <= 9.5e-69)
		tmp = exp(((-z - b) * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Power[N[(z / N[Exp[t], $MachinePrecision]), $MachinePrecision], y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -480.0], t$95$1, If[LessEqual[y, 9.5e-69], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -480 or 9.50000000000000094e-69 < y

   1. Initial program 96.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 91.7

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 91.7%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   if -480 < y < 9.50000000000000094e-69

   1. Initial program 97.6%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 89.2

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 89.2%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{\left(-1 \cdot z - b\right) \cdot a} \]
   7. Step-by-step derivation

   8. Applied rewrites 89.2%

      \[\leadsto x \cdot e^{\left(\left(-z\right) - b\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -480:\\ \;\;\;\;{\left(\frac{z}{e^{t}}\right)}^{y} \cdot x\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-69}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{z}{e^{t}}\right)}^{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-245}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-64}:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= t -3.4e+29)
     t_1
     (if (<= t 3.6e-245)
       (* (exp (* (- b) a)) x)
       (if (<= t 2e-64) (* (pow z y) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-t * y)) * x;
	double tmp;
	if (t <= -3.4e+29) {
		tmp = t_1;
	} else if (t <= 3.6e-245) {
		tmp = exp((-b * a)) * x;
	} else if (t <= 2e-64) {
		tmp = pow(z, y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-t * y)) * x
    if (t <= (-3.4d+29)) then
        tmp = t_1
    else if (t <= 3.6d-245) then
        tmp = exp((-b * a)) * x
    else if (t <= 2d-64) then
        tmp = (z ** y) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-t * y)) * x;
	double tmp;
	if (t <= -3.4e+29) {
		tmp = t_1;
	} else if (t <= 3.6e-245) {
		tmp = Math.exp((-b * a)) * x;
	} else if (t <= 2e-64) {
		tmp = Math.pow(z, y) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-t * y)) * x
	tmp = 0
	if t <= -3.4e+29:
		tmp = t_1
	elif t <= 3.6e-245:
		tmp = math.exp((-b * a)) * x
	elif t <= 2e-64:
		tmp = math.pow(z, y) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-t) * y)) * x)
	tmp = 0.0
	if (t <= -3.4e+29)
		tmp = t_1;
	elseif (t <= 3.6e-245)
		tmp = Float64(exp(Float64(Float64(-b) * a)) * x);
	elseif (t <= 2e-64)
		tmp = Float64((z ^ y) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-t * y)) * x;
	tmp = 0.0;
	if (t <= -3.4e+29)
		tmp = t_1;
	elseif (t <= 3.6e-245)
		tmp = exp((-b * a)) * x;
	elseif (t <= 2e-64)
		tmp = (z ^ y) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -3.4e+29], t$95$1, If[LessEqual[t, 3.6e-245], N[(N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t, 2e-64], N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.39999999999999981e29 or 1.99999999999999993e-64 < t

   1. Initial program 96.3%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 83.1

         \[\leadsto x \cdot e^{\color{blue}{\left(-t\right)} \cdot y} \]

   5. Applied rewrites 83.1%

      \[\leadsto x \cdot e^{\color{blue}{\left(-t\right) \cdot y}} \]

   if -3.39999999999999981e29 < t < 3.59999999999999999e-245

   1. Initial program 98.8%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{b \cdot a}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot a}} \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot a}} \]

      5. lower-neg.f64 77.9

         \[\leadsto x \cdot e^{\color{blue}{\left(-b\right)} \cdot a} \]

   5. Applied rewrites 77.9%

      \[\leadsto x \cdot e^{\color{blue}{\left(-b\right) \cdot a}} \]

   if 3.59999999999999999e-245 < t < 1.99999999999999993e-64

   1. Initial program 94.4%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 77.8

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 77.8%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.8%

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+29}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-245}:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-64}:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-21}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (exp (* (- t) y)) x)))
   (if (<= t -7.2e+29)
     t_1
     (if (<= t 2e-21) (* (exp (* (- (- z) b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = exp((-t * y)) * x;
	double tmp;
	if (t <= -7.2e+29) {
		tmp = t_1;
	} else if (t <= 2e-21) {
		tmp = exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = exp((-t * y)) * x
    if (t <= (-7.2d+29)) then
        tmp = t_1
    else if (t <= 2d-21) then
        tmp = exp(((-z - b) * a)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.exp((-t * y)) * x;
	double tmp;
	if (t <= -7.2e+29) {
		tmp = t_1;
	} else if (t <= 2e-21) {
		tmp = Math.exp(((-z - b) * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.exp((-t * y)) * x
	tmp = 0
	if t <= -7.2e+29:
		tmp = t_1
	elif t <= 2e-21:
		tmp = math.exp(((-z - b) * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(exp(Float64(Float64(-t) * y)) * x)
	tmp = 0.0
	if (t <= -7.2e+29)
		tmp = t_1;
	elseif (t <= 2e-21)
		tmp = Float64(exp(Float64(Float64(Float64(-z) - b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = exp((-t * y)) * x;
	tmp = 0.0;
	if (t <= -7.2e+29)
		tmp = t_1;
	elseif (t <= 2e-21)
		tmp = exp(((-z - b) * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Exp[N[((-t) * y), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -7.2e+29], t$95$1, If[LessEqual[t, 2e-21], N[(N[Exp[N[(N[((-z) - b), $MachinePrecision] * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.19999999999999952e29 or 1.99999999999999982e-21 < t

   1. Initial program 96.1%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 84.7

         \[\leadsto x \cdot e^{\color{blue}{\left(-t\right)} \cdot y} \]

   5. Applied rewrites 84.7%

      \[\leadsto x \cdot e^{\color{blue}{\left(-t\right) \cdot y}} \]

   if -7.19999999999999952e29 < t < 1.99999999999999982e-21

   1. Initial program 97.7%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x \cdot e^{\color{blue}{a \cdot \left(\log \left(1 - z\right) - b\right)}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right) \cdot a}} \]

      3. lower--.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\log \left(1 - z\right) - b\right)} \cdot a} \]

      4. sub-neg N/A

         \[\leadsto x \cdot e^{\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(z\right)\right)\right)} - b\right) \cdot a} \]

      5. lower-log1p.f64 N/A

         \[\leadsto x \cdot e^{\left(\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(z\right)\right)} - b\right) \cdot a} \]

      6. lower-neg.f64 78.1

         \[\leadsto x \cdot e^{\left(\mathsf{log1p}\left(\color{blue}{-z}\right) - b\right) \cdot a} \]

   5. Applied rewrites 78.1%

      \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{log1p}\left(-z\right) - b\right) \cdot a}} \]

   6. Taylor expanded in z around 0

      \[\leadsto x \cdot e^{\left(-1 \cdot z - b\right) \cdot a} \]
   7. Step-by-step derivation

   8. Applied rewrites 78.1%

      \[\leadsto x \cdot e^{\left(\left(-z\right) - b\right) \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{+29}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-21}:\\ \;\;\;\;e^{\left(\left(-z\right) - b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-t\right) \cdot y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {z}^{y} \cdot x\\ \mathbf{if}\;y \leq -1900:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.086:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (pow z y) x)))
   (if (<= y -1900.0) t_1 (if (<= y 0.086) (* (exp (* (- b) a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = pow(z, y) * x;
	double tmp;
	if (y <= -1900.0) {
		tmp = t_1;
	} else if (y <= 0.086) {
		tmp = exp((-b * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z ** y) * x
    if (y <= (-1900.0d0)) then
        tmp = t_1
    else if (y <= 0.086d0) then
        tmp = exp((-b * a)) * x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = Math.pow(z, y) * x;
	double tmp;
	if (y <= -1900.0) {
		tmp = t_1;
	} else if (y <= 0.086) {
		tmp = Math.exp((-b * a)) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.pow(z, y) * x
	tmp = 0
	if y <= -1900.0:
		tmp = t_1
	elif y <= 0.086:
		tmp = math.exp((-b * a)) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64((z ^ y) * x)
	tmp = 0.0
	if (y <= -1900.0)
		tmp = t_1;
	elseif (y <= 0.086)
		tmp = Float64(exp(Float64(Float64(-b) * a)) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (z ^ y) * x;
	tmp = 0.0;
	if (y <= -1900.0)
		tmp = t_1;
	elseif (y <= 0.086)
		tmp = exp((-b * a)) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -1900.0], t$95$1, If[LessEqual[y, 0.086], N[(N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1900 or 0.085999999999999993 < y

   1. Initial program 96.5%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. *-commutative N/A

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

      2. exp-prod N/A

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

      3. lower-pow.f64 N/A

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

      4. exp-diff N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

      5. rem-exp-log N/A

         \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

      6. lower-/.f64 N/A

         \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

      7. lower-exp.f64 92.3

         \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

   5. Applied rewrites 92.3%

      \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

   6. Taylor expanded in t around 0

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 71.8%

      \[\leadsto x \cdot {z}^{\color{blue}{y}} \]

   if -1900 < y < 0.085999999999999993

   1. Initial program 97.2%

      \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto x \cdot e^{\color{blue}{-1 \cdot \left(a \cdot b\right)}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x \cdot e^{\color{blue}{\mathsf{neg}\left(a \cdot b\right)}} \]

      2. *-commutative N/A

         \[\leadsto x \cdot e^{\mathsf{neg}\left(\color{blue}{b \cdot a}\right)} \]

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

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot a}} \]

      4. lower-*.f64 N/A

         \[\leadsto x \cdot e^{\color{blue}{\left(\mathsf{neg}\left(b\right)\right) \cdot a}} \]

      5. lower-neg.f64 80.7

         \[\leadsto x \cdot e^{\color{blue}{\left(-b\right)} \cdot a} \]

   5. Applied rewrites 80.7%

      \[\leadsto x \cdot e^{\color{blue}{\left(-b\right) \cdot a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1900:\\ \;\;\;\;{z}^{y} \cdot x\\ \mathbf{elif}\;y \leq 0.086:\\ \;\;\;\;e^{\left(-b\right) \cdot a} \cdot x\\ \mathbf{else}:\\ \;\;\;\;{z}^{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ {z}^{y} \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* (pow z y) x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return pow(z, y) * x;
}

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return Math.pow(z, y) * x;
}

Python

def code(x, y, z, t, a, b):
	return math.pow(z, y) * x

Julia

function code(x, y, z, t, a, b)
	return Float64((z ^ y) * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = (z ^ y) * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[Power[z, y], $MachinePrecision] * x), $MachinePrecision]

Derivation

1. Initial program 96.9%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. exp-diff N/A

      \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

   5. rem-exp-log N/A

      \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

   6. lower-/.f64 N/A

      \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

   7. lower-exp.f64 71.4

      \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

5. Applied rewrites 71.4%

   \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

6. Taylor expanded in t around 0

   \[\leadsto x \cdot {z}^{\color{blue}{y}} \]
7. Step-by-step derivation

8. Applied rewrites 53.3%

   \[\leadsto x \cdot {z}^{\color{blue}{y}} \]

9. Final simplification 53.3%

   \[\leadsto {z}^{y} \cdot x \]
10. Add Preprocessing

Alternative 7: 19.0% accurate, 54.7× speedup

Math

\[\begin{array}{l} \\ 1 \cdot x \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 (* 1.0 x))

C

double code(double x, double y, double z, double t, double a, double b) {
	return 1.0 * x;
}

Fortran

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

Java

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

Python

def code(x, y, z, t, a, b):
	return 1.0 * x

Julia

function code(x, y, z, t, a, b)
	return Float64(1.0 * x)
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = 1.0 * x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(1.0 * x), $MachinePrecision]

Derivation

1. Initial program 96.9%

   \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. *-commutative N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. exp-diff N/A

      \[\leadsto x \cdot {\color{blue}{\left(\frac{e^{\log z}}{e^{t}}\right)}}^{y} \]

   5. rem-exp-log N/A

      \[\leadsto x \cdot {\left(\frac{\color{blue}{z}}{e^{t}}\right)}^{y} \]

   6. lower-/.f64 N/A

      \[\leadsto x \cdot {\color{blue}{\left(\frac{z}{e^{t}}\right)}}^{y} \]

   7. lower-exp.f64 71.4

      \[\leadsto x \cdot {\left(\frac{z}{\color{blue}{e^{t}}}\right)}^{y} \]

5. Applied rewrites 71.4%

   \[\leadsto x \cdot \color{blue}{{\left(\frac{z}{e^{t}}\right)}^{y}} \]

6. Taylor expanded in y around 0

   \[\leadsto x \cdot 1 \]
7. Step-by-step derivation

8. Applied rewrites 22.6%

   \[\leadsto x \cdot 1 \]

9. Final simplification 22.6%

   \[\leadsto 1 \cdot x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024296 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))