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

Percentage Accurate: 96.4% → 99.3%

Time: 8.3s

Alternatives: 7

Speedup: 1.5×

• 42.1% 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: 96.4% 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: 99.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-out N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-log.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 95.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -260 \lor \neg \left(t \leq 1.08 \cdot 10^{-114}\right):\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -260.0) (not (<= t 1.08e-114)))
   (* x (exp (fma (- b) a (* (- t) y))))
   (* x (exp (fma (- b) a (* (log z) y))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -260.0) || !(t <= 1.08e-114)) {
		tmp = x * exp(fma(-b, a, (-t * y)));
	} else {
		tmp = x * exp(fma(-b, a, (log(z) * y)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -260.0) || !(t <= 1.08e-114))
		tmp = Float64(x * exp(fma(Float64(-b), a, Float64(Float64(-t) * y))));
	else
		tmp = Float64(x * exp(fma(Float64(-b), a, Float64(log(z) * y))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -260.0], N[Not[LessEqual[t, 1.08e-114]], $MachinePrecision]], N[(x * N[Exp[N[((-b) * a + N[((-t) * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-b) * a + N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -260 or 1.08e-114 < t

   1. Initial program 95.7%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 96.8%

      \[\leadsto x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)} \]

   if -260 < t < 1.08e-114

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 96.7%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -260 \lor \neg \left(t \leq 1.08 \cdot 10^{-114}\right):\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\mathsf{fma}\left(-b, a, \log z \cdot y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 z around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 97.0

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

5. Applied rewrites 97.0%

   \[\leadsto x \cdot e^{\color{blue}{\mathsf{fma}\left(-b, a, \left(\log z - t\right) \cdot y\right)}} \]
6. Add Preprocessing

Alternative 4: 72.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -580 \lor \neg \left(t \leq 2.6 \cdot 10^{+54}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -580.0) (not (<= t 2.6e+54)))
   (* x (exp (* (- y) t)))
   (* x (exp (* (- b) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -580.0) || !(t <= 2.6e+54)) {
		tmp = x * exp((-y * t));
	} else {
		tmp = x * exp((-b * a));
	}
	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) :: tmp
    if ((t <= (-580.0d0)) .or. (.not. (t <= 2.6d+54))) then
        tmp = x * exp((-y * t))
    else
        tmp = x * exp((-b * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -580.0) || !(t <= 2.6e+54)) {
		tmp = x * Math.exp((-y * t));
	} else {
		tmp = x * Math.exp((-b * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -580.0) or not (t <= 2.6e+54):
		tmp = x * math.exp((-y * t))
	else:
		tmp = x * math.exp((-b * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -580.0) || !(t <= 2.6e+54))
		tmp = Float64(x * exp(Float64(Float64(-y) * t)));
	else
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -580.0) || ~((t <= 2.6e+54)))
		tmp = x * exp((-y * t));
	else
		tmp = x * exp((-b * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -580.0], N[Not[LessEqual[t, 2.6e+54]], $MachinePrecision]], N[(x * N[Exp[N[((-y) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -580 or 2.60000000000000007e54 < t

   1. Initial program 95.4%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 82.2

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

   5. Applied rewrites 82.2%

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

   if -580 < t < 2.60000000000000007e54

   1. Initial program 97.4%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 67.1%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -580 \lor \neg \left(t \leq 2.6 \cdot 10^{+54}\right):\\ \;\;\;\;x \cdot e^{\left(-y\right) \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-101} \lor \neg \left(b \leq 6.5 \cdot 10^{-108}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-z\right) \cdot a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= b -1.05e-101) (not (<= b 6.5e-108)))
   (* x (exp (* (- b) a)))
   (* x (exp (* (- z) a)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.05e-101) || !(b <= 6.5e-108)) {
		tmp = x * exp((-b * a));
	} else {
		tmp = x * exp((-z * a));
	}
	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) :: tmp
    if ((b <= (-1.05d-101)) .or. (.not. (b <= 6.5d-108))) then
        tmp = x * exp((-b * a))
    else
        tmp = x * exp((-z * a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((b <= -1.05e-101) || !(b <= 6.5e-108)) {
		tmp = x * Math.exp((-b * a));
	} else {
		tmp = x * Math.exp((-z * a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (b <= -1.05e-101) or not (b <= 6.5e-108):
		tmp = x * math.exp((-b * a))
	else:
		tmp = x * math.exp((-z * a))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((b <= -1.05e-101) || !(b <= 6.5e-108))
		tmp = Float64(x * exp(Float64(Float64(-b) * a)));
	else
		tmp = Float64(x * exp(Float64(Float64(-z) * a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((b <= -1.05e-101) || ~((b <= 6.5e-108)))
		tmp = x * exp((-b * a));
	else
		tmp = x * exp((-z * a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[b, -1.05e-101], N[Not[LessEqual[b, 6.5e-108]], $MachinePrecision]], N[(x * N[Exp[N[((-b) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x * N[Exp[N[((-z) * a), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.05000000000000008e-101 or 6.5000000000000002e-108 < b

   1. Initial program 98.3%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-log.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.9%

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

   if -1.05000000000000008e-101 < b < 6.5000000000000002e-108

   1. Initial program 92.7%

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

   3. Taylor expanded in z around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-*r* N/A

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

      5. distribute-lft-out N/A

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

      6. lower-fma.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 N/A

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

      13. lower-log.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 47.2%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-101} \lor \neg \left(b \leq 6.5 \cdot 10^{-108}\right):\\ \;\;\;\;x \cdot e^{\left(-b\right) \cdot a}\\ \mathbf{else}:\\ \;\;\;\;x \cdot e^{\left(-z\right) \cdot a}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (* x (exp (fma (- b) a (* (- t) y)))))

C

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

Julia

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

Wolfram

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

Derivation

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 z around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-fma.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower-log.f64 97.0

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

5. Applied rewrites 97.0%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 82.1%

   \[\leadsto x \cdot e^{\mathsf{fma}\left(-b, a, \left(-t\right) \cdot y\right)} \]
9. Add Preprocessing

Alternative 7: 33.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ x \cdot e^{\left(-z\right) \cdot a} \end{array} \]

FPCore

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

C

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

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((-z * a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 z around 0

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-lft-out N/A

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

   6. lower-fma.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 N/A

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

   13. lower-log.f64 100.0

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 34.4%

   \[\leadsto x \cdot e^{\left(-z\right) \cdot \color{blue}{a}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(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))))))