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

Percentage Accurate: 96.7% → 96.7%

Time: 3.8s

Alternatives: 8

Speedup: 1.0×

• 41.7% 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: 96.7% 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: 96.7% 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]

Derivation

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. Add Preprocessing

Alternative 2: 54.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(1 - z\right)\\ t_2 := a \cdot \left(t\_1 - b\right)\\ t_3 := x \cdot e^{y \cdot \left(\log z - t\right) + t\_2}\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+59}:\\ \;\;\;\;x \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (log (- 1.0 z)))
        (t_2 (* a (- t_1 b)))
        (t_3 (* x (exp (+ (* y (- (log z) t)) t_2)))))
   (if (<= t_3 -2e+59) (* x t_2) (if (<= t_3 5e+168) t_1 (exp t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log((1.0 - z));
	double t_2 = a * (t_1 - b);
	double t_3 = x * exp(((y * (log(z) - t)) + t_2));
	double tmp;
	if (t_3 <= -2e+59) {
		tmp = x * t_2;
	} else if (t_3 <= 5e+168) {
		tmp = t_1;
	} else {
		tmp = exp(t_2);
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = log((1.0d0 - z))
    t_2 = a * (t_1 - b)
    t_3 = x * exp(((y * (log(z) - t)) + t_2))
    if (t_3 <= (-2d+59)) then
        tmp = x * t_2
    else if (t_3 <= 5d+168) then
        tmp = t_1
    else
        tmp = exp(t_2)
    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.log((1.0 - z));
	double t_2 = a * (t_1 - b);
	double t_3 = x * Math.exp(((y * (Math.log(z) - t)) + t_2));
	double tmp;
	if (t_3 <= -2e+59) {
		tmp = x * t_2;
	} else if (t_3 <= 5e+168) {
		tmp = t_1;
	} else {
		tmp = Math.exp(t_2);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.log((1.0 - z))
	t_2 = a * (t_1 - b)
	t_3 = x * math.exp(((y * (math.log(z) - t)) + t_2))
	tmp = 0
	if t_3 <= -2e+59:
		tmp = x * t_2
	elif t_3 <= 5e+168:
		tmp = t_1
	else:
		tmp = math.exp(t_2)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = log(Float64(1.0 - z))
	t_2 = Float64(a * Float64(t_1 - b))
	t_3 = Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + t_2)))
	tmp = 0.0
	if (t_3 <= -2e+59)
		tmp = Float64(x * t_2);
	elseif (t_3 <= 5e+168)
		tmp = t_1;
	else
		tmp = exp(t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = log((1.0 - z));
	t_2 = a * (t_1 - b);
	t_3 = x * exp(((y * (log(z) - t)) + t_2));
	tmp = 0.0;
	if (t_3 <= -2e+59)
		tmp = x * t_2;
	elseif (t_3 <= 5e+168)
		tmp = t_1;
	else
		tmp = exp(t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+59], N[(x * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 5e+168], t$95$1, N[Exp[t$95$2], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))) < -1.99999999999999994e59

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 21.6%

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

   if -1.99999999999999994e59 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))) < 4.99999999999999967e168

   1. Initial program 96.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) + y \cdot \left(\frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right)\right)} \]

   5. Applied rewrites 67.8%

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

   if 4.99999999999999967e168 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))))

   1. Initial program 90.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 90.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 31.3%

       \[\leadsto e^{\log z - t} \]

   12. Taylor expanded in y around -inf

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

   14. Applied rewrites 49.4%

       \[\leadsto e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 65.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \left(\log \left(1 - z\right) - b\right)\\ t_2 := e^{y \cdot \left(\log z - t\right) + t\_1}\\ \mathbf{if}\;x \cdot t\_2 \leq -5 \cdot 10^{-274}:\\ \;\;\;\;x \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* a (- (log (- 1.0 z)) b)))
        (t_2 (exp (+ (* y (- (log z) t)) t_1))))
   (if (<= (* x t_2) -5e-274) (* x t_1) t_2)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a * (log((1.0 - z)) - b);
	double t_2 = exp(((y * (log(z) - t)) + t_1));
	double tmp;
	if ((x * t_2) <= -5e-274) {
		tmp = x * t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = a * (log((1.0d0 - z)) - b)
    t_2 = exp(((y * (log(z) - t)) + t_1))
    if ((x * t_2) <= (-5d-274)) then
        tmp = x * t_1
    else
        tmp = t_2
    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 = a * (Math.log((1.0 - z)) - b);
	double t_2 = Math.exp(((y * (Math.log(z) - t)) + t_1));
	double tmp;
	if ((x * t_2) <= -5e-274) {
		tmp = x * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a * (math.log((1.0 - z)) - b)
	t_2 = math.exp(((y * (math.log(z) - t)) + t_1))
	tmp = 0
	if (x * t_2) <= -5e-274:
		tmp = x * t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a * Float64(log(Float64(1.0 - z)) - b))
	t_2 = exp(Float64(Float64(y * Float64(log(z) - t)) + t_1))
	tmp = 0.0
	if (Float64(x * t_2) <= -5e-274)
		tmp = Float64(x * t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a * (log((1.0 - z)) - b);
	t_2 = exp(((y * (log(z) - t)) + t_1));
	tmp = 0.0;
	if ((x * t_2) <= -5e-274)
		tmp = x * t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a * N[(N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(x * t$95$2), $MachinePrecision], -5e-274], N[(x * t$95$1), $MachinePrecision], t$95$2]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))) < -5e-274

   1. Initial program 98.6%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 17.8%

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

   if -5e-274 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))))

   1. Initial program 95.2%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 84.1%

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

Alternative 4: 42.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(1 - z\right)\\ t_2 := a \cdot \left(t\_1 - b\right)\\ \mathbf{if}\;x \cdot e^{y \cdot \left(\log z - t\right) + t\_2} \leq 10^{+211}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (log (- 1.0 z))) (t_2 (* a (- t_1 b))))
   (if (<= (* x (exp (+ (* y (- (log z) t)) t_2))) 1e+211) t_1 t_2)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log((1.0 - z));
	double t_2 = a * (t_1 - b);
	double tmp;
	if ((x * exp(((y * (log(z) - t)) + t_2))) <= 1e+211) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = log((1.0d0 - z))
    t_2 = a * (t_1 - b)
    if ((x * exp(((y * (log(z) - t)) + t_2))) <= 1d+211) then
        tmp = t_1
    else
        tmp = t_2
    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.log((1.0 - z));
	double t_2 = a * (t_1 - b);
	double tmp;
	if ((x * Math.exp(((y * (Math.log(z) - t)) + t_2))) <= 1e+211) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.log((1.0 - z))
	t_2 = a * (t_1 - b)
	tmp = 0
	if (x * math.exp(((y * (math.log(z) - t)) + t_2))) <= 1e+211:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = log(Float64(1.0 - z))
	t_2 = Float64(a * Float64(t_1 - b))
	tmp = 0.0
	if (Float64(x * exp(Float64(Float64(y * Float64(log(z) - t)) + t_2))) <= 1e+211)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = log((1.0 - z));
	t_2 = a * (t_1 - b);
	tmp = 0.0;
	if ((x * exp(((y * (log(z) - t)) + t_2))) <= 1e+211)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * N[Exp[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e+211], t$95$1, t$95$2]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))) < 9.9999999999999996e210

   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 y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) + y \cdot \left(\frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right)\right)} \]

   5. Applied rewrites 50.3%

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

   if 9.9999999999999996e210 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))))

   1. Initial program 91.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 91.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 24.5%

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

Alternative 5: 43.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(1 - z\right)\\ t_2 := y \cdot \left(\log z - t\right)\\ \mathbf{if}\;x \cdot e^{t\_2 + a \cdot \left(t\_1 - b\right)} \leq 5 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (log (- 1.0 z))) (t_2 (* y (- (log z) t))))
   (if (<= (* x (exp (+ t_2 (* a (- t_1 b))))) 5e+168) t_1 t_2)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log((1.0 - z));
	double t_2 = y * (log(z) - t);
	double tmp;
	if ((x * exp((t_2 + (a * (t_1 - b))))) <= 5e+168) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = log((1.0d0 - z))
    t_2 = y * (log(z) - t)
    if ((x * exp((t_2 + (a * (t_1 - b))))) <= 5d+168) then
        tmp = t_1
    else
        tmp = t_2
    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.log((1.0 - z));
	double t_2 = y * (Math.log(z) - t);
	double tmp;
	if ((x * Math.exp((t_2 + (a * (t_1 - b))))) <= 5e+168) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.log((1.0 - z))
	t_2 = y * (math.log(z) - t)
	tmp = 0
	if (x * math.exp((t_2 + (a * (t_1 - b))))) <= 5e+168:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = log(Float64(1.0 - z))
	t_2 = Float64(y * Float64(log(z) - t))
	tmp = 0.0
	if (Float64(x * exp(Float64(t_2 + Float64(a * Float64(t_1 - b))))) <= 5e+168)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = log((1.0 - z));
	t_2 = y * (log(z) - t);
	tmp = 0.0;
	if ((x * exp((t_2 + (a * (t_1 - b))))) <= 5e+168)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * N[Exp[N[(t$95$2 + N[(a * N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+168], t$95$1, t$95$2]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))))) < 4.99999999999999967e168

   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

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) + y \cdot \left(\frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right)\right)} \]

   5. Applied rewrites 50.5%

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

   if 4.99999999999999967e168 < (*.f64 x (exp.f64 (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))))

   1. Initial program 90.1%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 14.9%

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

Alternative 6: 48.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(1 - z\right)\\ t_2 := a \cdot \left(t\_1 - b\right)\\ \mathbf{if}\;y \cdot \left(\log z - t\right) + t\_2 \leq -1 \cdot 10^{-179}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (log (- 1.0 z))) (t_2 (* a (- t_1 b))))
   (if (<= (+ (* y (- (log z) t)) t_2) -1e-179) t_1 (* x t_2))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = log((1.0 - z));
	double t_2 = a * (t_1 - b);
	double tmp;
	if (((y * (log(z) - t)) + t_2) <= -1e-179) {
		tmp = t_1;
	} else {
		tmp = x * t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = log((1.0d0 - z))
    t_2 = a * (t_1 - b)
    if (((y * (log(z) - t)) + t_2) <= (-1d-179)) then
        tmp = t_1
    else
        tmp = x * t_2
    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.log((1.0 - z));
	double t_2 = a * (t_1 - b);
	double tmp;
	if (((y * (Math.log(z) - t)) + t_2) <= -1e-179) {
		tmp = t_1;
	} else {
		tmp = x * t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = math.log((1.0 - z))
	t_2 = a * (t_1 - b)
	tmp = 0
	if ((y * (math.log(z) - t)) + t_2) <= -1e-179:
		tmp = t_1
	else:
		tmp = x * t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = log(Float64(1.0 - z))
	t_2 = Float64(a * Float64(t_1 - b))
	tmp = 0.0
	if (Float64(Float64(y * Float64(log(z) - t)) + t_2) <= -1e-179)
		tmp = t_1;
	else
		tmp = Float64(x * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = log((1.0 - z));
	t_2 = a * (t_1 - b);
	tmp = 0.0;
	if (((y * (log(z) - t)) + t_2) <= -1e-179)
		tmp = t_1;
	else
		tmp = x * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(t$95$1 - b), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(y * N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], -1e-179], t$95$1, N[(x * t$95$2), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b))) < -1e-179

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) + y \cdot \left(\frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right)\right)} \]

   5. Applied rewrites 77.2%

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

   if -1e-179 < (+.f64 (*.f64 y (-.f64 (log.f64 z) t)) (*.f64 a (-.f64 (log.f64 (-.f64 #s(literal 1 binary64) z)) b)))

   1. Initial program 96.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 20.9%

      \[\leadsto x \cdot \color{blue}{\left(a \cdot \left(\log \left(1 - z\right) - b\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 40.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \log \left(1 - z\right) \end{array} \]

FPCore

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

C

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

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 = log((1.0d0 - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[Log[N[(1.0 - z), $MachinePrecision]], $MachinePrecision]

Derivation

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 y around 0

   \[\leadsto \color{blue}{x \cdot e^{a \cdot \left(\log \left(1 - z\right) - b\right)} + y \cdot \left(x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot \left(\log z - t\right)\right) + y \cdot \left(\frac{1}{6} \cdot \left(x \cdot \left(y \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(x \cdot \left(e^{a \cdot \left(\log \left(1 - z\right) - b\right)} \cdot {\left(\log z - t\right)}^{2}\right)\right)\right)\right)} \]

5. Applied rewrites 41.3%

   \[\leadsto \color{blue}{\log \left(1 - z\right)} \]
6. Add Preprocessing

Alternative 8: 2.7% accurate, 82.0× speedup

Math

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

FPCore

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

C

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

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 - z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 y around 0

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

5. Applied rewrites 61.3%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 2.8%

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

Reproduce

herbie shell --seed 2024313 
(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))))))