Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Average Error: 1.8 → 1.6

Time: 26.9s

Precision: binary64

Cost: 33736

Math

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

↓

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -705:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)}}{y} \cdot x\\ \mathbf{elif}\;t_1 \leq -391:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(t_1 + y \cdot \log z\right) - b}}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* (+ t -1.0) (log a))))
   (if (<= t_1 -705.0)
     (* (/ (pow a (+ t -1.0)) y) x)
     (if (<= t_1 -391.0)
       (/ x (* y (+ a (* a b))))
       (/ (* x (exp (- (+ t_1 (* y (log z))) b))) y)))))

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * log(a);
	double tmp;
	if (t_1 <= -705.0) {
		tmp = (pow(a, (t + -1.0)) / y) * x;
	} else if (t_1 <= -391.0) {
		tmp = x / (y * (a + (a * b)));
	} else {
		tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y;
	}
	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
    code = (x * exp((((y * log(z)) + ((t - 1.0d0) * log(a))) - b))) / y
end function

↓

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 = (t + (-1.0d0)) * log(a)
    if (t_1 <= (-705.0d0)) then
        tmp = ((a ** (t + (-1.0d0))) / y) * x
    else if (t_1 <= (-391.0d0)) then
        tmp = x / (y * (a + (a * b)))
    else
        tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y
    end if
    code = tmp
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 - 1.0) * Math.log(a))) - b))) / y;
}

↓

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t + -1.0) * Math.log(a);
	double tmp;
	if (t_1 <= -705.0) {
		tmp = (Math.pow(a, (t + -1.0)) / y) * x;
	} else if (t_1 <= -391.0) {
		tmp = x / (y * (a + (a * b)));
	} else {
		tmp = (x * Math.exp(((t_1 + (y * Math.log(z))) - b))) / y;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z, t, a, b):
	t_1 = (t + -1.0) * math.log(a)
	tmp = 0
	if t_1 <= -705.0:
		tmp = (math.pow(a, (t + -1.0)) / y) * x
	elif t_1 <= -391.0:
		tmp = x / (y * (a + (a * b)))
	else:
		tmp = (x * math.exp(((t_1 + (y * math.log(z))) - b))) / y
	return tmp

Julia

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

↓

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t + -1.0) * log(a))
	tmp = 0.0
	if (t_1 <= -705.0)
		tmp = Float64(Float64((a ^ Float64(t + -1.0)) / y) * x);
	elseif (t_1 <= -391.0)
		tmp = Float64(x / Float64(y * Float64(a + Float64(a * b))));
	else
		tmp = Float64(Float64(x * exp(Float64(Float64(t_1 + Float64(y * log(z))) - b))) / y);
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t + -1.0) * log(a);
	tmp = 0.0;
	if (t_1 <= -705.0)
		tmp = ((a ^ (t + -1.0)) / y) * x;
	elseif (t_1 <= -391.0)
		tmp = x / (y * (a + (a * b)));
	else
		tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -705.0], N[(N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$1, -391.0], N[(x / N[(y * N[(a + N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Exp[N[(N[(t$95$1 + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]

Target

Original	1.8
Target	11.2
Herbie	1.6

\[\begin{array}{l} \mathbf{if}\;t < -0.8845848504127471:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t < 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array} \]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t 1) (log.f64 a)) < -705

   1. Initial program 0.0

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

   2. Taylor expanded in y around 0 1.0

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

   3. Simplified 9.8

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

      [Start] 1.0	\[ \frac{e^{\left(t - 1\right) \cdot \log a - b} \cdot x}{y} \]
      associate-/l* [=>] 9.4	\[ \color{blue}{\frac{e^{\left(t - 1\right) \cdot \log a - b}}{\frac{y}{x}}} \]
      associate-/r/ [=>] 1.0	\[ \color{blue}{\frac{e^{\left(t - 1\right) \cdot \log a - b}}{y} \cdot x} \]
      exp-diff [=>] 9.8	\[ \frac{\color{blue}{\frac{e^{\left(t - 1\right) \cdot \log a}}{e^{b}}}}{y} \cdot x \]
      log-pow [<=] 9.8	\[ \frac{\frac{e^{\color{blue}{\log \left({a}^{\left(t - 1\right)}\right)}}}{e^{b}}}{y} \cdot x \]
      sub-neg [=>] 9.8	\[ \frac{\frac{e^{\log \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}}\right)}}{e^{b}}}{y} \cdot x \]
      metadata-eval [=>] 9.8	\[ \frac{\frac{e^{\log \left({a}^{\left(t + \color{blue}{-1}\right)}\right)}}{e^{b}}}{y} \cdot x \]
      rem-exp-log [=>] 9.8	\[ \frac{\frac{\color{blue}{{a}^{\left(t + -1\right)}}}{e^{b}}}{y} \cdot x \]

   4. Taylor expanded in b around 0 0.1

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

   if -705 < (*.f64 (-.f64 t 1) (log.f64 a)) < -391

   1. Initial program 6.8

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

   2. Taylor expanded in y around 0 14.8

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

   3. Simplified 5.4

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

      [Start] 14.8	\[ \frac{e^{\left(t - 1\right) \cdot \log a - b} \cdot x}{y} \]
      associate-/l* [=>] 18.7	\[ \color{blue}{\frac{e^{\left(t - 1\right) \cdot \log a - b}}{\frac{y}{x}}} \]
      associate-/r/ [=>] 7.3	\[ \color{blue}{\frac{e^{\left(t - 1\right) \cdot \log a - b}}{y} \cdot x} \]
      exp-diff [=>] 7.3	\[ \frac{\color{blue}{\frac{e^{\left(t - 1\right) \cdot \log a}}{e^{b}}}}{y} \cdot x \]
      log-pow [<=] 7.3	\[ \frac{\frac{e^{\color{blue}{\log \left({a}^{\left(t - 1\right)}\right)}}}{e^{b}}}{y} \cdot x \]
      sub-neg [=>] 7.3	\[ \frac{\frac{e^{\log \left({a}^{\color{blue}{\left(t + \left(-1\right)\right)}}\right)}}{e^{b}}}{y} \cdot x \]
      metadata-eval [=>] 7.3	\[ \frac{\frac{e^{\log \left({a}^{\left(t + \color{blue}{-1}\right)}\right)}}{e^{b}}}{y} \cdot x \]
      rem-exp-log [=>] 5.4	\[ \frac{\frac{\color{blue}{{a}^{\left(t + -1\right)}}}{e^{b}}}{y} \cdot x \]

   4. Taylor expanded in t around 0 5.7

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

   5. Taylor expanded in b around 0 14.1

      \[\leadsto \frac{x}{\color{blue}{a \cdot y + a \cdot \left(y \cdot b\right)}} \]

   6. Simplified 4.8

      \[\leadsto \frac{x}{\color{blue}{y \cdot \left(a + a \cdot b\right)}} \]
      Proof

      [Start] 14.1	\[ \frac{x}{a \cdot y + a \cdot \left(y \cdot b\right)} \]
      *-commutative [=>] 14.1	\[ \frac{x}{a \cdot y + a \cdot \color{blue}{\left(b \cdot y\right)}} \]
      associate-*r* [=>] 10.4	\[ \frac{x}{a \cdot y + \color{blue}{\left(a \cdot b\right) \cdot y}} \]
      distribute-rgt-out [=>] 4.8	\[ \frac{x}{\color{blue}{y \cdot \left(a + a \cdot b\right)}} \]

   if -391 < (*.f64 (-.f64 t 1) (log.f64 a))

   1. Initial program 2.0

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -705:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)}}{y} \cdot x\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -391:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(\left(t + -1\right) \cdot \log a + y \cdot \log z\right) - b}}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.4
Cost	33805

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -705:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)}}{y} \cdot x\\ \mathbf{elif}\;t_1 \leq -185 \lor \neg \left(t_1 \leq 2 \cdot 10^{+108}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}\\ \end{array} \]

Alternative 2
Error	1.8
Cost	33480

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -705:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)}}{y} \cdot x\\ \mathbf{elif}\;t_1 \leq -391:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \end{array} \]

Alternative 3
Error	18.5
Cost	7705

\[\begin{array}{l} t_1 := \frac{x}{\frac{y}{\frac{{z}^{y}}{a}}}\\ \mathbf{if}\;y \leq -6.5 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -5.3 \cdot 10^{-160}:\\ \;\;\;\;\frac{x}{a \cdot \left(\left(y + y \cdot b\right) + \left(y \cdot 0.5\right) \cdot \left(b \cdot b\right)\right)}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{-281}:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)}}{y} \cdot x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+143} \lor \neg \left(y \leq 9.5 \cdot 10^{+172}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 4
Error	10.6
Cost	7308

\[\begin{array}{l} t_1 := \frac{\frac{x}{\frac{y}{{a}^{t}}}}{a}\\ \mathbf{if}\;b \leq 4.8 \cdot 10^{-139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-56}:\\ \;\;\;\;\frac{x}{\frac{y}{\frac{{z}^{y}}{a}}}\\ \mathbf{elif}\;b \leq 185:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 5
Error	12.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \cdot 10^{-242}:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{a \cdot y}\\ \mathbf{elif}\;b \leq 400:\\ \;\;\;\;\frac{{a}^{t}}{y} \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 6
Error	20.9
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{-274}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}\\ \mathbf{elif}\;b \leq 9 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 7
Error	21.3
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-274}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 8
Error	12.8
Cost	7044

\[\begin{array}{l} \mathbf{if}\;b \leq 176:\\ \;\;\;\;\frac{{a}^{t}}{y} \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 9
Error	10.0
Cost	7044

\[\begin{array}{l} \mathbf{if}\;b \leq 370:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)}}{y} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 10
Error	26.9
Cost	1352

\[\begin{array}{l} \mathbf{if}\;b \leq -1.8 \cdot 10^{-273}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(\left(y + y \cdot b\right) + \left(y \cdot 0.5\right) \cdot \left(b \cdot b\right)\right)}\\ \end{array} \]

Alternative 11
Error	35.8
Cost	844

\[\begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-273}:\\ \;\;\;\;\frac{1}{\frac{a \cdot y}{x}}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-226}:\\ \;\;\;\;\frac{b}{a \cdot y} \cdot \left(-x\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 12
Error	36.0
Cost	844

\[\begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-273}:\\ \;\;\;\;\frac{1}{\frac{a \cdot y}{x}}\\ \mathbf{elif}\;b \leq 5.3 \cdot 10^{-226}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-b}{a}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 13
Error	34.0
Cost	841

\[\begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{-273} \lor \neg \left(b \leq 7.2 \cdot 10^{-226}\right):\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-b}{a}\\ \end{array} \]

Alternative 14
Error	30.7
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+57} \lor \neg \left(y \leq 10^{-28}\right):\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot \left(1 + b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a + a \cdot b}}{y}\\ \end{array} \]

Alternative 15
Error	38.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{a \cdot y}{x}}\\ \end{array} \]

Alternative 16
Error	36.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;b \leq 3600000000:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 17
Error	41.5
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 4.3 \cdot 10^{+186}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]

Alternative 18
Error	38.4
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 1.15 \cdot 10^{-48}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]

Alternative 19
Error	41.2
Cost	320

\[\frac{x}{a \cdot y} \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))