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

Average Accuracy: 97.3% → 97.5%

Time: 27.6s

Precision: binary64

Cost: 33736

Math

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

↓

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -2000:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{elif}\;t_1 \leq -360:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(t_1 + y \cdot \log z\right) - b}}{y}\\ \end{array} \]

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 -2000.0)
     (/ (* (pow a (+ t -1.0)) x) y)
     (if (<= t_1 -360.0)
       (/ 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 <= -2000.0) {
		tmp = (pow(a, (t + -1.0)) * x) / y;
	} else if (t_1 <= -360.0) {
		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 <= (-2000.0d0)) then
        tmp = ((a ** (t + (-1.0d0))) * x) / y
    else if (t_1 <= (-360.0d0)) 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 <= -2000.0) {
		tmp = (Math.pow(a, (t + -1.0)) * x) / y;
	} else if (t_1 <= -360.0) {
		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 <= -2000.0:
		tmp = (math.pow(a, (t + -1.0)) * x) / y
	elif t_1 <= -360.0:
		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 <= -2000.0)
		tmp = Float64(Float64((a ^ Float64(t + -1.0)) * x) / y);
	elseif (t_1 <= -360.0)
		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 <= -2000.0)
		tmp = ((a ^ (t + -1.0)) * x) / y;
	elseif (t_1 <= -360.0)
		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, -2000.0], N[(N[(N[Power[a, N[(t + -1.0), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, -360.0], 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	97.3%
Target	82.4%
Herbie	97.5%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 t 1) (log.f64 a)) < -2e3

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 98.6%

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

   3. Taylor expanded in b around 0 100.0%

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

   if -2e3 < (*.f64 (-.f64 t 1) (log.f64 a)) < -360

   1. Initial program 90.3%

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

   2. Simplified 75.9%

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

      [Start] 90.3	\[ \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
      associate-*l/ [<=] 82.8	\[ \color{blue}{\frac{x}{y} \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}} \]
      *-commutative [=>] 82.8	\[ \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]
      sub-neg [=>] 82.8	\[ e^{\color{blue}{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) + \left(-b\right)}} \cdot \frac{x}{y} \]
      exp-sum [=>] 73.8	\[ \color{blue}{\left(e^{y \cdot \log z + \left(t - 1\right) \cdot \log a} \cdot e^{-b}\right)} \cdot \frac{x}{y} \]

   3. Taylor expanded in t around 0 89.9%

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

   4. Simplified 83.0%

      \[\leadsto \color{blue}{\frac{{z}^{y}}{y} \cdot \frac{\frac{x}{a}}{e^{b}}} \]
      Proof

      [Start] 89.9	\[ \frac{{z}^{y} \cdot x}{a \cdot \left(y \cdot e^{b}\right)} \]
      *-commutative [=>] 89.9	\[ \frac{{z}^{y} \cdot x}{a \cdot \color{blue}{\left(e^{b} \cdot y\right)}} \]
      associate-*r* [=>] 89.9	\[ \frac{{z}^{y} \cdot x}{\color{blue}{\left(a \cdot e^{b}\right) \cdot y}} \]
      *-commutative [<=] 89.9	\[ \frac{{z}^{y} \cdot x}{\color{blue}{y \cdot \left(a \cdot e^{b}\right)}} \]
      times-frac [=>] 83.1	\[ \color{blue}{\frac{{z}^{y}}{y} \cdot \frac{x}{a \cdot e^{b}}} \]
      associate-/r* [=>] 83.0	\[ \frac{{z}^{y}}{y} \cdot \color{blue}{\frac{\frac{x}{a}}{e^{b}}} \]

   5. Taylor expanded in y around 0 91.1%

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

   6. Taylor expanded in b around 0 92.1%

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

   if -360 < (*.f64 (-.f64 t 1) (log.f64 a))

   1. Initial program 97.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -2000:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{elif}\;\left(t + -1\right) \cdot \log a \leq -360:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(\left(t + -1\right) \cdot \log a + y \cdot \log z\right) - b}}{y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.5%
Cost	33804

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := x \cdot {z}^{y}\\ \mathbf{if}\;t_1 \leq -670:\\ \;\;\;\;\frac{x \cdot e^{t_1 - b}}{y}\\ \mathbf{elif}\;t_1 \leq -55:\\ \;\;\;\;\frac{t_2}{a \cdot y}\\ \mathbf{elif}\;t_1 \leq 350:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t + -1\right)}}{e^{b}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \]

Alternative 2
Accuracy	97.2%
Cost	33480

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -2000:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{elif}\;t_1 \leq -360:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \end{array} \]

Alternative 3
Accuracy	82.3%
Cost	13836

\[\begin{array}{l} t_1 := \frac{x \cdot {z}^{y}}{a \cdot y}\\ t_2 := {a}^{\left(t + -1\right)}\\ \mathbf{if}\;b \leq -1.36 \cdot 10^{-142}:\\ \;\;\;\;\frac{t_2 \cdot x}{y}\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-204}:\\ \;\;\;\;x \cdot \frac{\frac{t_2}{e^{b}}}{y}\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.66:\\ \;\;\;\;\frac{t_2}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \]

Alternative 4
Accuracy	82.2%
Cost	7572

\[\begin{array}{l} t_1 := \frac{x \cdot {z}^{y}}{a \cdot y}\\ t_2 := \frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{if}\;b \leq -6.5 \cdot 10^{-147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -1.35 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.45 \cdot 10^{-300}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 125:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \]

Alternative 5
Accuracy	81.7%
Cost	7572

\[\begin{array}{l} t_1 := \frac{x \cdot {z}^{y}}{a \cdot y}\\ t_2 := {a}^{\left(t + -1\right)}\\ t_3 := \frac{t_2 \cdot x}{y}\\ \mathbf{if}\;b \leq -1.4 \cdot 10^{-142}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -1.3 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.16 \cdot 10^{-300}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 128:\\ \;\;\;\;\frac{t_2}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \]

Alternative 6
Accuracy	66.3%
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \]

Alternative 7
Accuracy	83.3%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;b \leq 12:\\ \;\;\;\;\frac{{a}^{\left(t + -1\right)} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \]

Alternative 8
Accuracy	66.6%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-41}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 9
Accuracy	43.7%
Cost	712

\[\begin{array}{l} \mathbf{if}\;b \leq 7 \cdot 10^{-180}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \mathbf{elif}\;b \leq 0.000195:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 10
Accuracy	50.0%
Cost	708

\[\begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{a}}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \end{array} \]

Alternative 11
Accuracy	36.8%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.35 \cdot 10^{-172} \lor \neg \left(y \leq 2 \cdot 10^{-138}\right):\\ \;\;\;\;\frac{x}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]

Alternative 12
Accuracy	40.0%
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 10^{-32}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]

Alternative 13
Accuracy	34.8%
Cost	320

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

Reproduce

herbie shell --seed 2023140 
(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))