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

Average Accuracy: 96.9% → 97.1%

Time: 25.5s

Precision: binary64

Cost: 33737.00

Math

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

↓

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -690 \lor \neg \left(t_1 \leq -222\right):\\ \;\;\;\;\frac{x \cdot e^{\left(t_1 + y \cdot \log z\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{{a}^{\left(1 - t\right)}}}{y \cdot e^{b}}\\ \end{array} \]

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 (or (<= t_1 -690.0) (not (<= t_1 -222.0)))
     (/ (* x (exp (- (+ t_1 (* y (log z))) b))) y)
     (* x (/ (/ (pow z y) (pow a (- 1.0 t))) (* y (exp b)))))))

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 <= -690.0) || !(t_1 <= -222.0)) {
		tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y;
	} else {
		tmp = x * ((pow(z, y) / pow(a, (1.0 - t))) / (y * exp(b)));
	}
	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 <= (-690.0d0)) .or. (.not. (t_1 <= (-222.0d0)))) then
        tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y
    else
        tmp = x * (((z ** y) / (a ** (1.0d0 - t))) / (y * exp(b)))
    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 <= -690.0) || !(t_1 <= -222.0)) {
		tmp = (x * Math.exp(((t_1 + (y * Math.log(z))) - b))) / y;
	} else {
		tmp = x * ((Math.pow(z, y) / Math.pow(a, (1.0 - t))) / (y * Math.exp(b)));
	}
	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 <= -690.0) or not (t_1 <= -222.0):
		tmp = (x * math.exp(((t_1 + (y * math.log(z))) - b))) / y
	else:
		tmp = x * ((math.pow(z, y) / math.pow(a, (1.0 - t))) / (y * math.exp(b)))
	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 <= -690.0) || !(t_1 <= -222.0))
		tmp = Float64(Float64(x * exp(Float64(Float64(t_1 + Float64(y * log(z))) - b))) / y);
	else
		tmp = Float64(x * Float64(Float64((z ^ y) / (a ^ Float64(1.0 - t))) / Float64(y * exp(b))));
	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 <= -690.0) || ~((t_1 <= -222.0)))
		tmp = (x * exp(((t_1 + (y * log(z))) - b))) / y;
	else
		tmp = x * (((z ^ y) / (a ^ (1.0 - t))) / (y * exp(b)));
	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[Or[LessEqual[t$95$1, -690.0], N[Not[LessEqual[t$95$1, -222.0]], $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], N[(x * N[(N[(N[Power[z, y], $MachinePrecision] / N[Power[a, N[(1.0 - t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(y * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	96.9%
Target	82.4%
Herbie	97.1%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 t 1) (log.f64 a)) < -690 or -222 < (*.f64 (-.f64 t 1) (log.f64 a))

   1. Initial program 98.8%

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

   if -690 < (*.f64 (-.f64 t 1) (log.f64 a)) < -222

   1. Initial program 88.9%

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

   2. Simplified 89.8%

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

      [Start] 88.9	\[ \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
      associate-*r/ [<=] 96.4	\[ \color{blue}{x \cdot \frac{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \]
      exp-diff [=>] 86.7	\[ x \cdot \frac{\color{blue}{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{e^{b}}}}{y} \]
      associate-/l/ [=>] 86.6	\[ x \cdot \color{blue}{\frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a}}{y \cdot e^{b}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -690 \lor \neg \left(\left(t + -1\right) \cdot \log a \leq -222\right):\\ \;\;\;\;\frac{x \cdot e^{\left(\left(t + -1\right) \cdot \log a + y \cdot \log z\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{{a}^{\left(1 - t\right)}}}{y \cdot e^{b}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.7%
Cost	33804.00

\[\begin{array}{l} t_1 := \left(t + -1\right) \cdot \log a\\ t_2 := a \cdot e^{b}\\ \mathbf{if}\;t_1 \leq -690:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;t_1 \leq 30:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y \cdot t_2}\\ \mathbf{elif}\;t_1 \leq 350:\\ \;\;\;\;\frac{{a}^{t}}{y} \cdot \frac{x}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y \cdot e^{b}}\\ \end{array} \]

Alternative 2
Accuracy	96.0%
Cost	26692.00

\[\begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x \cdot {a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \end{array} \]

Alternative 3
Accuracy	65.3%
Cost	7508.00

\[\begin{array}{l} t_1 := \frac{1}{y} + \frac{b}{y}\\ \mathbf{if}\;b \leq -1.06 \cdot 10^{-271}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-244}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.6 \cdot 10^{-195}:\\ \;\;\;\;\frac{1 - b}{y} \cdot \frac{t_1}{t_1 \cdot \frac{a}{x}}\\ \mathbf{elif}\;b \leq 3.05 \cdot 10^{-136}:\\ \;\;\;\;\frac{b}{y} \cdot \left(-\frac{x}{a}\right)\\ \mathbf{elif}\;b \leq 1.05 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 4
Accuracy	65.9%
Cost	7244.00

\[\begin{array}{l} t_1 := \frac{1}{y} + \frac{b}{y}\\ \mathbf{if}\;b \leq -3.5 \cdot 10^{-271}:\\ \;\;\;\;\frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-245}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-b}{a}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-198}:\\ \;\;\;\;\frac{1 - b}{y} \cdot \frac{t_1}{t_1 \cdot \frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot e^{b}\right)}\\ \end{array} \]

Alternative 5
Accuracy	80.7%
Cost	7044.00

\[\begin{array}{l} \mathbf{if}\;b \leq 3.3 \cdot 10^{-31}:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 6
Accuracy	44.2%
Cost	2392.00

\[\begin{array}{l} t_1 := \frac{1}{y} + \frac{b}{y}\\ t_2 := \frac{1 - b}{y} \cdot \frac{t_1}{t_1 \cdot \frac{a}{x}}\\ t_3 := a + a \cdot b\\ t_4 := \frac{x}{y \cdot t_3}\\ \mathbf{if}\;b \leq -4.3 \cdot 10^{-269}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-245}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.55 \cdot 10^{-195}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{-135}:\\ \;\;\;\;\frac{b}{y} \cdot \left(-\frac{x}{a}\right)\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{+35}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;b \leq 5.7 \cdot 10^{+132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 2.45 \cdot 10^{+180}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t_3}}{y}\\ \end{array} \]

Alternative 7
Accuracy	44.4%
Cost	1108.00

\[\begin{array}{l} \mathbf{if}\;b \leq -13000:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq -4.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-269}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-247}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-b}{a}\\ \mathbf{elif}\;b \leq 0.075:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 8
Accuracy	45.2%
Cost	1104.00

\[\begin{array}{l} t_1 := \frac{x}{y \cdot \left(a + a \cdot b\right)}\\ \mathbf{if}\;b \leq -3.6 \cdot 10^{-271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-250}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-b}{a}\\ \mathbf{elif}\;b \leq 1.52 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-136}:\\ \;\;\;\;\frac{b}{y} \cdot \left(-\frac{x}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	45.2%
Cost	844.00

\[\begin{array}{l} \mathbf{if}\;b \leq -800:\\ \;\;\;\;\frac{x}{y \cdot \left(a \cdot b\right)}\\ \mathbf{elif}\;b \leq -1 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;b \leq 1:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 10
Accuracy	45.3%
Cost	840.00

\[\begin{array}{l} t_1 := a + a \cdot b\\ \mathbf{if}\;b \leq -5.8 \cdot 10^{-271}:\\ \;\;\;\;\frac{x}{y \cdot t_1}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-244}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{-b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t_1}}{y}\\ \end{array} \]

Alternative 11
Accuracy	39.7%
Cost	580.00

\[\begin{array}{l} \mathbf{if}\;a \leq 1.3 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]

Alternative 12
Accuracy	43.7%
Cost	580.00

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

Alternative 13
Accuracy	39.8%
Cost	452.00

\[\begin{array}{l} \mathbf{if}\;a \leq 2.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]

Alternative 14
Accuracy	34.7%
Cost	320.00

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

Reproduce

herbie shell --seed 2023096 
(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))