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

Percentage Accurate: 98.4% → 98.4%

Time: 22.5s

Precision: binary64

Cost: 20160

• 21.6% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

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

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
    code = (x * exp((((y * log(z)) + ((t + (-1.0d0)) * log(a))) - b))) / y
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) {
	return (x * Math.exp((((y * Math.log(z)) + ((t + -1.0) * Math.log(a))) - b))) / y;
}

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

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)
	return Float64(Float64(x * exp(Float64(Float64(Float64(y * log(z)) + Float64(Float64(t + -1.0) * log(a))) - b))) / y)
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 = code(x, y, z, t, a, b)
	tmp = (x * exp((((y * log(z)) + ((t + -1.0) * log(a))) - b))) / y;
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_] := 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]

Target

Original	98.4%
Target	72.2%
Herbie	98.4%

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

Derivation

1. Initial program 98.5%

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

2. Final simplification 98.5%

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

Alternatives

Alternative 1
Accuracy	98.4%
Cost	20160

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

Alternative 2
Accuracy	87.3%
Cost	20432

\[\begin{array}{l} t_1 := \frac{e^{\left(y \cdot \log z - \log a\right) - b}}{\frac{y}{x}}\\ t_2 := \frac{x}{\frac{y}{e^{t \cdot \log a - b}}}\\ \mathbf{if}\;t \leq -2.55 \cdot 10^{-27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-291}:\\ \;\;\;\;\frac{x \cdot {z}^{y}}{y \cdot \left(a \cdot e^{b}\right)}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	87.2%
Cost	20296

\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{y \cdot \log z - b}}}\\ \mathbf{elif}\;b \leq 3.8 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{{a}^{t}}{a}}{\frac{y}{{z}^{y}}}}{e^{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{t \cdot \log a - b}}}\\ \end{array} \]

Alternative 4
Accuracy	83.3%
Cost	13772

\[\begin{array}{l} t_1 := \frac{x}{\frac{y}{e^{y \cdot \log z - b}}}\\ \mathbf{if}\;y \leq -860000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{-232}:\\ \;\;\;\;\frac{x}{\frac{y}{e^{t \cdot \log a - b}}}\\ \mathbf{elif}\;y \leq 0.68:\\ \;\;\;\;\frac{x}{\frac{a}{\frac{{a}^{t}}{y}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	86.2%
Cost	13705

\[\begin{array}{l} \mathbf{if}\;y \leq -1350000000 \lor \neg \left(y \leq 29000000\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{y \cdot \log z - b}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{{a}^{t}}{a}}{y}}{e^{b}}\\ \end{array} \]

Alternative 6
Accuracy	85.6%
Cost	13705

\[\begin{array}{l} \mathbf{if}\;y \leq -820000000 \lor \neg \left(y \leq 3600000\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{y \cdot \log z - b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{t}}{a} \cdot \frac{x}{y \cdot e^{b}}\\ \end{array} \]

Alternative 7
Accuracy	79.4%
Cost	13641

\[\begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+42} \lor \neg \left(t \leq 1.3 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x}{\frac{y}{e^{t \cdot \log a - b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \]

Alternative 8
Accuracy	58.1%
Cost	7312

\[\begin{array}{l} t_1 := \frac{x \cdot e^{-b}}{y}\\ \mathbf{if}\;b \leq -12:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -6 \cdot 10^{-254}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-285}:\\ \;\;\;\;\frac{x \cdot \left(-b\right)}{y}\\ \mathbf{elif}\;b \leq 1250000000:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	53.8%
Cost	7248

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{e^{b}}\\ \mathbf{if}\;b \leq -160:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-254}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-285}:\\ \;\;\;\;\frac{x \cdot \left(-b\right)}{y}\\ \mathbf{elif}\;b \leq 1250000000:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	64.2%
Cost	7180

\[\begin{array}{l} t_1 := \frac{x \cdot e^{-b}}{y}\\ \mathbf{if}\;b \leq -1.15:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-163}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{elif}\;b \leq 880:\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	73.3%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -720 \lor \neg \left(b \leq 3500\right):\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{{a}^{t}}{y \cdot a}\\ \end{array} \]

Alternative 12
Accuracy	73.3%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -700 \lor \neg \left(b \leq 90000\right):\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a}^{t}}{y} \cdot \frac{x}{a}\\ \end{array} \]

Alternative 13
Accuracy	75.8%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;b \leq -700 \lor \neg \left(b \leq 3700000\right):\\ \;\;\;\;\frac{x \cdot e^{-b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{\frac{{a}^{t}}{y}}}\\ \end{array} \]

Alternative 14
Accuracy	75.5%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{+46} \lor \neg \left(t \leq 2.1 \cdot 10^{+31}\right):\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{z}^{y}}{a}}{y}\\ \end{array} \]

Alternative 15
Accuracy	69.9%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+158} \lor \neg \left(t \leq 4.9 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{x \cdot {a}^{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{e^{b} \cdot \left(y \cdot a\right)}\\ \end{array} \]

Alternative 16
Accuracy	32.1%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;t \leq 3.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - \frac{y}{x} \cdot \left(x \cdot b\right)}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 17
Accuracy	33.8%
Cost	844

\[\begin{array}{l} t_1 := \frac{x \cdot \left(-b\right)}{y}\\ \mathbf{if}\;b \leq -2.55 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-254}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{a}}{y}\\ \end{array} \]

Alternative 18
Accuracy	33.7%
Cost	780

\[\begin{array}{l} t_1 := \frac{x \cdot \left(-b\right)}{y}\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-250}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]

Alternative 19
Accuracy	33.8%
Cost	780

\[\begin{array}{l} t_1 := \frac{x \cdot \left(-b\right)}{y}\\ \mathbf{if}\;b \leq -9.8 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4 \cdot 10^{-250}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-285}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]

Alternative 20
Accuracy	29.5%
Cost	649

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-77} \lor \neg \left(y \leq 1.42\right):\\ \;\;\;\;x \cdot \frac{-b}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]

Alternative 21
Accuracy	31.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-103}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]

Alternative 22
Accuracy	31.5%
Cost	320

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

Alternative 23
Accuracy	15.9%
Cost	192

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

Reproduce

herbie shell --seed 2023165 
(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))