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

Average Error: 1.9 → 1.9

Time: 29.0s

Precision: binary64

Cost: 20160

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	1.9
Target	11.0
Herbie	1.9

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

Derivation

1. Initial program 1.9

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

2. Final simplification 1.9

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

Alternatives

Alternative 1
Error	2.4
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -580:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \end{array} \]

Alternative 2
Error	7.5
Cost	20228

\[\begin{array}{l} \mathbf{if}\;\left(t + -1\right) \cdot \log a \leq -551:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{{z}^{y}}{y} \cdot \frac{x}{a \cdot e^{b}}\\ \end{array} \]

Alternative 3
Error	27.7
Cost	7244

\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \frac{y - \frac{y}{b}}{a \cdot \left(\left(-y\right) \cdot \frac{y}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{b}}}{y \cdot a}\\ \end{array} \]

Alternative 4
Error	24.6
Cost	7244

\[\begin{array}{l} \mathbf{if}\;b \leq -4.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-204}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 9.2 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{y - \frac{y}{b}}{a \cdot \left(\left(-y\right) \cdot \frac{y}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 5
Error	10.3
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq 5.6 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;b \leq 0.0055:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 6
Error	10.4
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq 7.2 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{elif}\;b \leq 0.016:\\ \;\;\;\;\frac{{z}^{y}}{y} \cdot \frac{x}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 7
Error	10.5
Cost	7044

\[\begin{array}{l} \mathbf{if}\;b \leq 1.16 \cdot 10^{-27}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t + -1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 8
Error	33.0
Cost	1864

\[\begin{array}{l} t_1 := y + y \cdot b\\ \mathbf{if}\;y \leq -1.65 \cdot 10^{+127}:\\ \;\;\;\;\frac{\frac{x}{t_1}}{a}\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{+84}:\\ \;\;\;\;\frac{y \cdot y - \left(y \cdot b\right) \cdot \left(y \cdot b\right)}{\frac{a}{x} \cdot \left(\left(y \cdot y\right) \cdot t_1\right)}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - y \cdot b}{\frac{a \cdot \left(y \cdot y\right)}{x}}\\ \end{array} \]

Alternative 9
Error	37.5
Cost	1420

\[\begin{array}{l} t_1 := \frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{if}\;b \leq -1 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-198}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 6.4 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \frac{y - \frac{y}{b}}{a \cdot \left(\left(-y\right) \cdot \frac{y}{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	36.8
Cost	1360

\[\begin{array}{l} t_1 := \frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{if}\;b \leq -6.8 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-278}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{-36}:\\ \;\;\;\;\frac{-x}{\frac{a}{\frac{-1}{y}}}\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{+54}:\\ \;\;\;\;\frac{y - y \cdot b}{\frac{a \cdot \left(y \cdot y\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	37.6
Cost	1360

\[\begin{array}{l} t_1 := \frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{if}\;b \leq -4.6 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{-200}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.22 \cdot 10^{-89}:\\ \;\;\;\;\frac{x}{a} \cdot \frac{\frac{y}{b} - y}{y \cdot \frac{y}{b}}\\ \mathbf{elif}\;b \leq 2.35 \cdot 10^{+55}:\\ \;\;\;\;\frac{y - y \cdot b}{\frac{a \cdot \left(y \cdot y\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	37.5
Cost	1360

\[\begin{array}{l} t_1 := \frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{if}\;b \leq -4.9 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-204}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.36 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{y}{b} - y}{\frac{a}{x} \cdot \left(y \cdot \frac{y}{b}\right)}\\ \mathbf{elif}\;b \leq 1.75 \cdot 10^{+55}:\\ \;\;\;\;\frac{y - y \cdot b}{\frac{a \cdot \left(y \cdot y\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	37.0
Cost	1236

\[\begin{array}{l} t_1 := \frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-278}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{-x}{\frac{a}{\frac{-1}{y}}}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-30}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{+149}:\\ \;\;\;\;\frac{\frac{x}{y + y \cdot b}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	36.8
Cost	1104

\[\begin{array}{l} t_1 := \frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{if}\;b \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-276}:\\ \;\;\;\;\frac{x}{y} \cdot \left(-\frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 1.7 \cdot 10^{-51}:\\ \;\;\;\;\frac{-x}{\frac{a}{\frac{-1}{y}}}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	38.9
Cost	844

\[\begin{array}{l} \mathbf{if}\;a \leq 2 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{elif}\;a \leq 5.9 \cdot 10^{+225}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+292}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 16
Error	40.0
Cost	777

\[\begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{+56} \lor \neg \left(y \leq 3.7 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(-\frac{b}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{a}{x}}\\ \end{array} \]

Alternative 17
Error	38.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{y \cdot a}\\ \end{array} \]

Alternative 18
Error	40.4
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-84} \lor \neg \left(y \leq 4.9 \cdot 10^{-57}\right):\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]

Alternative 19
Error	38.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 10^{-26}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 20
Error	41.9
Cost	320

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

Reproduce

herbie shell --seed 2023039 
(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))