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

Average Error: 1.8 → 1.8

Time: 48.2s

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	1.8
Target	11.0
Herbie	1.8

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

Derivation

1. Initial program 1.8

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

2. Final simplification 1.8

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

Alternatives

Alternative 1
Error	5.5
Cost	27016

\[\begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;t_1 \leq -674.5:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{t}}{a}}{y}\\ \mathbf{elif}\;t_1 \leq -134.5:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{{z}^{y}}{a}}{e^{b}}}{y}\\ \end{array} \]

Alternative 2
Error	2.2
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -50000000000:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{t}}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot e^{\left(y \cdot \log z - \log a\right) - b}}{y}\\ \end{array} \]

Alternative 3
Error	12.2
Cost	14364

\[\begin{array}{l} t_1 := \frac{x \cdot \frac{{a}^{t}}{a \cdot e^{b}}}{y}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-31}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{t}}{a}}{y}\\ \mathbf{elif}\;b \leq -4.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{{z}^{y}}{y \cdot \frac{a}{x}}\\ \mathbf{elif}\;b \leq -5.8 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{{a}^{t}}{\frac{a}{x}}}{y}\\ \mathbf{elif}\;b \leq -3.9 \cdot 10^{-258}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{y}}{a}\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{-124}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 4
Error	12.6
Cost	7836

\[\begin{array}{l} t_1 := \frac{x \cdot \frac{{a}^{t}}{a}}{y}\\ t_2 := \frac{\frac{{a}^{t}}{\frac{a}{x}}}{y}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4.8 \cdot 10^{-135}:\\ \;\;\;\;\frac{{z}^{y}}{y \cdot \frac{a}{x}}\\ \mathbf{elif}\;b \leq -5.8 \cdot 10^{-165}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -3.9 \cdot 10^{-258}:\\ \;\;\;\;x \cdot \frac{\frac{{z}^{y}}{y}}{a}\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{-187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{-124}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 5
Error	12.9
Cost	7572

\[\begin{array}{l} t_1 := \frac{x \cdot \frac{{a}^{t}}{a}}{y}\\ \mathbf{if}\;b \leq -4 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-188}:\\ \;\;\;\;\frac{{z}^{y}}{y \cdot \frac{a}{x}}\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{-187}:\\ \;\;\;\;\frac{\frac{{a}^{t}}{\frac{a}{x}}}{y}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{-124}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{elif}\;b \leq 4.9 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 6
Error	18.9
Cost	7508

\[\begin{array}{l} t_1 := \left(1 + \frac{x}{y \cdot a}\right) + -1\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-272}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{x}{y}}}\\ \mathbf{elif}\;b \leq 1.02 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 7
Error	13.9
Cost	7440

\[\begin{array}{l} t_1 := \frac{{z}^{y}}{y \cdot \frac{a}{x}}\\ t_2 := \frac{\frac{{a}^{t}}{\frac{a}{x}}}{y}\\ \mathbf{if}\;b \leq -3.7 \cdot 10^{-30}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-188}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{-187}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;b \leq 205:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 8
Error	13.7
Cost	7440

\[\begin{array}{l} t_1 := \frac{\frac{{a}^{t}}{\frac{a}{x}}}{y}\\ \mathbf{if}\;b \leq -3.7 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -9 \cdot 10^{-188}:\\ \;\;\;\;\frac{{z}^{y}}{y \cdot \frac{a}{x}}\\ \mathbf{elif}\;b \leq 9.6 \cdot 10^{-187}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 205:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 9
Error	12.8
Cost	7044

\[\begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{{a}^{t}}{\frac{a}{x}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot e^{b}\right)}\\ \end{array} \]

Alternative 10
Error	30.4
Cost	1236

\[\begin{array}{l} t_1 := \left(1 + \frac{x}{y \cdot a}\right) + -1\\ \mathbf{if}\;b \leq -1.45 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.4 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;b \leq -2.1 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{-272}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{x}{y}}}\\ \mathbf{elif}\;b \leq 1.396406642252593 \cdot 10^{+128}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot \left(y \cdot b\right)}\\ \end{array} \]

Alternative 11
Error	34.9
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{a \cdot \left(y \cdot b\right)}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-194}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	34.9
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{a \cdot \left(y \cdot b\right)}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-194}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	34.9
Cost	976

\[\begin{array}{l} t_1 := \frac{x}{a \cdot \left(y \cdot b\right)}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5 \cdot 10^{-194}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{a}\\ \mathbf{elif}\;b \leq 4.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	27.3
Cost	840

\[\begin{array}{l} t_1 := \left(1 + \frac{x}{y \cdot a}\right) + -1\\ \mathbf{if}\;y \leq -7 \cdot 10^{-36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{x}{a + a \cdot b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	38.6
Cost	584

\[\begin{array}{l} t_1 := \frac{x}{y \cdot a}\\ \mathbf{if}\;x \leq -6521737883515.281:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.414142615934211 \cdot 10^{-65}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	38.6
Cost	580

\[\begin{array}{l} \mathbf{if}\;a \leq 10^{-80}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 17
Error	40.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 5.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{x}{y}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]

Alternative 18
Error	42.6
Cost	320

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

Reproduce

herbie shell --seed 2022316 
(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))