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

Average Error: 1.8 → 2.3

Time: 53.5s

Precision: binary64

Cost: 20164

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{e^{\left(t + -1\right) \cdot \log a + y \cdot \log z} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

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
 (if (<= b 8.5e-11)
   (/ (* (exp (+ (* (+ t -1.0) (log a)) (* y (log z)))) x) y)
   (/ (/ x (* a (exp 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 tmp;
	if (b <= 8.5e-11) {
		tmp = (exp((((t + -1.0) * log(a)) + (y * log(z)))) * x) / y;
	} else {
		tmp = (x / (a * exp(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) :: tmp
    if (b <= 8.5d-11) then
        tmp = (exp((((t + (-1.0d0)) * log(a)) + (y * log(z)))) * x) / y
    else
        tmp = (x / (a * exp(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 tmp;
	if (b <= 8.5e-11) {
		tmp = (Math.exp((((t + -1.0) * Math.log(a)) + (y * Math.log(z)))) * x) / y;
	} else {
		tmp = (x / (a * Math.exp(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):
	tmp = 0
	if b <= 8.5e-11:
		tmp = (math.exp((((t + -1.0) * math.log(a)) + (y * math.log(z)))) * x) / y
	else:
		tmp = (x / (a * math.exp(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)
	tmp = 0.0
	if (b <= 8.5e-11)
		tmp = Float64(Float64(exp(Float64(Float64(Float64(t + -1.0) * log(a)) + Float64(y * log(z)))) * x) / y);
	else
		tmp = Float64(Float64(x / Float64(a * exp(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)
	tmp = 0.0;
	if (b <= 8.5e-11)
		tmp = (exp((((t + -1.0) * log(a)) + (y * log(z)))) * x) / y;
	else
		tmp = (x / (a * exp(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_] := If[LessEqual[b, 8.5e-11], N[(N[(N[Exp[N[(N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] + N[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], N[(N[(x / N[(a * N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Target

Original	1.8
Target	11.4
Herbie	2.3

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

Derivation

1. Split input into 2 regimes

2. if b < 8.50000000000000037e-11

   1. Initial program 2.9

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

   2. Taylor expanded in b around 0 3.1

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

   3. Simplified 3.1

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

      [Start] 3.1	\[ \frac{e^{y \cdot \log z + \left(t - 1\right) \cdot \log a} \cdot x}{y} \]
      exponential.json-simplify-1 [=>] 11.5	\[ \frac{\color{blue}{\left(e^{y \cdot \log z} \cdot e^{\left(t - 1\right) \cdot \log a}\right)} \cdot x}{y} \]
      rational.json-simplify-4 [<=] 11.5	\[ \frac{\left(e^{y \cdot \log z} \cdot e^{\color{blue}{\left(t - 1\right) \cdot \log a + 0}}\right) \cdot x}{y} \]
      rational.json-simplify-4 [<=] 11.5	\[ \frac{\left(e^{y \cdot \log z} \cdot e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + 0\right)} + 0}\right) \cdot x}{y} \]
      rational.json-simplify-1 [<=] 11.5	\[ \frac{\left(e^{y \cdot \log z} \cdot e^{\color{blue}{\left(0 + \left(t - 1\right) \cdot \log a\right)} + 0}\right) \cdot x}{y} \]
      exponential.json-simplify-1 [<=] 3.1	\[ \frac{\color{blue}{e^{y \cdot \log z + \left(\left(0 + \left(t - 1\right) \cdot \log a\right) + 0\right)}} \cdot x}{y} \]
      rational.json-simplify-1 [=>] 3.1	\[ \frac{e^{y \cdot \log z + \color{blue}{\left(0 + \left(0 + \left(t - 1\right) \cdot \log a\right)\right)}} \cdot x}{y} \]
      rational.json-simplify-1 [=>] 3.1	\[ \frac{e^{y \cdot \log z + \left(0 + \color{blue}{\left(\left(t - 1\right) \cdot \log a + 0\right)}\right)} \cdot x}{y} \]
      rational.json-simplify-4 [=>] 3.1	\[ \frac{e^{y \cdot \log z + \left(0 + \color{blue}{\left(t - 1\right) \cdot \log a}\right)} \cdot x}{y} \]
      rational.json-simplify-41 [<=] 3.1	\[ \frac{e^{\color{blue}{\left(t - 1\right) \cdot \log a + \left(y \cdot \log z + 0\right)}} \cdot x}{y} \]
      rational.json-simplify-15 [<=] 3.1	\[ \frac{e^{\color{blue}{\left(t + -1\right)} \cdot \log a + \left(y \cdot \log z + 0\right)} \cdot x}{y} \]
      rational.json-simplify-4 [=>] 3.1	\[ \frac{e^{\left(t + -1\right) \cdot \log a + \color{blue}{y \cdot \log z}} \cdot x}{y} \]

   if 8.50000000000000037e-11 < b

   1. Initial program 0.1

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

   2. Taylor expanded in t around 0 1.3

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

   3. Simplified 7.5

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

      [Start] 1.3	\[ \frac{t \cdot \left(e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b} \cdot \left(x \cdot \log a\right)\right) + e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b} \cdot x}{y} \]
      rational.json-simplify-2 [=>] 1.3	\[ \frac{t \cdot \color{blue}{\left(\left(x \cdot \log a\right) \cdot e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b}\right)} + e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b} \cdot x}{y} \]
      rational.json-simplify-2 [<=] 1.3	\[ \frac{t \cdot \left(\color{blue}{\left(\log a \cdot x\right)} \cdot e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b}\right) + e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b} \cdot x}{y} \]
      rational.json-simplify-43 [<=] 7.5	\[ \frac{\color{blue}{e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b} \cdot \left(t \cdot \left(\log a \cdot x\right)\right)} + e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b} \cdot x}{y} \]
      rational.json-simplify-2 [<=] 7.5	\[ \frac{e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b} \cdot \color{blue}{\left(\left(\log a \cdot x\right) \cdot t\right)} + e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b} \cdot x}{y} \]
      rational.json-simplify-2 [=>] 7.5	\[ \frac{e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b} \cdot \left(\left(\log a \cdot x\right) \cdot t\right) + \color{blue}{x \cdot e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b}}}{y} \]
      rational.json-simplify-47 [=>] 7.5	\[ \frac{\color{blue}{e^{\left(y \cdot \log z + -1 \cdot \log a\right) - b} \cdot \left(x + \left(\log a \cdot x\right) \cdot t\right)}}{y} \]
      rational.json-simplify-2 [=>] 7.5	\[ \frac{e^{\left(y \cdot \log z + \color{blue}{\log a \cdot -1}\right) - b} \cdot \left(x + \left(\log a \cdot x\right) \cdot t\right)}{y} \]
      rational.json-simplify-9 [=>] 7.5	\[ \frac{e^{\left(y \cdot \log z + \color{blue}{\left(-\log a\right)}\right) - b} \cdot \left(x + \left(\log a \cdot x\right) \cdot t\right)}{y} \]

   4. Taylor expanded in y around 0 7.4

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

   5. Simplified 7.2

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

      [Start] 7.4	\[ \frac{e^{-\left(b + \log a\right)} \cdot \left(t \cdot \left(x \cdot \log a\right) + x\right)}{y} \]
      exponential.json-simplify-2 [=>] 7.4	\[ \frac{\color{blue}{\frac{1}{e^{b + \log a}}} \cdot \left(t \cdot \left(x \cdot \log a\right) + x\right)}{y} \]
      rational.json-simplify-1 [=>] 7.4	\[ \frac{\frac{1}{e^{\color{blue}{\log a + b}}} \cdot \left(t \cdot \left(x \cdot \log a\right) + x\right)}{y} \]
      exponential.json-simplify-1 [=>] 7.4	\[ \frac{\frac{1}{\color{blue}{e^{\log a} \cdot e^{b}}} \cdot \left(t \cdot \left(x \cdot \log a\right) + x\right)}{y} \]
      exponential.json-simplify-7 [=>] 7.4	\[ \frac{\frac{1}{\color{blue}{a} \cdot e^{b}} \cdot \left(t \cdot \left(x \cdot \log a\right) + x\right)}{y} \]
      rational.json-simplify-1 [=>] 7.4	\[ \frac{\frac{1}{a \cdot e^{b}} \cdot \color{blue}{\left(x + t \cdot \left(x \cdot \log a\right)\right)}}{y} \]
      rational.json-simplify-43 [<=] 7.2	\[ \frac{\frac{1}{a \cdot e^{b}} \cdot \left(x + \color{blue}{\log a \cdot \left(t \cdot x\right)}\right)}{y} \]

   6. Taylor expanded in t around 0 0.8

      \[\leadsto \frac{\color{blue}{\frac{x}{a \cdot e^{b}}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 2.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{e^{\left(t + -1\right) \cdot \log a + y \cdot \log z} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.3
Cost	26756

\[\begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -625:\\ \;\;\;\;\left(0 - \left(-1 - \frac{{a}^{\left(t + -1\right)} \cdot x}{y}\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(y \cdot \log z + \left(-\log a\right)\right) - b} \cdot x}{y}\\ \end{array} \]

Alternative 2
Error	1.8
Cost	20160

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

Alternative 3
Error	10.2
Cost	13768

\[\begin{array}{l} t_1 := \left(0 - \left(-1 - \frac{{a}^{\left(t + -1\right)} \cdot x}{y}\right)\right) - 1\\ \mathbf{if}\;b \leq -2.15 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.6 \cdot 10^{-262}:\\ \;\;\;\;\frac{\left(e^{y \cdot \log z} \cdot \frac{1}{a}\right) \cdot x}{y}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 4
Error	11.6
Cost	7564

\[\begin{array}{l} \mathbf{if}\;b \leq -1.26 \cdot 10^{-282}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{0.5 \cdot \frac{{b}^{2} \cdot x}{a}}{y}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{{a}^{t} \cdot \left(\left(1 - b\right) \cdot x\right)}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 5
Error	9.9
Cost	7428

\[\begin{array}{l} \mathbf{if}\;b \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\left(0 - \left(-1 - \frac{{a}^{\left(t + -1\right)} \cdot x}{y}\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 6
Error	14.0
Cost	7308

\[\begin{array}{l} t_1 := \frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{if}\;b \leq -7.5 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{-155}:\\ \;\;\;\;\left(0 - \left(-1 - \frac{\frac{x}{a}}{y}\right)\right) - 1\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 7
Error	14.0
Cost	7308

\[\begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{-116}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-155}:\\ \;\;\;\;\left(0 - \left(-1 - \frac{\frac{x}{a}}{y}\right)\right) - 1\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{{a}^{t} \cdot x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 8
Error	12.0
Cost	7308

\[\begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-283}:\\ \;\;\;\;\frac{{a}^{\left(t - 1\right)} \cdot x}{y}\\ \mathbf{elif}\;b \leq 1.25 \cdot 10^{-145}:\\ \;\;\;\;\frac{0.5 \cdot \frac{{b}^{2} \cdot x}{a}}{y}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{{a}^{t} \cdot x}{a}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 9
Error	16.4
Cost	7112

\[\begin{array}{l} \mathbf{if}\;b \leq -1.32 \cdot 10^{-27}:\\ \;\;\;\;\frac{{a}^{t} \cdot x}{y}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-38}:\\ \;\;\;\;\left(0 - \left(-1 - \frac{\frac{x}{a}}{y}\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a \cdot e^{b}}}{y}\\ \end{array} \]

Alternative 10
Error	16.0
Cost	7048

\[\begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-26}:\\ \;\;\;\;\frac{{a}^{t} \cdot x}{y}\\ \mathbf{elif}\;b \leq 660:\\ \;\;\;\;\left(0 - \left(-1 - \frac{\frac{x}{a}}{y}\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-b} \cdot x}{y}\\ \end{array} \]

Alternative 11
Error	16.6
Cost	6916

\[\begin{array}{l} \mathbf{if}\;b \leq 660:\\ \;\;\;\;\left(0 - \left(-1 - \frac{\frac{x}{a}}{y}\right)\right) - 1\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-b} \cdot x}{y}\\ \end{array} \]

Alternative 12
Error	40.9
Cost	716

\[\begin{array}{l} t_1 := \frac{\frac{x}{a}}{y}\\ \mathbf{if}\;a \leq 3.8 \cdot 10^{-210}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{+229}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	40.9
Cost	716

\[\begin{array}{l} \mathbf{if}\;a \leq 5.9 \cdot 10^{-205}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;a \leq 10^{-94}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot x}{y}\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{+229}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \end{array} \]

Alternative 14
Error	29.2
Cost	704

\[\left(0 - \left(-1 - \frac{\frac{x}{a}}{y}\right)\right) - 1 \]

Alternative 15
Error	40.6
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{-34}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{a \cdot y}\\ \end{array} \]

Alternative 16
Error	54.4
Cost	192

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

Reproduce

herbie shell --seed 2023077 
(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))