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

Average Error: 2.1 → 1.4

Time: 21.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \log z \cdot y + \log a \cdot t\\ \mathbf{if}\;\log a \leq -54.988702570892116:\\ \;\;\;\;\frac{e^{t_1 - \left(\log a + b\right)} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;e^{t_1 - b} \cdot \frac{x}{a \cdot y}\\ \end{array} \]

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
 (let* ((t_1 (+ (* (log z) y) (* (log a) t))))
   (if (<= (log a) -54.988702570892116)
     (/ (* (exp (- t_1 (+ (log a) b))) x) y)
     (* (exp (- t_1 b)) (/ x (* a 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 t_1 = (log(z) * y) + (log(a) * t);
	double tmp;
	if (log(a) <= -54.988702570892116) {
		tmp = (exp((t_1 - (log(a) + b))) * x) / y;
	} else {
		tmp = exp((t_1 - b)) * (x / (a * 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) :: t_1
    real(8) :: tmp
    t_1 = (log(z) * y) + (log(a) * t)
    if (log(a) <= (-54.988702570892116d0)) then
        tmp = (exp((t_1 - (log(a) + b))) * x) / y
    else
        tmp = exp((t_1 - b)) * (x / (a * 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 t_1 = (Math.log(z) * y) + (Math.log(a) * t);
	double tmp;
	if (Math.log(a) <= -54.988702570892116) {
		tmp = (Math.exp((t_1 - (Math.log(a) + b))) * x) / y;
	} else {
		tmp = Math.exp((t_1 - b)) * (x / (a * 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):
	t_1 = (math.log(z) * y) + (math.log(a) * t)
	tmp = 0
	if math.log(a) <= -54.988702570892116:
		tmp = (math.exp((t_1 - (math.log(a) + b))) * x) / y
	else:
		tmp = math.exp((t_1 - b)) * (x / (a * 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)
	t_1 = Float64(Float64(log(z) * y) + Float64(log(a) * t))
	tmp = 0.0
	if (log(a) <= -54.988702570892116)
		tmp = Float64(Float64(exp(Float64(t_1 - Float64(log(a) + b))) * x) / y);
	else
		tmp = Float64(exp(Float64(t_1 - b)) * Float64(x / Float64(a * 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)
	t_1 = (log(z) * y) + (log(a) * t);
	tmp = 0.0;
	if (log(a) <= -54.988702570892116)
		tmp = (exp((t_1 - (log(a) + b))) * x) / y;
	else
		tmp = exp((t_1 - b)) * (x / (a * 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_] := Block[{t$95$1 = N[(N[(N[Log[z], $MachinePrecision] * y), $MachinePrecision] + N[(N[Log[a], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Log[a], $MachinePrecision], -54.988702570892116], N[(N[(N[Exp[N[(t$95$1 - N[(N[Log[a], $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * x), $MachinePrecision] / y), $MachinePrecision], N[(N[Exp[N[(t$95$1 - b), $MachinePrecision]], $MachinePrecision] * N[(x / N[(a * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	2.1
Target	10.6
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if (log.f64 a) < -54.988702570892116

   1. Initial program 0.7

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

   2. Taylor expanded in x around 0 0.7

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

   if -54.988702570892116 < (log.f64 a)

   1. Initial program 3.2

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

   2. Taylor expanded in y around inf 3.2

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

   3. Simplified 18.1

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

   4. Applied egg-rr 15.9

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{a \cdot e^{b}}{{a}^{t}} \cdot \frac{y}{{z}^{y}}}} \]

   5. Taylor expanded in a around inf 15.9

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

   6. Simplified 1.9

      \[\leadsto \color{blue}{e^{\left(\log z \cdot y - t \cdot \left(-\log a\right)\right) - b} \cdot \frac{x}{y \cdot a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log a \leq -54.988702570892116:\\ \;\;\;\;\frac{e^{\left(\log z \cdot y + \log a \cdot t\right) - \left(\log a + b\right)} \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log z \cdot y + \log a \cdot t\right) - b} \cdot \frac{x}{a \cdot y}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(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))