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

Average Error: 2.1 → 1.2

Time: 50.9s

Precision: binary64

Cost: 20552

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \log z\\ t_2 := \frac{\frac{e^{t_1 + \left(\left(t + -1\right) \cdot \log a - b\right)}}{y}}{\frac{1}{x}}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-33}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+75}:\\ \;\;\;\;e^{\left(t_1 + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (* y (log z)))
        (t_2 (/ (/ (exp (+ t_1 (- (* (+ t -1.0) (log a)) b))) y) (/ 1.0 x))))
   (if (<= x -5e-33)
     t_2
     (if (<= x 2.9e+75)
       (* (exp (- (+ t_1 (* (- t 1.0) (log a))) b)) (/ x y))
       t_2))))

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 = y * log(z);
	double t_2 = (exp((t_1 + (((t + -1.0) * log(a)) - b))) / y) / (1.0 / x);
	double tmp;
	if (x <= -5e-33) {
		tmp = t_2;
	} else if (x <= 2.9e+75) {
		tmp = exp(((t_1 + ((t - 1.0) * log(a))) - b)) * (x / y);
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = y * log(z)
    t_2 = (exp((t_1 + (((t + (-1.0d0)) * log(a)) - b))) / y) / (1.0d0 / x)
    if (x <= (-5d-33)) then
        tmp = t_2
    else if (x <= 2.9d+75) then
        tmp = exp(((t_1 + ((t - 1.0d0) * log(a))) - b)) * (x / y)
    else
        tmp = t_2
    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 = y * Math.log(z);
	double t_2 = (Math.exp((t_1 + (((t + -1.0) * Math.log(a)) - b))) / y) / (1.0 / x);
	double tmp;
	if (x <= -5e-33) {
		tmp = t_2;
	} else if (x <= 2.9e+75) {
		tmp = Math.exp(((t_1 + ((t - 1.0) * Math.log(a))) - b)) * (x / y);
	} else {
		tmp = t_2;
	}
	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 = y * math.log(z)
	t_2 = (math.exp((t_1 + (((t + -1.0) * math.log(a)) - b))) / y) / (1.0 / x)
	tmp = 0
	if x <= -5e-33:
		tmp = t_2
	elif x <= 2.9e+75:
		tmp = math.exp(((t_1 + ((t - 1.0) * math.log(a))) - b)) * (x / y)
	else:
		tmp = t_2
	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(y * log(z))
	t_2 = Float64(Float64(exp(Float64(t_1 + Float64(Float64(Float64(t + -1.0) * log(a)) - b))) / y) / Float64(1.0 / x))
	tmp = 0.0
	if (x <= -5e-33)
		tmp = t_2;
	elseif (x <= 2.9e+75)
		tmp = Float64(exp(Float64(Float64(t_1 + Float64(Float64(t - 1.0) * log(a))) - b)) * Float64(x / y));
	else
		tmp = t_2;
	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 = y * log(z);
	t_2 = (exp((t_1 + (((t + -1.0) * log(a)) - b))) / y) / (1.0 / x);
	tmp = 0.0;
	if (x <= -5e-33)
		tmp = t_2;
	elseif (x <= 2.9e+75)
		tmp = exp(((t_1 + ((t - 1.0) * log(a))) - b)) * (x / y);
	else
		tmp = t_2;
	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[(y * N[Log[z], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[N[(t$95$1 + N[(N[(N[(t + -1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision] / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e-33], t$95$2, If[LessEqual[x, 2.9e+75], N[(N[Exp[N[(N[(t$95$1 + N[(N[(t - 1.0), $MachinePrecision] * N[Log[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision]], $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Target

Original	2.1
Target	10.9
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000028e-33 or 2.8999999999999998e75 < x

   1. Initial program 0.8

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

   2. Simplified 0.9

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

      [Start] 0.8	\[ \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
      rational.json-simplify-2 [=>] 0.8	\[ \frac{\color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot x}}{y} \]
      rational.json-simplify-49 [=>] 0.9	\[ \color{blue}{x \cdot \frac{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}} \]
      rational.json-simplify-1 [=>] 0.9	\[ x \cdot \frac{e^{\color{blue}{\left(\left(t - 1\right) \cdot \log a + y \cdot \log z\right)} - b}}{y} \]
      rational.json-simplify-48 [=>] 0.9	\[ x \cdot \frac{e^{\color{blue}{y \cdot \log z + \left(\left(t - 1\right) \cdot \log a - b\right)}}}{y} \]

   3. Applied egg-rr 0.9

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

   if -5.00000000000000028e-33 < x < 2.8999999999999998e75

   1. Initial program 3.1

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

   2. Simplified 1.5

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

      [Start] 3.1	\[ \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y} \]
      rational.json-simplify-49 [=>] 1.5	\[ \color{blue}{e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{e^{y \cdot \log z + \left(\left(t + -1\right) \cdot \log a - b\right)}}{y}}{\frac{1}{x}}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+75}:\\ \;\;\;\;e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{e^{y \cdot \log z + \left(\left(t + -1\right) \cdot \log a - b\right)}}{y}}{\frac{1}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	26692

\[\begin{array}{l} t_1 := y \cdot \log z\\ t_2 := \left(t - 1\right) \cdot \log a\\ \mathbf{if}\;\log a \leq -75:\\ \;\;\;\;\frac{x \cdot e^{\left(t_1 + t_2\right) - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{e^{t_1 + \left(t_2 - b\right)}}{y}\\ \end{array} \]

Alternative 2
Error	1.2
Cost	20424

\[\begin{array}{l} t_1 := \left(t - 1\right) \cdot \log a\\ t_2 := y \cdot \log z\\ t_3 := x \cdot \frac{e^{t_2 + \left(t_1 - b\right)}}{y}\\ \mathbf{if}\;x \leq -4 \cdot 10^{-33}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+75}:\\ \;\;\;\;e^{\left(t_2 + t_1\right) - b} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	2.4
Cost	20420

\[\begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -600:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{e^{y \cdot \log z - b}}{y}}{a}}{\frac{1}{x}}\\ \end{array} \]

Alternative 4
Error	4.2
Cost	20292

\[\begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \leq -500:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{a}{e^{y \cdot \log z - b}}}\\ \end{array} \]

Alternative 5
Error	2.3
Cost	20160

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

Alternative 6
Error	8.0
Cost	13768

\[\begin{array}{l} t_1 := \frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{if}\;y \leq -2.15 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{+50}:\\ \;\;\;\;x \cdot \frac{e^{\left(t - 1\right) \cdot \log a - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	7.1
Cost	13768

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \cdot 10^{+98}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+80}:\\ \;\;\;\;x \cdot \frac{e^{\left(t - 1\right) \cdot \log a - b}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{y \cdot \log z - b}}{a \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 8
Error	11.9
Cost	13576

\[\begin{array}{l} \mathbf{if}\;b \leq 5.2 \cdot 10^{-265}:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{elif}\;b \leq 3.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\left(-\log a\right) - b} \cdot x}{y}\\ \end{array} \]

Alternative 9
Error	11.5
Cost	7308

\[\begin{array}{l} t_1 := x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{if}\;b \leq 4.4 \cdot 10^{-262}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-66}:\\ \;\;\;\;\frac{{z}^{y}}{a} \cdot \frac{x}{y}\\ \mathbf{elif}\;b \leq 460:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{b}}}{y}\\ \end{array} \]

Alternative 10
Error	22.7
Cost	7248

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{a}\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{-17}:\\ \;\;\;\;x \cdot \left(\frac{2}{y \cdot \left(a \cdot \left(a + a\right)\right)} \cdot a\right)\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -5.3 \cdot 10^{-276}:\\ \;\;\;\;x \cdot \left(a \cdot \frac{\frac{1}{a}}{a \cdot y}\right)\\ \mathbf{elif}\;b \leq 110:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{b}}}{y}\\ \end{array} \]

Alternative 11
Error	10.4
Cost	7044

\[\begin{array}{l} \mathbf{if}\;b \leq 380:\\ \;\;\;\;x \cdot \frac{{a}^{\left(t - 1\right)}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{e^{b}}}{y}\\ \end{array} \]

Alternative 12
Error	35.9
Cost	1360

\[\begin{array}{l} t_1 := \frac{1}{a \cdot \frac{y}{x}}\\ t_2 := x \cdot \left(\frac{2}{y \cdot \left(a \cdot \left(a + a\right)\right)} \cdot a\right)\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-229}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-254}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 13
Error	35.9
Cost	1360

\[\begin{array}{l} t_1 := \frac{1}{a \cdot \frac{y}{x}}\\ t_2 := x \cdot \left(\frac{2}{y \cdot \left(a \cdot \left(a + a\right)\right)} \cdot a\right)\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-289}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-254}:\\ \;\;\;\;\frac{\left(\frac{-1}{b} + 1\right) \cdot \left(\frac{x}{a} \cdot \left(-b\right)\right)}{y}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 14
Error	37.3
Cost	1232

\[\begin{array}{l} t_1 := \frac{1}{a \cdot \frac{y}{x}}\\ t_2 := x \cdot \left(a \cdot \frac{\frac{1}{a}}{a \cdot y}\right)\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -2.4 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-253}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Error	38.7
Cost	976

\[\begin{array}{l} t_1 := \frac{1}{a \cdot \frac{y}{x}}\\ t_2 := \frac{x}{y \cdot a}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+42}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.32 \cdot 10^{-265}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-249}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;x \leq 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Error	38.7
Cost	976

\[\begin{array}{l} t_1 := \frac{1}{a \cdot \frac{y}{x}}\\ t_2 := \frac{x}{y \cdot a}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-229}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-254}:\\ \;\;\;\;\frac{1}{\frac{y}{\frac{x}{a}}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Error	39.4
Cost	848

\[\begin{array}{l} t_1 := \frac{\frac{x}{y}}{a}\\ t_2 := \frac{x}{y \cdot a}\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{+95}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-228}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-242}:\\ \;\;\;\;\frac{\frac{x}{a}}{y}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Error	39.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq 1.7 \cdot 10^{-303}:\\ \;\;\;\;\frac{1}{a \cdot \frac{y}{x}}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{1}{y}}{\frac{a}{x}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(x \cdot \frac{b}{a \cdot y}\right)\\ \end{array} \]

Alternative 19
Error	40.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;a \leq 7.4 \cdot 10^{-59}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot a}\\ \end{array} \]

Alternative 20
Error	54.7
Cost	192

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

Reproduce

herbie shell --seed 2023063 
(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))