Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G

Percentage Accurate: 84.7% → 99.0%

Time: 14.4s

Precision: binary64

Cost: 7049

• could not determine a ground truth

  1. x = -4.3403007608152746e+129
  2. y = 1.7948342671767444e+245
  3. z = 1.6312374134177418e+99

Math

\[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -53000000 \lor \neg \left(y \leq 2.4 \cdot 10^{-35}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -53000000.0) (not (<= y 2.4e-35)))
   (+ x (/ (exp (- z)) y))
   (+ x (/ 1.0 y))))

C

double code(double x, double y, double z) {
	return x + (exp((y * log((y / (z + y))))) / y);
}

↓

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -53000000.0) || !(y <= 2.4e-35)) {
		tmp = x + (exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + (exp((y * log((y / (z + y))))) / y)
end function

↓

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-53000000.0d0)) .or. (.not. (y <= 2.4d-35))) then
        tmp = x + (exp(-z) / y)
    else
        tmp = x + (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	return x + (Math.exp((y * Math.log((y / (z + y))))) / y);
}

↓

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -53000000.0) || !(y <= 2.4e-35)) {
		tmp = x + (Math.exp(-z) / y);
	} else {
		tmp = x + (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	return x + (math.exp((y * math.log((y / (z + y))))) / y)

↓

def code(x, y, z):
	tmp = 0
	if (y <= -53000000.0) or not (y <= 2.4e-35):
		tmp = x + (math.exp(-z) / y)
	else:
		tmp = x + (1.0 / y)
	return tmp

Julia

function code(x, y, z)
	return Float64(x + Float64(exp(Float64(y * log(Float64(y / Float64(z + y))))) / y))
end

↓

function code(x, y, z)
	tmp = 0.0
	if ((y <= -53000000.0) || !(y <= 2.4e-35))
		tmp = Float64(x + Float64(exp(Float64(-z)) / y));
	else
		tmp = Float64(x + Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (exp((y * log((y / (z + y))))) / y);
end

↓

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -53000000.0) || ~((y <= 2.4e-35)))
		tmp = x + (exp(-z) / y);
	else
		tmp = x + (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[Exp[N[(y * N[Log[N[(y / N[(z + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := If[Or[LessEqual[y, -53000000.0], N[Not[LessEqual[y, 2.4e-35]], $MachinePrecision]], N[(x + N[(N[Exp[(-z)], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Target

Original	84.7%
Target	91.9%
Herbie	99.0%

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z + y} < 7.11541576 \cdot 10^{-315}:\\ \;\;\;\;x + \frac{e^{\frac{-1}{z}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{\log \left({\left(\frac{y}{y + z}\right)}^{y}\right)}}{y}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < -5.3e7 or 2.4000000000000001e-35 < y

   1. Initial program 89.4%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]

   2. Simplified 89.4%

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
      Step-by-step derivation

      [Start] 89.4%	\[ x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
      *-commutative [=>] 89.4%	\[ x + \frac{e^{\color{blue}{\log \left(\frac{y}{z + y}\right) \cdot y}}}{y} \]
      exp-prod [=>] 89.4%	\[ x + \frac{\color{blue}{{\left(e^{\log \left(\frac{y}{z + y}\right)}\right)}^{y}}}{y} \]
      rem-exp-log [=>] 89.4%	\[ x + \frac{{\color{blue}{\left(\frac{y}{z + y}\right)}}^{y}}{y} \]
      +-commutative [=>] 89.4%	\[ x + \frac{{\left(\frac{y}{\color{blue}{y + z}}\right)}^{y}}{y} \]

   3. Taylor expanded in y around inf 99.6%

      \[\leadsto x + \frac{\color{blue}{e^{-1 \cdot z}}}{y} \]

   4. Simplified 99.6%

      \[\leadsto x + \frac{\color{blue}{e^{-z}}}{y} \]
      Step-by-step derivation

      [Start] 99.6%	\[ x + \frac{e^{-1 \cdot z}}{y} \]
      mul-1-neg [=>] 99.6%	\[ x + \frac{e^{\color{blue}{-z}}}{y} \]

   if -5.3e7 < y < 2.4000000000000001e-35

   1. Initial program 91.9%

      \[x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]

   2. Simplified 99.9%

      \[\leadsto \color{blue}{x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}} \]
      Step-by-step derivation

      [Start] 91.9%	\[ x + \frac{e^{y \cdot \log \left(\frac{y}{z + y}\right)}}{y} \]
      exp-prod [=>] 99.9%	\[ x + \frac{\color{blue}{{\left(e^{y}\right)}^{\log \left(\frac{y}{z + y}\right)}}}{y} \]
      sqr-pow [=>] 99.9%	\[ x + \frac{\color{blue}{{\left(e^{y}\right)}^{\left(\frac{\log \left(\frac{y}{z + y}\right)}{2}\right)} \cdot {\left(e^{y}\right)}^{\left(\frac{\log \left(\frac{y}{z + y}\right)}{2}\right)}}}{y} \]
      sqr-pow [<=] 99.9%	\[ x + \frac{\color{blue}{{\left(e^{y}\right)}^{\log \left(\frac{y}{z + y}\right)}}}{y} \]
      +-commutative [=>] 99.9%	\[ x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{\color{blue}{y + z}}\right)}}{y} \]

   3. Taylor expanded in y around inf 99.1%

      \[\leadsto \color{blue}{\frac{1}{y} + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -53000000 \lor \neg \left(y \leq 2.4 \cdot 10^{-35}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.0%
Cost	7049

\[\begin{array}{l} \mathbf{if}\;y \leq -53000000 \lor \neg \left(y \leq 2.4 \cdot 10^{-35}\right):\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 2
Accuracy	95.7%
Cost	19972

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+51}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}\\ \end{array} \]

Alternative 3
Accuracy	90.4%
Cost	6788

\[\begin{array}{l} \mathbf{if}\;z \leq -780000000:\\ \;\;\;\;\frac{e^{-z}}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 4
Accuracy	90.4%
Cost	964

\[\begin{array}{l} \mathbf{if}\;y \leq -55000000:\\ \;\;\;\;x + \left(\frac{1}{y} + 0.5 \cdot \frac{z \cdot z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 5
Accuracy	85.0%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+199}:\\ \;\;\;\;0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 6
Accuracy	87.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+171}:\\ \;\;\;\;0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y}\\ \end{array} \]

Alternative 7
Accuracy	64.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+35}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	85.9%
Cost	320

\[x + \frac{1}{y} \]

Alternative 9
Accuracy	43.2%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023272 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, G"
  :precision binary64

  :herbie-target
  (if (< (/ y (+ z y)) 7.11541576e-315) (+ x (/ (exp (/ -1.0 z)) y)) (+ x (/ (exp (log (pow (/ y (+ y z)) y))) y)))

  (+ x (/ (exp (* y (log (/ y (+ z y))))) y)))