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

Average Error: 5.9 → 1.2

Time: 18.0s

Precision: binary64

Cost: 7300

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{y}{{\left(\frac{y}{y + z}\right)}^{y}}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-40) (+ (/ 1.0 y) x) (+ x (/ 1.0 (/ y (pow (/ y (+ y z)) 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 <= 5e-40) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = x + (1.0 / (y / pow((y / (y + z)), 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 <= 5d-40) then
        tmp = (1.0d0 / y) + x
    else
        tmp = x + (1.0d0 / (y / ((y / (y + z)) ** 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 <= 5e-40) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = x + (1.0 / (y / Math.pow((y / (y + z)), 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 <= 5e-40:
		tmp = (1.0 / y) + x
	else:
		tmp = x + (1.0 / (y / math.pow((y / (y + z)), 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 <= 5e-40)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = Float64(x + Float64(1.0 / Float64(y / (Float64(y / Float64(y + z)) ^ 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 <= 5e-40)
		tmp = (1.0 / y) + x;
	else
		tmp = x + (1.0 / (y / ((y / (y + z)) ^ 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[LessEqual[y, 5e-40], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(1.0 / N[(y / N[Power[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision], y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	5.9
Target	1.0
Herbie	1.2

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

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999965e-40

   1. Initial program 8.1

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

   2. Taylor expanded in z around 0 1.0

      \[\leadsto \color{blue}{\frac{1}{y} + x} \]

   if 4.99999999999999965e-40 < y

   1. Initial program 1.4

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

   2. Applied egg-rr 1.4

      \[\leadsto x + \color{blue}{\left(\frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y} + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y}\right)} \]

   3. Simplified 1.4

      \[\leadsto x + \color{blue}{2 \cdot \frac{0.5}{\frac{y}{{\left(\frac{y}{z + y}\right)}^{y}}}} \]
      Proof

      [Start] 1.4	\[ x + \left(\frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y} + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y}\right) \]
      rational.json-simplify-6 [<=] 1.4	\[ x + \left(\color{blue}{1 \cdot \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y}} + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y}\right) \]
      rational.json-simplify-2 [<=] 1.4	\[ x + \left(\color{blue}{\frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y} \cdot 1} + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y}\right) \]
      rational.json-simplify-6 [<=] 1.4	\[ x + \left(\frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y} \cdot 1 + \color{blue}{1 \cdot \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y}}\right) \]
      rational.json-simplify-51 [=>] 1.4	\[ x + \color{blue}{\frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y} \cdot \left(1 + 1\right)} \]
      metadata-eval [=>] 1.4	\[ x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y} \cdot \color{blue}{2} \]
      rational.json-simplify-2 [<=] 1.4	\[ x + \color{blue}{2 \cdot \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + y}} \]
      rational.json-simplify-7 [<=] 1.4	\[ x + 2 \cdot \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y + \color{blue}{\frac{y}{1}}} \]
      rational.json-simplify-30 [<=] 1.4	\[ x + 2 \cdot \frac{{\left(\frac{y}{y + z}\right)}^{y}}{\color{blue}{\left(1 + 1\right) \cdot \frac{y}{1}}} \]
      metadata-eval [=>] 1.4	\[ x + 2 \cdot \frac{{\left(\frac{y}{y + z}\right)}^{y}}{\color{blue}{2} \cdot \frac{y}{1}} \]
      rational.json-simplify-7 [=>] 1.4	\[ x + 2 \cdot \frac{{\left(\frac{y}{y + z}\right)}^{y}}{2 \cdot \color{blue}{y}} \]
      rational.json-simplify-47 [<=] 1.4	\[ x + 2 \cdot \color{blue}{\frac{\frac{{\left(\frac{y}{y + z}\right)}^{y}}{2}}{y}} \]
      rational.json-simplify-44 [=>] 1.4	\[ x + 2 \cdot \color{blue}{\frac{\frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}}{2}} \]
      rational.json-simplify-7 [<=] 1.4	\[ x + 2 \cdot \frac{\frac{{\left(\frac{y}{y + z}\right)}^{y}}{\color{blue}{\frac{y}{1}}}}{2} \]
      rational.json-simplify-61 [=>] 1.4	\[ x + 2 \cdot \frac{\color{blue}{\frac{1}{\frac{y}{{\left(\frac{y}{y + z}\right)}^{y}}}}}{2} \]
      rational.json-simplify-44 [=>] 1.4	\[ x + 2 \cdot \color{blue}{\frac{\frac{1}{2}}{\frac{y}{{\left(\frac{y}{y + z}\right)}^{y}}}} \]
      metadata-eval [=>] 1.4	\[ x + 2 \cdot \frac{\color{blue}{0.5}}{\frac{y}{{\left(\frac{y}{y + z}\right)}^{y}}} \]
      rational.json-simplify-1 [=>] 1.4	\[ x + 2 \cdot \frac{0.5}{\frac{y}{{\left(\frac{y}{\color{blue}{z + y}}\right)}^{y}}} \]

   4. Applied egg-rr 1.4

      \[\leadsto x + \color{blue}{\frac{1}{\frac{y}{{\left(\frac{y}{y + z}\right)}^{y}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{\frac{y}{{\left(\frac{y}{y + z}\right)}^{y}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	7172

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-40}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{y}{z + y}\right)}^{y}}{y}\\ \end{array} \]

Alternative 2
Error	14.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	2.2
Cost	320

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

Alternative 4
Error	28.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023069 
(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)))