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

Average Error: 6.0 → 1.0

Time: 10.1s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y \cdot e^{z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-28) (+ x (/ 1.0 y)) (+ x (/ 1.0 (* y (exp z))))))

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-28) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (1.0 / (y * exp(z)));
	}
	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-28) then
        tmp = x + (1.0d0 / y)
    else
        tmp = x + (1.0d0 / (y * exp(z)))
    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-28) {
		tmp = x + (1.0 / y);
	} else {
		tmp = x + (1.0 / (y * Math.exp(z)));
	}
	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-28:
		tmp = x + (1.0 / y)
	else:
		tmp = x + (1.0 / (y * math.exp(z)))
	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-28)
		tmp = Float64(x + Float64(1.0 / y));
	else
		tmp = Float64(x + Float64(1.0 / Float64(y * exp(z))));
	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-28)
		tmp = x + (1.0 / y);
	else
		tmp = x + (1.0 / (y * exp(z)));
	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-28], N[(x + N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(x + N[(1.0 / N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.0
Target	1.1
Herbie	1.0

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

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000002e-28

   1. Initial program 8.0

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

   2. Simplified 0.8

      \[\leadsto \color{blue}{x + \frac{{\left(e^{y}\right)}^{\log \left(\frac{y}{y + z}\right)}}{y}} \]
      Proof
      (+.f64 x (/.f64 (pow.f64 (exp.f64 y) (log.f64 (/.f64 y (+.f64 y z)))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (pow.f64 (exp.f64 y) (log.f64 (/.f64 y (Rewrite<= +-commutative_binary64 (+.f64 z y))))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 (exp.f64 y) (/.f64 (log.f64 (/.f64 y (+.f64 z y))) 2)) (pow.f64 (exp.f64 y) (/.f64 (log.f64 (/.f64 y (+.f64 z y))) 2)))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 (exp.f64 y) (log.f64 (/.f64 y (+.f64 z y))))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y)))))) y)): 17 points increase in error, 1 points decrease in error

   3. Taylor expanded in y around inf 1.0

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

   if 5.0000000000000002e-28 < y

   1. Initial program 1.6

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

   2. Simplified 1.6

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
      Proof
      (+.f64 x (/.f64 (pow.f64 (/.f64 y (+.f64 y z)) y) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (pow.f64 (/.f64 y (Rewrite<= +-commutative_binary64 (+.f64 z y))) y) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 (/.f64 y (+.f64 z y)) (/.f64 y 2)) (pow.f64 (/.f64 y (+.f64 z y)) (/.f64 y 2)))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 (/.f64 y (+.f64 z y)) y)) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (Rewrite<= exp-to-pow_binary64 (exp.f64 (*.f64 (log.f64 (/.f64 y (+.f64 z y))) y))) y)): 0 points increase in error, 0 points decrease in error
      (+.f64 x (/.f64 (exp.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (log.f64 (/.f64 y (+.f64 z y)))))) y)): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 1.0

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

   4. Simplified 1.0

      \[\leadsto x + \frac{\color{blue}{e^{-z}}}{y} \]
      Proof
      (exp.f64 (neg.f64 z)): 0 points increase in error, 0 points decrease in error
      (exp.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 z))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 1.0

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

   6. Taylor expanded in x around 0 1.0

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

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{1}{y \cdot e^{z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	19840

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

Alternative 2
Error	0.9
Cost	6916

\[\begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-21}:\\ \;\;\;\;x + \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{e^{-z}}{y}\\ \end{array} \]

Alternative 3
Error	15.8
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -62000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{-30}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+20}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	2.4
Cost	320

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

Alternative 5
Error	28.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(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)))