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

Average Error: 6.1 → 1.4

Time: 6.3s

Precision: binary64

Cost: 7172

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.2e-51) (+ (/ 1.0 y) x) (+ x (/ (pow (/ y (+ y z)) y) 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 <= 3.2e-51) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = x + (pow((y / (y + z)), y) / 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 <= 3.2d-51) then
        tmp = (1.0d0 / y) + x
    else
        tmp = x + (((y / (y + z)) ** y) / 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 <= 3.2e-51) {
		tmp = (1.0 / y) + x;
	} else {
		tmp = x + (Math.pow((y / (y + z)), y) / 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 <= 3.2e-51:
		tmp = (1.0 / y) + x
	else:
		tmp = x + (math.pow((y / (y + z)), y) / 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 <= 3.2e-51)
		tmp = Float64(Float64(1.0 / y) + x);
	else
		tmp = Float64(x + Float64((Float64(y / Float64(y + z)) ^ y) / 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 <= 3.2e-51)
		tmp = (1.0 / y) + x;
	else
		tmp = x + (((y / (y + z)) ^ y) / 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, 3.2e-51], N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[Power[N[(y / N[(y + z), $MachinePrecision]), $MachinePrecision], y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.1
Target	1.1
Herbie	1.4

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

Derivation

1. Split input into 2 regimes

2. if y < 3.2e-51

   1. Initial program 8.2

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

   2. Simplified 8.2

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
      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 z around 0 1.2

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

   if 3.2e-51 < y

   1. Initial program 1.8

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

   2. Simplified 1.8

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+95}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-58}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-115}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-10}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	2.5
Cost	320

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

Alternative 3
Error	28.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022291 
(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)))