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

Average Error: 6.2 → 2.9

Time: 11.4s

Precision: binary64

Cost: 584

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{y} + x\\ \mathbf{if}\;z \leq 0.23:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (/ 1.0 y) x))) (if (<= z 0.23) t_0 (if (<= z 3.2e+71) x t_0))))

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 t_0 = (1.0 / y) + x;
	double tmp;
	if (z <= 0.23) {
		tmp = t_0;
	} else if (z <= 3.2e+71) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (1.0d0 / y) + x
    if (z <= 0.23d0) then
        tmp = t_0
    else if (z <= 3.2d+71) then
        tmp = x
    else
        tmp = t_0
    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 t_0 = (1.0 / y) + x;
	double tmp;
	if (z <= 0.23) {
		tmp = t_0;
	} else if (z <= 3.2e+71) {
		tmp = x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (1.0 / y) + x
	tmp = 0
	if z <= 0.23:
		tmp = t_0
	elif z <= 3.2e+71:
		tmp = x
	else:
		tmp = t_0
	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)
	t_0 = Float64(Float64(1.0 / y) + x)
	tmp = 0.0
	if (z <= 0.23)
		tmp = t_0;
	elseif (z <= 3.2e+71)
		tmp = x;
	else
		tmp = t_0;
	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)
	t_0 = (1.0 / y) + x;
	tmp = 0.0;
	if (z <= 0.23)
		tmp = t_0;
	elseif (z <= 3.2e+71)
		tmp = x;
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(1.0 / y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, 0.23], t$95$0, If[LessEqual[z, 3.2e+71], x, t$95$0]]]

Target

Original	6.2
Target	1.1
Herbie	2.9

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

Derivation

1. Split input into 2 regimes

2. if z < 0.23000000000000001 or 3.20000000000000023e71 < z

   1. Initial program 6.0

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

   2. Simplified 6.0

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
      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)): 1 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, 1 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 1.9

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

   if 0.23000000000000001 < z < 3.20000000000000023e71

   1. Initial program 10.1

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

   2. Simplified 10.1

      \[\leadsto \color{blue}{x + \frac{{\left(\frac{y}{y + z}\right)}^{y}}{y}} \]
      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)): 1 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, 1 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 x around inf 17.2

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.23:\\ \;\;\;\;\frac{1}{y} + x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} + x\\ \end{array} \]

Alternatives

Alternative 1
Error	14.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+69}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	28.3
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022308 
(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)))