Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A

Average Error: 3.0 → 0.1

Time: 8.3s

Precision: binary64

Cost: 7104

Math

\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

↓

\[x + \frac{-1}{x + \frac{e^{z}}{y} \cdot -1.1283791670955126} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

↓

(FPCore (x y z)
 :precision binary64
 (+ x (/ -1.0 (+ x (* (/ (exp z) y) -1.1283791670955126)))))

C

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

↓

double code(double x, double y, double z) {
	return x + (-1.0 / (x + ((exp(z) / y) * -1.1283791670955126)));
}

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 + (y / ((1.1283791670955126d0 * exp(z)) - (x * 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
    code = x + ((-1.0d0) / (x + ((exp(z) / y) * (-1.1283791670955126d0))))
end function

Java

public static double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * Math.exp(z)) - (x * y)));
}

↓

public static double code(double x, double y, double z) {
	return x + (-1.0 / (x + ((Math.exp(z) / y) * -1.1283791670955126)));
}

Python

def code(x, y, z):
	return x + (y / ((1.1283791670955126 * math.exp(z)) - (x * y)))

↓

def code(x, y, z):
	return x + (-1.0 / (x + ((math.exp(z) / y) * -1.1283791670955126)))

Julia

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

↓

function code(x, y, z)
	return Float64(x + Float64(-1.0 / Float64(x + Float64(Float64(exp(z) / y) * -1.1283791670955126))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
end

↓

function tmp = code(x, y, z)
	tmp = x + (-1.0 / (x + ((exp(z) / y) * -1.1283791670955126)));
end

Wolfram

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(x + N[(-1.0 / N[(x + N[(N[(N[Exp[z], $MachinePrecision] / y), $MachinePrecision] * -1.1283791670955126), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	3.0
Target	0.1
Herbie	0.1

\[x + \frac{1}{\frac{1.1283791670955126}{y} \cdot e^{z} - x} \]

Derivation

1. Initial program 3.0

   \[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

2. Simplified 0.1

   \[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(-1.1283791670955126, \frac{e^{z}}{y}, x\right)}} \]
   Proof
   (+.f64 x (/.f64 -1 (fma.f64 -5641895835477563/5000000000000000 (/.f64 (exp.f64 z) y) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (Rewrite<= metadata-eval (neg.f64 5641895835477563/5000000000000000)) (/.f64 (exp.f64 z) y) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 (neg.f64 5641895835477563/5000000000000000) (/.f64 (exp.f64 z) y)) x)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (neg.f64 5641895835477563/5000000000000000) (exp.f64 z)) y)) x))): 4 points increase in error, 3 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (/.f64 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)))) y) x))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (+.f64 (*.f64 x (Rewrite<= metadata-eval (neg.f64 -1))) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (Rewrite<= fma-udef_binary64 (fma.f64 x (neg.f64 -1) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 x 1)) (neg.f64 -1) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (Rewrite<= metadata-eval (neg.f64 -1))) (neg.f64 -1) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (Rewrite=> metadata-eval 1) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (Rewrite<= *-inverses_binary64 (/.f64 y y)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)))) (*.f64 -1 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (/.f64 (*.f64 -1 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)))) (Rewrite<= neg-mul-1_binary64 (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (*.f64 (/.f64 x (Rewrite=> metadata-eval 1)) (/.f64 y y)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (*.f64 (Rewrite=> /-rgt-identity_binary64 x) (/.f64 y y)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x y) y)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y))))): 14 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (/.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 5641895835477563/5000000000000000) (exp.f64 z))) (neg.f64 y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (/.f64 (*.f64 (neg.f64 5641895835477563/5000000000000000) (exp.f64 z)) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (neg.f64 5641895835477563/5000000000000000) -1) (/.f64 (exp.f64 z) y)))))): 3 points increase in error, 4 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (*.f64 (/.f64 (Rewrite=> metadata-eval -5641895835477563/5000000000000000) -1) (/.f64 (exp.f64 z) y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (*.f64 (Rewrite=> metadata-eval 5641895835477563/5000000000000000) (/.f64 (exp.f64 z) y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) y))))): 4 points increase in error, 3 points decrease in error
   (+.f64 x (/.f64 -1 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 x y) (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 1 points decrease in error
   (+.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 y) (-.f64 (*.f64 x y) (*.f64 5641895835477563/5000000000000000 (exp.f64 z)))))): 9 points increase in error, 10 points decrease in error
   (+.f64 x (/.f64 (*.f64 -1 y) (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 x y) (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (*.f64 -1 y) (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (*.f64 x y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (*.f64 -1 y) (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)))) (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (*.f64 -1 y) (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (*.f64 -1 y) (Rewrite=> sub0-neg_binary64 (neg.f64 (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (/.f64 (*.f64 -1 y) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -1 -1) (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (*.f64 (Rewrite=> metadata-eval 1) (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))): 0 points increase in error, 0 points decrease in error
   (+.f64 x (Rewrite=> *-lft-identity_binary64 (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.1

   \[\leadsto x + \frac{-1}{\color{blue}{\frac{e^{z}}{y} \cdot -1.1283791670955126 + x}} \]

4. Final simplification 0.1

   \[\leadsto x + \frac{-1}{x + \frac{e^{z}}{y} \cdot -1.1283791670955126} \]

Alternatives

Alternative 1
Error	0.2
Cost	19912

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126}\\ \end{array} \]

Alternative 2
Error	0.3
Cost	1096

\[\begin{array}{l} \mathbf{if}\;z \leq -95:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	9.7
Cost	848

\[\begin{array}{l} t_0 := x + \frac{-1}{x}\\ t_1 := x + y \cdot 0.8862269254527579\\ \mathbf{if}\;z \leq -3.35 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	9.6
Cost	848

\[\begin{array}{l} t_0 := x + \frac{-1}{x}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.85 \cdot 10^{-238}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-17}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	0.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -155:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	0.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -390:\\ \;\;\;\;x + \frac{-1}{x}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	17.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-124}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.04 \cdot 10^{-78}:\\ \;\;\;\;x + y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	20.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-218}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-85}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	20.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	19.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022329 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:invErfc from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (+ x (/ 1.0 (- (* (/ 1.1283791670955126 y) (exp z)) x)))

  (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))